Applications of linear regression models in exploring the relationship between media attention, economic policy uncertainty and corporate green innovation

Yang Xu; Conghao Zhu; Runze Yang; Qiying Ran; Xiaodong Yang; Yang Xu; Conghao Zhu; Runze Yang; Qiying Ran; Xiaodong Yang

doi:10.3934/math.2023954

AIMS Mathematics

2023, Volume 8, Issue 8: 18734-18761. doi: 10.3934/math.2023954

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Research article Special Issues

Applications of linear regression models in exploring the relationship between media attention, economic policy uncertainty and corporate green innovation

1.
School of Economics and Management, Xinjiang University, Urumqi, China
2.
International Business School, Tianjin Foreign Studies University, Tianjin, China
3.
School of Finance, Shanghai University of Finance and Economics, Shanghai, China
4.
Shanghai Business School, Shanghai, China
5.
Center for Innovation Management, Xinjiang University, Urumqi, China

Received: 17 April 2023 Revised: 08 May 2023 Accepted: 12 May 2023 Published: 02 June 2023
MSC : 62P20, 62P25, 90A19, 90A80

The media plays a dual role of "supervision" and "collusion" in governance mechanisms. This study investigates the impact of media attention and economic policy uncertainty on green innovation by analyzing A-share industrial listed enterprises data between 2011 and 2020. The results show that media attention can effectively promote green innovation and that this impact is significantly heterogeneous. Media attention significantly affects green innovation in non-state-owned enterprises and manufacturing companies positively, but it is insignificant for state-owned enterprises and mining and energy supply industries. Moreover, the results indicate that external economic policy uncertainty can lead enterprises to take early measures to hedge risks, thereby positively regulating the promotion effect of media attention on green innovation during economic fluctuations. Finally, media attention can promote green innovation by increasing environmental regulation intensity, reducing corporate financing constraints, and enhancing corporate social responsibility. Therefore, paying full attention to the media as an institutional subject outside of laws and regulations, gradually forming a pressure-driven mechanism for corporate green innovation, and reducing information opacity, is a pivotal way to promote enterprises' green innovation.

Keywords:

Citation: Yang Xu, Conghao Zhu, Runze Yang, Qiying Ran, Xiaodong Yang. Applications of linear regression models in exploring the relationship between media attention, economic policy uncertainty and corporate green innovation[J]. AIMS Mathematics, 2023, 8(8): 18734-18761. doi: 10.3934/math.2023954

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Abstract

1. Introduction

Let $\mathbb{A}$ stand for the collection of functions $\mathcal{G}$ of the type

$\begin{equation} \mathcal{G}(\xi ) = \xi +\sum\limits_{j = 2}^{\infty }a_{j}\xi ^{j}, \end{equation}$

(1.1)

that are holomorphic in the open unit disk $\Lambda : = \left\{ \xi \in \mathbb{C}:\left\vert \xi \right\vert < 1\right\}$ of the complex plane, and let $\mathfrak{S}$ indicate the subclass of functions of $\mathbb{A}$ which are univalent in $\Lambda$ . For functions $\mathcal{G}\in \mathbb{A}$ given by (1.1) and $\mathcal{H}\in \mathbb{A}$ given by $\mathcal{H} (\zeta) = \zeta +\sum\limits_{j = 2}^{\infty }b_{j}\zeta ^{j},$ we define the convolution product (or Hadamard) of $\mathcal{G}$ and $\mathcal{H}$ by

$\begin{equation} (\mathcal{G}\ast \mathcal{H})(\zeta ) = (\mathcal{H}\ast \mathcal{G})(\zeta ) = \zeta +\sum\limits_{j = 2}^{\infty }a_{j}b_{j}\zeta ^{j}, \, \, \, \, \zeta \in \Lambda . \end{equation}$

(1.2)

Let $\mathcal{G}$ and $\mathcal{F}$ be two holomorphic functions in $\Lambda$ . The function $\mathcal{G}$ is said to be subordinated to $\mathcal{F}$ if there are Schwarz function $w(\xi)$ , that is, holomorphic in $\Lambda$ with $w(0) = 0$ and $\left\vert w(\xi)\right\vert < 1$ , $\xi \in \Lambda$ , such as $\mathcal{G}(\xi) = \mathcal{F}(w(\xi))$ for all $\xi \in \Lambda$ . This subordination notion is indicated by

$\begin{equation*} \mathcal{G}\prec \mathcal{F}\quad \text{or}\quad \mathcal{G}(\xi )\prec \mathcal{F}(\xi ). \end{equation*}$

If the function $\mathcal{F}$ is univalent in $\Lambda$ , then we have the inclusion equivalence

$\begin{equation*} \mathcal{G}(\xi )\prec \mathcal{F}(\xi )\Leftrightarrow \mathcal{G}(0) = \mathcal{F}(0)\quad \text{and}\quad \mathcal{G}(\Lambda )\subset \mathcal{F} (\Lambda ). \end{equation*}$

The subfamilies of $\mathfrak{S}$ which are the starlike and the convex function in $\Lambda$ defined by

$\begin{equation} \mathfrak{S}^{\ast }: = \left\{ \mathcal{G}\in \mathbb{A}: {Re}\frac{\xi \mathcal{G}^{^{\prime }}(\xi )}{\mathcal{G}(\xi )} > 0, \;\xi \in \Lambda \right\} \end{equation}$

(1.3)

and

$\begin{equation} \mathfrak{C}: = \left\{ \mathcal{G}\in \mathbb{A}: {Re}\frac{(\xi \mathcal{ G}^{^{\prime }}(\xi ))^{\prime }}{\mathcal{G}^{^{\prime }}(\xi )} > 0, \;\xi \in \Lambda \right\} , \end{equation}$

(1.4)

respectively. Equivalently, we have

$\begin{equation*} \mathfrak{S}^{\ast }(\varphi ) = \left\{ \mathcal{G}\in \mathbb{A}:\frac{\xi \mathcal{G}^{^{\prime }}(\xi )}{\mathcal{G}(\xi )}\prec \varphi (\xi )\right\} , \quad \mathfrak{C}(\varphi ) = \left\{ \mathcal{G}\in \mathbb{A}: \frac{(\xi \mathcal{G}^{^{\prime }}(\xi ))^{\prime }}{\mathcal{G}^{^{\prime }}(\xi )}\prec \varphi (\xi )\right\} , \end{equation*}$

where

$\begin{equation} \varphi (\xi ) = \frac{1+\xi }{1-\xi }. \end{equation}$

(1.5)

Janowski defined in ^[4] the extended function family $\mathfrak{S} ^{\ast }\left[\mathcal{A}, \mathcal{B}\right]$ of starlike functions called the Janwoski class of functions as follows: A function $\mathcal{ G}\in \mathbb{A}$ is in the family $\mathfrak{S}^{\ast }\left[\mathcal{A}, \mathcal{B}\right]$ if

$\begin{equation*} \frac{\xi \mathcal{G}^{^{\prime }}(\xi )}{\mathcal{G}(\xi )}\prec \frac{1+ \mathcal{A}\xi }{1+\mathcal{B}\xi }\quad \left( -1\leq \mathcal{B} < \mathcal{A }\leq 1\right) . \end{equation*}$

The above subordination could be written as

$\begin{equation} \frac{\xi \mathcal{G}^{^{\prime }}(\xi )}{\mathcal{G}(\xi )} = \frac{1+ \mathcal{A}p\left( \xi \right) }{1+\mathcal{B}p\left( \xi \right) }\quad \left( -1\leq \mathcal{B} < \mathcal{A}\leq 1\right) , \end{equation}$

(1.6)

where an analytical function with a real positive part in $\Lambda$ is denoted by $p(\xi)$ .

The Janowski convex and Janowski starlike functions are obtained by reducing the above-described classes to the requirement $-1\leq \mathcal{B} < \mathcal{A }\leq 1$ . For the special cases $\mathcal{A}: = 1-2\alpha$ and $\mathcal{B} : = -1$ , where $0\leq \alpha < 1$ , we obtain the families, namely the family of starlike and convex functions of order $\alpha$ $(0\leq \alpha < 1)$ previously defined by Robertson in ^[6], and considered respectively by

$\begin{eqnarray*} \mathfrak{S}^{\ast }(\alpha ) &:& = \left\{ \mathcal{G}\in \mathbb{A}: {Re} \frac{\xi \mathcal{G}^{^{\prime }}(\xi )}{\mathcal{G}(\xi )} > \alpha , \;\xi \in \Lambda \right\} , \quad \\ \mathfrak{C}(\alpha ) &:& = \left\{ \mathcal{G}\in \mathbb{A}: {Re}\frac{ (\xi \mathcal{G}^{^{\prime }}(\xi ))^{\prime }}{\mathcal{G}^{^{\prime }}(\xi )} > \alpha , \;\xi \in \Lambda \right\} . \end{eqnarray*}$

Babalola defined the operator $\mathcal{I}_{\upsilon }^{\rho }:\; \mathbb{A} \rightarrow \mathbb{A}\;$ as

$\begin{equation} \mathcal{I}_{\upsilon }^{\sigma }\mathcal{G}\left( \zeta \right) = \left( \rho _{\sigma }\ast \rho _{\sigma , \upsilon }^{-1}\ast \mathcal{G}\right) \left( \zeta \right) , \end{equation}$

(1.7)

where

$\begin{equation*} \rho _{\sigma , \upsilon }\left( \zeta \right) = \frac{\zeta }{\left( 1-\zeta \right) ^{\sigma -\upsilon +1}}, \ \ \ \sigma -\upsilon +1 > 0, \ \ \rho _{\sigma } = \rho _{\sigma , 0}, \end{equation*}$

and $\rho _{\sigma, \upsilon }^{-1}$ is

$\begin{equation*} \left( \rho _{\sigma , \upsilon }\ast \rho _{\sigma , \upsilon }^{-1}\right) \left( \zeta \right) = \frac{\zeta }{1-\zeta }\; \ \ \ \left( \sigma , \upsilon \in \mathbb{N} = \{1, 2, 3, ...\}\right) . \end{equation*}$

For $\mathcal{G}\in \mathbb{A},$ then (1.7) is equivalent to

$\begin{equation*} \mathcal{I}_{\upsilon }^{\sigma }\mathcal{G}\left( \zeta \right) = \zeta +\sum\limits_{j = 2}^{\infty }\left[ \tfrac{\Gamma (\sigma +j)}{\Gamma (\sigma +1)}\cdot \tfrac{(\sigma -\upsilon )!}{(\sigma +j-\upsilon -1)!} \right] a_{j}\; \zeta ^{j}. \end{equation*}$

Making use the binomial series

$\begin{equation*} \left( 1-\delta \right) ^{t} = \sum\limits_{i = 0}^{t}\left( \begin{array}{c} t \\ i \end{array} \right) \left( -1\right) ^{i}\ \delta ^{i}\ \ \ \ \ \left( t\in \mathbb{N} \right) , \end{equation*}$

for $\mathcal{G}\in \mathbb{A}, \$ El-Deeb ^[3] introduced the linear differential operator as follows:

$\begin{equation*} \mathcal{D}_{m, \delta , \upsilon }^{\sigma , 0}\mathcal{G}\left( \zeta \right) = \mathcal{G}\left( \zeta \right) , \end{equation*}$

$\begin{eqnarray} \mathcal{D}_{t, \delta , \upsilon }^{\sigma , 1}\mathcal{G}\left( \zeta \right) & = &\mathcal{D}_{t, \delta , \upsilon }^{\sigma }\mathcal{G}\left( \zeta \right) = \left( 1-\delta \right) ^{t}\mathcal{I}_{\upsilon }^{\sigma } \mathcal{G}\left( \zeta \right) +\left[ 1-\left( 1-\delta \right) ^{t}\right] \zeta \left( \mathcal{I}_{\upsilon }^{\sigma }\mathcal{G}\right) ^{^{\prime }}\left( \zeta \right) \\ & = &\zeta +\sum\limits_{j = 2}^{\infty }\left[ 1+\left( j-1\right) c^{t}(\delta )\right] \left[ \tfrac{\Gamma (\sigma +j)}{\Gamma (\sigma +1)} \cdot \tfrac{(\sigma -\upsilon )!}{(\sigma +j-\upsilon -1)!}\right] a_{j}\; \zeta ^{j} \\ &&. \\ &&. \\ &&. \\ \mathcal{D}_{t, \delta , \upsilon }^{\sigma , n}\mathcal{G}\left( \zeta \right) & = &\mathcal{D}_{t, \delta , \upsilon }^{\sigma }\left( \mathcal{D}_{t, \delta , \upsilon }^{\sigma , n-1}\mathcal{G}\left( \zeta \right) \right) \\ & = &\left( 1-\delta \right) ^{t}\mathcal{D}_{t, \delta , \upsilon }^{\sigma , n-1}\mathcal{G}\left( \zeta \right) +\left[ 1-\left( 1-\delta \right) ^{t} \right] \zeta \left( \mathcal{D}_{t, \delta , \upsilon }^{\sigma , n-1}\mathcal{ G}\left( \zeta \right) \right) ^{^{\prime }} \\ & = &\zeta +\sum\limits_{j = 2}^{\infty }\left[ 1+\left( j-1\right) c^{t}(\delta )\right] ^{n}\left[ \tfrac{\Gamma (\sigma +j)}{\Gamma (\sigma +1)}\cdot \tfrac{(\sigma -\upsilon )!}{(\sigma +j-\upsilon -1)!}\right] a_{j}\; \zeta ^{j} \\ & = &\zeta +\sum\limits_{j = 2}^{\infty }\psi _{j}^{n}\left[ \tfrac{\Gamma (\sigma +j)}{\Gamma (\sigma +1)}\cdot \tfrac{(\sigma -\upsilon )!}{(\sigma +j-\upsilon -1)!}\right] a_{j}\; \zeta ^{j}, \ \ \\ &&\left( \delta > 0;\ t, \sigma , \upsilon \in \mathbb{N} ;\ n\in \mathbb{N} _{0} = \mathbb{N} \cup \{0\}\right) , \end{eqnarray}$

(1.8)

where

$\begin{equation} \psi _{j}^{n} = \left[ 1+\left( j-1\right) c^{t}(\delta )\right] ^{n}, \end{equation}$

(1.9)

and

$\begin{equation*} c^{t}(\delta ) = \sum\limits_{i = 1}^{t}\left( \begin{array}{c} t \\ i \end{array} \right) \left( -1\right) ^{i+1}\ \delta ^{i}\ \ \ \ \ \left( t\in \mathbb{N} \right) . \end{equation*}$

From (1.8), we obtain that

$\begin{equation} c^{t}(\delta )\ \zeta \ \left( \mathcal{D}_{t, \delta , \upsilon }^{\sigma , n} \mathcal{G}\left( \zeta \right) \right) ^{^{^{\prime }}} = \mathcal{D} _{t, \delta , \upsilon }^{\sigma , n+1}\mathcal{G}\left( \zeta \right) -\left[ 1-c^{t}(\delta )\right] \mathcal{D}_{t, \delta , \upsilon }^{\sigma , n} \mathcal{G}\left( \zeta \right) . \end{equation}$

(1.10)

In this article using the El-Deeb operator defined in (1.8), we define a new sub-family of $\mathbb{A}$ :

$\begin{equation} \mathcal{R}_{t, \delta , \upsilon }^{m, n, \sigma }\left( \mathcal{A}, \mathcal{B} \right) = \left\{ \mathcal{G}\in \mathbb{A}:\frac{\mathcal{D}_{t, \delta , \upsilon }^{\sigma , m}\mathcal{G}\left( \zeta \right) }{\mathcal{D} _{t, \delta , \upsilon }^{\sigma , n}\mathcal{G}\left( \zeta \right) }\prec \frac{1+\mathcal{A}\xi }{1+\mathcal{B}\xi }\right\} , \end{equation}$

(1.11)

where $-1\leq \mathcal{A} < \mathcal{B}\leq 1$ ; $\ \delta > 0;\ t, \sigma, \upsilon \in \mathbb{N}$ and $n, m\in \mathbb{N} _{0}$ , that will lead us to the study of Fekete-Szegö problem. Further, coefficient estimates, characteristic properties and partial sums results will be established.

Specializing the values of $\mathcal{A}$ and $\mathcal{B}$ one can obtain the particular cases

$\begin{equation*} \text{(i)}\ \ \ \mathcal{R}_{t, \delta , \upsilon }^{m, n, \sigma }\left( 1-2\alpha , -1\right) = :\mathcal{W}_{t, \delta , \upsilon }^{m, n, \sigma }\left( \mathcal{\alpha }\right) = \left\{ \mathcal{G}\in \mathbb{A}: {Re}\left( \tfrac{\mathcal{D}_{t, \delta , \upsilon }^{\sigma , m}\mathcal{G}\left( \zeta \right) }{\mathcal{D}_{t, \delta , \upsilon }^{\sigma , n}\mathcal{G}\left( \zeta \right) }\right) > \alpha , \ (0\leq \alpha < 1)\right\} ; \end{equation*}$

and

$\begin{equation*} \text{(ii)}\ \ \ \mathcal{R}_{t, \delta , \upsilon }^{m, n, \sigma }\left( 1, -1\right) = :\mathcal{F}_{t, \delta , \upsilon }^{m, n, \sigma } = \left\{ \mathcal{G}\in \mathbb{A}: {Re}\left( \tfrac{\mathcal{D}_{t, \delta , \upsilon }^{\sigma , m}\mathcal{G}\left( \zeta \right) }{\mathcal{D} _{t, \delta , \upsilon }^{\sigma , n}\mathcal{G}\left( \zeta \right) }\right) > 0\right\} .\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{equation*}$

2. The Fekete-Szegö functional bounds for the class $\mathcal{R} _{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A }, \mathcal{B}\right)$

To solve the Fekete-Szegö type inequality for $\mathcal{G}\in \mathcal{R} _{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ we will use the next results (the first part is due to Carathéodory ^[1]):

Lemma 1. ^[1,5] If $P(\xi) = 1+p_{1}\xi +p_{2}\xi ^{2}+\dots \in \mathcal{P}$ where $\mathcal{P}$ the class of holomorphic functions with positive real part in $\Lambda$ , with $P(0) = 1$ , then

$\begin{equation} \left\vert p_{n}\right\vert \leq 2, \;n\geq 1, \end{equation}$

(2.1)

and for the complex number $\mu \in \mathbb{C}$ we have

$\begin{equation} \left\vert p_{2}-\mu p_{1}^{2}\right\vert \leq 2\max \left\{ 1;\left\vert 1-2\mu \right\vert \right\} . \end{equation}$

(2.2)

If $\mu$ is a real parameter, then

$\begin{equation} \left\vert p_{2}-\mu p_{1}^{2}\right\vert \leq \left\{ \begin{array}{lll} -4\mu +2, & \mathit{\text{if}} & \mu \leq 0, \\ 2 & \mathit{\text{if}} & 0\leq \mu \leq 1, \\ 4\mu -2 & \mathit{\text{if}} & \mu \geq 1. \end{array} \right. \end{equation}$

(2.3)

When $\mu > 1$ or $\mu < 0$ , equality (2.3) holds true if and only if $P_{1}(\xi) = \dfrac{1+\xi }{1-\xi }$ or one of its rotations. When $0 < \mu < 1$ , the equality (2.3) holds if and only if $P_{2}(\xi) = \dfrac{ 1+\xi ^{2}}{1-\xi ^{2}}$ or one of its rotations. When $\mu = 0$ , equality (2.3) holds if and only if

$\begin{equation*} P_{3}(\xi ) = \left( \frac{1+c}{2}\right) \frac{1+\xi }{-\xi +1}+\left( \frac{ 1-c}{2}\right) \frac{-\xi +1}{1+\xi }\quad \left( 0\leq c\leq 1\right) \end{equation*}$

or one of its rotations. When $\mu = 1$ , the equality (2.3) holds true if $P(\xi)$ is a reciprocal of one of the functions such that the equality holds true in the case when $\mu = 0$ .

Theorem 1. If $\mathcal{G}\in \mathbb{A}$ defined as (1.1), belongs to $\mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B} \right)$ , then

$\begin{align} \left\vert a_{2}\right\vert & \leq \frac{(\sigma -\upsilon +1)\left( \mathcal{A}-\mathcal{B}\right) }{(\sigma +1)|\psi _{2}^{m}-\psi _{2}^{n}|}, \end{align}$

(2.4)

$\begin{align} \left\vert a_{3}\right\vert & \leq \frac{(\sigma -\upsilon +1)(\sigma -\upsilon +2)\left( \mathcal{A}-\mathcal{B}\right) }{(\sigma +1)(\sigma +2)|\psi _{2}^{m}-\psi _{2}^{n}|}\times \\ & \max \left\{ 1;\left\vert -\mathcal{B}+\frac{(\mathcal{A}-\mathcal{B} )\left( \psi _{2}^{n+m}-\psi _{2}^{2n}\right) }{\left( \psi _{2}^{m}-\psi _{2}^{n}\right) ^{2}}\right\vert \right\} , \end{align}$

(2.5)

and for a complex number $\tau$ , we have

$\begin{equation} \left\vert a_{3}-\tau a_{2}^{2}\right\vert \leq \frac{(\sigma -\upsilon +1)(\sigma -\upsilon +2)\left( \mathcal{A}-\mathcal{B}\right) }{(\sigma +1)(\sigma +2)\left\vert \psi _{3}^{m}-\psi _{3}^{n}\right\vert }\max \left\{ 1;\left\vert \Omega \left( \tau , \sigma , \upsilon , \mathcal{A}, \mathcal{B}\right) \right\vert \right\} , \end{equation}$

(2.6)

where

$\begin{equation*} \Omega \left( \tau , \sigma , \upsilon , \mathcal{A}, \mathcal{B}\right) = 1-2\Theta \left( \tau , \sigma , \upsilon , \mathcal{A}, \mathcal{B}\right) , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{equation*}$

$\begin{equation*} \Theta \left( \tau , \sigma , \upsilon , \mathcal{A}, \mathcal{B}\right) = \frac{1 }{2}\left( 1+\mathcal{B-}\frac{(\mathcal{A}-\mathcal{B})\left( \psi _{2}^{n+m}-\psi _{2}^{2n}\right) }{\left( \psi _{2}^{m}-\psi _{2}^{n}\right) ^{2}}\right. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{equation*}$

$\begin{equation} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left. +\frac{\tau (\mathcal{A}-\mathcal{B})(\sigma +2)(\sigma -\upsilon +1)\left( \psi _{3}^{m}-\psi _{3}^{n}\right) }{(\sigma +1)(\sigma -\upsilon +2)\left( \psi _{2}^{m}-\psi _{2}^{n}\right) ^{2}} \right) , \end{equation}$

(2.7)

and $\psi _{j}^{n}$ is given by (1.9).

Proof. We will show that the relations $\left(2.4\right)$ – $\left(2.6\right)$ and $\left(2.16\right)$ hold true for $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ . If $\mathcal{G}\in \mathcal{R} _{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ , then

$\begin{equation*} \frac{\mathcal{D}_{t, \delta , \upsilon }^{\sigma , m}\mathcal{G}\left( \zeta \right) }{\mathcal{D}_{t, \delta , \upsilon }^{\sigma , n}\mathcal{G}\left( \zeta \right) }\prec \frac{1+\mathcal{A}\xi }{1+\mathcal{B}\xi } \end{equation*}$

which yields

$\begin{equation} \frac{\mathcal{D}_{t, \delta , \upsilon }^{\sigma , m}\mathcal{G}\left( \zeta \right) }{\mathcal{D}_{t, \delta , \upsilon }^{\sigma , n}\mathcal{G}\left( \zeta \right) }\prec \frac{1+\mathcal{A}w(\xi )}{1+\mathcal{B}w(\xi )} = G\left( w\left( \xi \right) \right) , \quad \left( -1\leq \mathcal{B} < \mathcal{A}\leq 1\right) . \end{equation}$

(2.8)

Since we can write $w\left(\xi \right)$ as

$\begin{equation*} w\left( \xi \right) = \frac{1-h\left( \xi \right) }{1+h\left( \xi \right) } = \frac{p_{1}\xi +p_{2}\xi ^{2}+p_{3}\xi ^{3}+\dots }{2+p_{1}\xi +p_{2}\xi ^{2}+p_{3}\xi ^{3}+\dots }, \end{equation*}$

where $h\left(\xi \right) \in \mathcal{P}$ and have the form $h\left(\xi \right) = 1+p_{1}\xi +p_{2}\xi ^{2}+p_{3}\xi ^{3}+\dots,$ so

$\begin{equation} G\left( w\left( \xi \right) \right) = 1+\frac{1}{2}\left( \mathcal{A}- \mathcal{B}\right) p_{1}\xi +\frac{\left( \mathcal{A}-\mathcal{B}\right) }{4} \left[ 2p_{2}-\left( 1+\mathcal{B}\right) p_{1}^{2}\right] \xi ^{2}+\dots , \end{equation}$

(2.9)

and therefore

$\begin{align} \frac{\mathcal{D}_{t, \delta , \upsilon }^{\sigma , m}\mathcal{G}\left( \zeta \right) }{\mathcal{D}_{t, \delta , \upsilon }^{\sigma , n}\mathcal{G}\left( \zeta \right) }& = 1+\frac{(\sigma +1)}{(\sigma -\upsilon +1)}\left( \psi _{2}^{m}-\psi _{2}^{n}\right) a_{2}\zeta \\ & +\left( \frac{(\sigma +1)(\sigma +2)}{(\sigma -\upsilon +1)(\sigma -\upsilon +2)}\left( \psi _{3}^{m}-\psi _{3}^{n}\right) a_{3}\right. \\ & \left. -\frac{(\sigma +1)^{2}}{(\sigma -\upsilon +1)^{2}}\left( \left( \psi _{2}^{n+m}-\psi _{2}^{2n}\right) \right) a_{2}^{2}\right) \zeta ^{2}+\dots . \end{align}$

(2.10)

If we compare the first coefficients of $\left(2.9\right)$ and $\left(2.10\right),$ we get

$\begin{eqnarray} &&a_{2} = \frac{(\sigma -\upsilon +1)\left( \mathcal{A}-\mathcal{B}\right) }{ 2(\sigma +1)\left( \psi _{2}^{m}-\psi _{2}^{n}\right) }p_{1}, \end{eqnarray}$

(2.11)

$\begin{eqnarray} &&a_{3} = \frac{(\sigma -\upsilon +1)(\sigma -\upsilon +2)\left( \mathcal{A}- \mathcal{B}\right) }{2(\sigma +1)(\sigma +2)\left( \psi _{3}^{m}-\psi _{3}^{n}\right) }\times \\ &&\left( p_{2}-\frac{p_{1}^{2}}{2}\left[ \frac{1+\mathcal{B}}{\mathcal{A}- \mathcal{B}}-\left( \frac{(\mathcal{A}-\mathcal{B})\left( \psi _{2}^{n+m}-\psi _{2}^{2n}\right) }{\left( \psi _{2}^{m}-\psi _{2}^{n}\right) ^{2}}\right) \right] \right) \\ && \end{eqnarray}$

(2.12)

and by using $\left(2.1\right)$ in $\left(2.11\right)$ and $\left(2.2\right)$ in $\left(2.12\right)$ , we get

$\begin{eqnarray} &&\left\vert a_{2}\right\vert \leq \frac{(\sigma -\upsilon +1)\left( \mathcal{A}-\mathcal{B}\right) }{(\sigma +1)\left\vert \psi _{2}^{m}-\psi _{2}^{n}\right\vert }, \end{eqnarray}$

(2.13)

$\begin{eqnarray} &&\left\vert a_{3}\right\vert \leq \frac{(\sigma -\upsilon +1)(\sigma -\upsilon +2)\left( \mathcal{A}-\mathcal{B}\right) }{(\sigma +1)(\sigma +2)\left\vert \psi _{3}^{m}-\psi _{3}^{n}\right\vert }\times \\ &&\max \left\{ 1;\left\vert -\mathcal{B}+\frac{(\mathcal{A}-\mathcal{B} )\left( \psi _{2}^{n+m}-\psi _{2}^{2n}\right) }{\left( \psi _{2}^{m}-\psi _{2}^{n}\right) ^{2}}\right\vert \right\} . \end{eqnarray}$

(2.14)

For a complex nubmer $\tau$ , and from $\left(2.11\right)$ together with $\left(2.12\right)$ , we have

$\begin{equation} \left\vert a_{3}-\tau a_{2}^{2}\right\vert = \frac{(\sigma -\upsilon +1)(\sigma -\upsilon +2)\left( \mathcal{A}-\mathcal{B}\right) }{2(\sigma +1)(\sigma +2)\left( \psi _{3}^{m}-\psi _{3}^{n}\right) }\left\vert p_{2}-\Theta \left( \tau , \sigma , \upsilon , \mathcal{A}, \mathcal{B}\right) p_{1}^{2}\right\vert , \end{equation}$

(2.15)

where $\Theta \left(\tau, \sigma, \upsilon, \mathcal{A}, \mathcal{B}\right)$ is denoted by (2.7). Now, we apply Lemma 1 to $\left(2.15 \right)$ and obtain the required results. □

Theorem 2. If the function $\mathcal{G}\in \mathbb{A}$ defined as (1.1) belongs to $\mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ , then for any real parameter $\tau$ we obtain

$\begin{equation} \left\vert a_{3}-\tau a_{2}^{2}\right\vert \leq \frac{(\sigma -\upsilon +1)(\sigma -\upsilon +2)\left( \mathcal{A}-\mathcal{B}\right) }{2(\sigma +1)(\sigma +2)\left\vert \psi _{3}^{m}-\psi _{3}^{n}\right\vert }\left\{ \begin{array}{lll} 1-2\Theta \left( \tau , \sigma , \upsilon , \mathcal{A}, \mathcal{B}\right) , & \mathit{\text{if}} & \tau < \varphi _{1}, \\ 1, & \mathit{\text{if}} & \varphi _{1}\leq \tau \leq \varphi _{2}, \\ 2\Theta \left( \tau , \sigma , \upsilon , \mathcal{A}, \mathcal{B}\right) -1, & \mathit{\text{if}} & \tau > \varphi _{2}, \end{array} \right. \end{equation}$

(2.16)

where $\Theta \left(\tau, \sigma, \upsilon, \mathcal{A}, \mathcal{B}\right)$ is given by (2.7),

$\begin{equation*} \varphi _{1} = \tfrac{(\sigma +1)(\sigma -\upsilon +2)\left( \psi _{2}^{m}-\psi _{2}^{n}\right) ^{2}}{(\mathcal{A}-\mathcal{B})(\sigma +2)(\sigma -\upsilon +1)\left( \psi _{3}^{m}-\psi _{3}^{n}\right) }\times \left( -1-\mathcal{B+}\tfrac{(\mathcal{A}-\mathcal{B})\left( \psi _{2}^{n+m}-\psi _{2}^{2n}\right) }{\left( \psi _{2}^{m}-\psi _{2}^{n}\right) ^{2}}\right) , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{equation*}$

and

$\begin{equation*} \varphi _{2} = \tfrac{(\sigma +1)(\sigma -\upsilon +2)\left( \psi _{2}^{m}-\psi _{2}^{n}\right) ^{2}}{(\mathcal{A}-\mathcal{B})(\sigma +2)(\sigma -\upsilon +1)\left( \psi _{3}^{m}-\psi _{3}^{n}\right) }\times \left( 1-\mathcal{B+}\tfrac{(\mathcal{A}-\mathcal{B})\left( \psi _{2}^{n+m}-\psi _{2}^{2n}\right) }{\left( \psi _{2}^{m}-\psi _{2}^{n}\right) ^{2}}\right) . \end{equation*}$

Proof. The proof can be produced directly by making use of Lemma 1 in $\left(2.15\right),$ so we choose to omit it. □

3. The coefficient inequalities for $\mathcal{G}^{-1}\in \mathcal{R} _{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{D }, \mathcal{E}\right)$

The "Koebe one quarter theorem" ^[2] ensures that the image of $\Lambda$ under each univalent function $\mathcal{G}\in \mathbb{A}$ consists a disk of radius $\dfrac{1}{4}$ . Thus each univalent function $\mathcal{G}$ has an inverse $\mathcal{G}^{-1}$ satisfying

$\begin{equation*} \mathcal{G}^{-1}\left( \mathcal{G}(\xi )\right) = \xi , \;(\xi \in \Lambda )\quad \text{and}\quad \mathcal{G}\left( \mathcal{G}^{-1}(w)\right) = w, \;\left( |w| < r_{0}(\mathcal{G}), \;r_{0}(\mathcal{G})\geq \frac{1}{4} \right) . \end{equation*}$

A function $\mathcal{G}\in \mathbb{A}$ is called bi-univalent in $\Lambda$ if both $\mathcal{G}$ and $\mathcal{G}^{-1}$ are univalent in $\Lambda$ . We mention that the collection of bi-univalent functions defined in the unit disk $\Lambda$ is not empty. For example, the functions $\xi$ , $\dfrac{\xi }{1-\xi }$ , $-\log (1-\xi)$ and $\dfrac{1}{2}\log \dfrac{1+\xi }{1-\xi }$ are members of bi-univalent function family, however the Koebe function is not a member.

Theorem 3. If $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ and the inverse function of $\mathcal{G}$ is $\mathcal{G}^{-1}(w) = w+\sum\limits_{j = 2}^{ \infty }d_{j}w^{j}$ , then

$\begin{eqnarray} |d_{2}| &\leq &\frac{(\sigma -\upsilon +1)\left( \mathcal{A}-\mathcal{B} \right) }{(\sigma +1)\left\vert \psi _{2}^{m}-\psi _{2}^{n}\right\vert } \end{eqnarray}$

(3.1)

$\begin{eqnarray} |d_{3}| &\leq &\frac{(\sigma -\upsilon +1)(\sigma -\upsilon +2)\left( \mathcal{A}-\mathcal{B}\right) }{(\sigma +1)(\sigma +2)\left\vert \psi _{3}^{m}-\psi _{3}^{n}\right\vert }\max \left\{ 1;\left\vert 2\Theta \left( 2, \sigma , \upsilon , \mathcal{A}, \mathcal{B}\right) -1\right\vert \right\} , \end{eqnarray}$

(3.2)

and for any $\mu \in \mathbb{C}$ , we have

$\begin{align*} \left\vert d_{3}-\mu d_{2}^{2}\right\vert & \leq \frac{(\sigma -\upsilon +1)(\sigma -\upsilon +2)\left( \mathcal{A}-\mathcal{B}\right) }{(\sigma +1)(\sigma +2)\left\vert \psi _{3}^{m}-\psi _{3}^{n}\right\vert }\times \\ & \max \left\{ 1;\left\vert 2\Theta \left( 2, \sigma , \upsilon , \mathcal{A}, \mathcal{B}\right) +\mu \tfrac{(\mathcal{A}-\mathcal{B})(\sigma +2)(\sigma -\upsilon +1)\left( \psi _{3}^{m}-\psi _{3}^{n}\right) }{(\sigma +1)(\sigma -\upsilon +2)\left( \psi _{2}^{m}-\psi _{2}^{n}\right) ^{2}}-1\right\vert \right\} , \end{align*}$

where $\Theta \left(2, \sigma, \upsilon, \mathcal{A}, \mathcal{B}\right)$ given by (2.7).

Proof. Since

$\begin{equation} \mathcal{G}^{-1}(w) = w+\sum\limits_{n = 2}^{\infty }d_{n}w^{n} \end{equation}$

(3.3)

is the inverse of the function $\mathcal{G}$ , it can be seen that

$\begin{equation} \xi = \mathcal{G}^{-1}\left( \mathcal{G}(\xi )\right) = \mathcal{G}\left( \mathcal{G}^{-1}(\xi )\right) , \;|\xi | < r_{0}(\mathcal{G}). \end{equation}$

(3.4)

From (1.1) and (3.4), we obtain that

$\begin{equation} \xi = \mathcal{G}^{-1}\left( \xi +\sum\limits_{n = 2}^{\infty }a_{n}\xi ^{n}\right) , \;|\xi | < r_{0}(\mathcal{G}). \end{equation}$

(3.5)

Therefore from (3.4) and (3.5) we get

$\begin{equation} \xi +(a_{2}+d_{2})\xi ^{2}+(a_{3}+2a_{2}d_{2}+d_{3})\xi ^{3}+\dots = \xi , \;|\xi | < r_{0}(\mathcal{G}). \end{equation}$

(3.6)

Equating the corresponding coefficients of the relation (3.6), we conclude that

$\begin{align} & d_{2} = -a_{2}, \end{align}$

(3.7)

$\begin{align} & d_{3} = 2a_{2}^{2}-a_{3}. \end{align}$

(3.8)

First, from the relations (2.11) and (3.7) we have

$\begin{equation} d_{2} = -\frac{(\sigma -\upsilon +1)\left( \mathcal{A}-\mathcal{B}\right) }{ 2(\sigma +1)\left( \psi _{2}^{m}-\psi _{2}^{n}\right) }p_{1}. \end{equation}$

(3.9)

To find $|d_{3}|$ , from (3.8) we have

$\begin{equation*} |d_{3}| = |a_{3}-2a_{2}^{2}|. \end{equation*}$

Hence, by using (2.15) for real $\tau = 2$ we deduce that

$\begin{align} |d_{3}|& = \left\vert a_{3}-2a_{2}^{2}\right\vert \\ & = \frac{(\sigma -\upsilon +1)(\sigma -\upsilon +2)\left( \mathcal{A}- \mathcal{B}\right) }{2(\sigma +1)(\sigma +2)\left\vert \psi _{3}^{m}-\psi _{3}^{n}\right\vert }\left\vert p_{2}-\Theta \left( 2, \sigma , \upsilon , \mathcal{A}, \mathcal{B}\right) p_{1}^{2}\right\vert \\ & = \frac{(\sigma -\upsilon +1)(\sigma -\upsilon +2)\left( \mathcal{A}- \mathcal{B}\right) }{(\sigma +1)(\sigma +2)\left\vert \psi _{3}^{m}-\psi _{3}^{n}\right\vert }\max \left\{ 1;\left\vert 2\Theta \left( 2, \sigma , \upsilon , \mathcal{A}, \mathcal{B}\right) \right\vert -1\right\} , \end{align}$

(3.10)

where $\Theta \left(2, \sigma, \upsilon, \mathcal{A}, \mathcal{B}\right)$ given by (2.7). For any complex number $\mu$ , a simple computation gives us that

$\begin{align} d_{3}-\mu d_{2}^{2}& = \frac{(\sigma -\upsilon +1)(\sigma -\upsilon +2)\left( \mathcal{A}-\mathcal{B}\right) }{2(\sigma +1)(\sigma +2)\left( \psi _{3}^{m}-\psi _{3}^{n}\right) }\left( p_{2}-\Theta \left( 2, \sigma , \upsilon , \mathcal{A}, \mathcal{B}\right) p_{1}^{2}\right) \\ & -\mu \frac{\left[ (\sigma -\upsilon +1)(\mathcal{A}-\mathcal{B})\right] ^{2}}{\left[ 2(\sigma +1)\left( \psi _{2}^{m}-\psi _{2}^{n}\right) \right] ^{2}}p_{1}^{2}. \\ & = \frac{(\sigma -\upsilon +1)(\sigma -\upsilon +2)\left( \mathcal{A}- \mathcal{B}\right) }{2(\sigma +1)(\sigma +2)\left( \psi _{3}^{m}-\psi _{3}^{n}\right) }\times \\ & \left( p_{2}-\frac{p_{1}^{2}}{2}\left[ 2\Theta \left( 2, \sigma , \upsilon , \mathcal{A}, \mathcal{B}\right) +\mu \frac{(\mathcal{A}-\mathcal{B})(\sigma +2)(\sigma -\upsilon +1)\left( \psi _{3}^{m}-\psi _{3}^{n}\right) }{(\sigma +1)(\sigma -\upsilon +2)\left( \psi _{2}^{m}-\psi _{2}^{n}\right) ^{2}} \right] \right) . \end{align}$

(3.11)

By taking modulus on both sides of (3.11) and applying Lemma 1 and (2.1), we find that

$\begin{align*} \left\vert d_{3}-\mu d_{2}^{2}\right\vert & \leq \frac{(\sigma -\upsilon +1)(\sigma -\upsilon +2)\left( \mathcal{A}-\mathcal{B}\right) }{(\sigma +1)(\sigma +2)\left\vert \psi _{3}^{m}-\psi _{3}^{n}\right\vert }\times \\ & \max \left\{ 1;\left\vert 2\Theta \left( 2, \sigma , \upsilon , \mathcal{A}, \mathcal{B}\right) +\mu \frac{(\mathcal{A}-\mathcal{B})(\sigma +2)(\sigma -\upsilon +1)\left( \psi _{3}^{m}-\psi _{3}^{n}\right) }{(\sigma +1)(\sigma -\upsilon +2)\left( \psi _{2}^{m}-\psi _{2}^{n}\right) ^{2}}-1\right\vert \right\} , \end{align*}$

and this completes our proof. □

4. Characterization properties

By applying the techniques introduced by Silverman in ^[7], we will introduce some characteristic properties of the functions $\mathcal{G} \in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ such as partial sums results, necessary and sufficient conditions, radii of close-to-convexity, distortion bounds, radii of starlikeness and convexity.

Theorem 4. If $\mathcal{G}\in \mathbb{A}$ and be defined as (1.1) belongs to $\mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ , then

$\begin{equation} \sum\limits_{j = 2}^{\infty }\left( (1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A}-1)\psi _{j}^{n}\right) (\tfrac{\sigma +1}{(\sigma -\upsilon +1)})\left\vert a_{j}\right\vert \leq \left( \mathcal{A}-\mathcal{B}\right) , \end{equation}$

(4.1)

where $\psi _{j}^{n}$ given by (1.9).

Proof. Letting $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ , by (1.11) we deduce that

$\begin{equation} \frac{\mathcal{D}_{t, \delta , \upsilon }^{\sigma , m}\mathcal{G}\left( \zeta \right) }{\mathcal{D}_{t, \delta , \upsilon }^{\sigma , n}\mathcal{G}\left( \zeta \right) } = \frac{1+\mathcal{A}w\left( \xi \right) }{1+\mathcal{B} w\left( \xi \right) }, \;\xi \in \Lambda , \end{equation}$

(4.2)

where $w\left(\xi \right)$ is a Schwarz function, or equivalently

$\begin{equation*} \left\vert \frac{\mathcal{D}_{t, \delta , \upsilon }^{\sigma , m}\mathcal{G} \left( \zeta \right) -\mathcal{D}_{t, \delta , \upsilon }^{\sigma , n}\mathcal{G }\left( \zeta \right) }{\mathcal{AD}_{t, \delta , \upsilon }^{\sigma , n} \mathcal{G}\left( \zeta \right) -\mathcal{BD}_{t, \delta , \upsilon }^{\sigma , m}\mathcal{G}\left( \zeta \right) }\right\vert < 1, \;\xi \in \Lambda . \end{equation*}$

Thus, the above relation leads us to

$\begin{eqnarray*} &&\left\vert \frac{\mathcal{D}_{t, \delta , \upsilon }^{\sigma , m}\mathcal{G} \left( \zeta \right) -\mathcal{D}_{t, \delta , \upsilon }^{\sigma , n}\mathcal{G }\left( \zeta \right) }{\mathcal{AD}_{t, \delta , \upsilon }^{\sigma , n} \mathcal{G}\left( \zeta \right) -\mathcal{BD}_{t, \delta , \upsilon }^{\sigma , m}\mathcal{G}\left( \zeta \right) }\right\vert \\ & = &\left\vert \frac{\sum\limits_{j = 2}^{\infty }\left( \psi _{j}^{m}-\psi _{j}^{n}\right) \left( \tfrac{\sigma +1}{(\sigma -\upsilon +1)}\right) a_{j}\xi ^{j}}{\left( \mathcal{A}-\mathcal{B}\right) \xi +\sum\limits_{j = 2}^{\infty }\left( \mathcal{A}\psi _{j}^{n}-\mathcal{B}\psi _{j}^{m}\right) \left( \tfrac{\sigma +1}{(\sigma -\upsilon +1)}\right) a_{j}\xi ^{j}}\right\vert \\ &\leq &\frac{\sum\limits_{j = 2}^{\infty }\left( \psi _{j}^{m}-\psi _{j}^{n}\right) \left( \tfrac{\sigma +1}{(\sigma -\upsilon +1)}\right) \left\vert a_{j}\right\vert r^{j-1}}{\left( \mathcal{A}-\mathcal{B}\right) -\sum\limits_{j = 2}^{\infty }\left( \mathcal{A}\psi _{j}^{n}-\mathcal{B}\psi _{j}^{m}\right) \left( \tfrac{\sigma +1}{(\sigma -\upsilon +1)}\right) \left\vert a_{j}\right\vert r^{j-1}} < 1, \end{eqnarray*}$

and taking $\left\vert \xi \right\vert = r\rightarrow 1^{-}$ simple computation yields $\left(4.1\right)$ . □

Example 1. For

$\begin{equation*} \mathcal{G}\left( \xi \right) = \xi +\sum\limits_{j = 2}^{\infty }\frac{\left( \mathcal{A}-\mathcal{B}\right) }{(1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A} -1)\psi _{j}^{n}}(\tfrac{\sigma -\upsilon +1}{\sigma +1})\ell _{j}\xi ^{j}, \;\xi \in \Lambda , \end{equation*}$

such that $\sum\limits_{j = 2}^{\infty }\ell _{j} = 1$ , we get

$\begin{eqnarray*} &&\sum\limits_{j = 2}^{\infty }\left( (1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A} -1)\psi _{j}^{n}\right) (\tfrac{\sigma +1}{\sigma -\upsilon +1})\left\vert a_{j}\right\vert \\ & = &\sum\limits_{j = 2}^{\infty }\left( (1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A} -1)\psi _{j}^{n}\right) (\tfrac{\sigma +1}{\sigma -\upsilon +1})\times \\ &&\frac{\left( \mathcal{A}-\mathcal{B}\right) }{(1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A}-1)\psi _{j}^{n}}(\tfrac{\sigma -\upsilon +1}{\sigma +1} )\ell _{j} \\ & = &\left( \mathcal{A}-\mathcal{B}\right) \sum\limits_{j = 2}^{\infty }\ell _{j} = \left( \mathcal{A}-\mathcal{B}\right) . \end{eqnarray*}$

Then $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ , and we note that the inequality $\left(4.1\right)$ is sharp.

Corollary 1. Let $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ given by (1.1). Then

$\begin{equation} \left\vert a_{j}\right\vert \leq \frac{\left( \mathcal{A}-\mathcal{B}\right) }{(1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A}-1)\psi _{j}^{n}}(\tfrac{\sigma -\upsilon +1}{\sigma +1}), \;\mathit{\text{for}}\;j\geq 2, \end{equation}$

(4.3)

where $\psi _{j}^{n}$ is defined by (1.9). The approximation is sharp for the function

$\begin{equation} \mathcal{G}_{\ast }\left( \xi \right) : = \xi -\frac{\left( \mathcal{A}- \mathcal{B}\right) }{(1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A}-1)\psi _{j}^{n}}(\tfrac{\sigma -\upsilon +1}{\sigma +1})\xi ^{j}, \;j\geq 2. \end{equation}$

(4.4)

Theorem 5. If $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ , then

$\begin{equation} r-\tfrac{\left( \mathcal{A}-\mathcal{B}\right) }{(1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A}-1)\psi _{j}^{n}}(\tfrac{\sigma -\upsilon +1}{\sigma +1} )r^{2}\leq \left\vert \mathcal{G}\left( \eta \right) \right\vert \leq r+ \tfrac{\left( \mathcal{A}-\mathcal{B}\right) }{(1-\mathcal{B})\psi _{j}^{m}+( \mathcal{A}-1)\psi _{j}^{n}}(\tfrac{\sigma -\upsilon +1}{\sigma +1})r^{2}. \end{equation}$

(4.5)

For the function defined by

$\begin{equation} \widehat{\mathcal{G}}\left( \xi \right) : = \xi -\frac{\left( \mathcal{A}- \mathcal{B}\right) }{(1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A}-1)\psi _{j}^{n}}(\tfrac{\sigma -\upsilon +1}{\sigma +1})\xi ^{2}, \;|\xi | = r < 1, \end{equation}$

(4.6)

the approximation is sharp.

Proof. For $\left\vert \xi \right\vert = r < 1$ we have

$\begin{equation*} \left\vert \mathcal{G}\left( \xi \right) \right\vert = \left\vert \xi +\sum\limits_{j = 2}^{\infty }a_{j}\xi ^{j}\right\vert \leq \left\vert \xi \right\vert +\sum\limits_{j = 2}^{\infty }a_{j}\left\vert \xi \right\vert ^{j} = r+\sum\limits_{j = 2}^{\infty }a_{j}\left\vert r\right\vert ^{j}. \end{equation*}$

Moreover, since for $\left\vert \xi \right\vert = r < 1$ we get $r^{j} < r^{2}$ for all $j\geq 2$ , the above relation implies that

$\begin{equation} \left\vert \mathcal{G}\left( \xi \right) \right\vert \leq r+r^{2}\sum\limits_{j = 2}^{\infty }\left\vert a_{j}\right\vert . \end{equation}$

(4.7)

Similarly, we get

$\begin{equation} \left\vert \mathcal{G}\left( \xi \right) \right\vert \geq r-r^{2}\sum\limits_{j = 2}^{\infty }\left\vert a_{j}\right\vert . \end{equation}$

(4.8)

From the relation $\left(4.1\right)$ we have

$\begin{equation*} \sum\limits_{j = 2}^{\infty }\left( (1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A}-1)\psi _{j}^{n}\right) (\tfrac{\sigma +1}{\sigma -\upsilon +1})\left\vert a_{j}\right\vert \leq \left( \mathcal{A}-\mathcal{B}\right) , \end{equation*}$

but

$\begin{equation*} \left( (1-\mathcal{B})\psi _{2}^{m}+(\mathcal{A}-1)\psi _{2}^{n}\right) ( \tfrac{\sigma +1}{\sigma -\upsilon +1})\sum\limits_{j = 2}^{\infty }\left\vert a_{j}\right\vert \leq \sum\limits_{j = 2}^{\infty }\left( (1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A}-1)\psi _{j}^{n}\right) (\tfrac{\sigma +1}{\sigma -\upsilon +1})\left\vert a_{j}\right\vert \leq \left( \mathcal{A}-\mathcal{B} \right) . \end{equation*}$

Therefore,

$\begin{equation} \sum\limits_{j = 2}^{\infty }a_{j}\leq \frac{(\tfrac{\sigma -\upsilon +1}{\sigma +1} )\left( \mathcal{A}-\mathcal{B}\right) }{(1-\mathcal{B})\psi _{2}^{m}+( \mathcal{A}-1)\psi _{2}^{n}}, \end{equation}$

(4.9)

and by using (4.3) in $\left(4.7\right)$ and $\left(4.8 \right)$ we get the desired result. □

The next distortion theorem for the family $\mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ could be similarly obtained:

Theorem 6. If $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ , then

$\begin{equation*} 1-\tfrac{2(\sigma -\upsilon +1)\left( \mathcal{A}-\mathcal{B}\right) }{ \left( \sigma +1\right) \left( (1-\mathcal{B})\psi _{2}^{m}+(\mathcal{A} -1)\psi _{2}^{n}\right) }r\leq \left\vert \mathcal{G}^{\prime }\left( \xi \right) \right\vert \leq 1+\tfrac{2(\sigma -\upsilon +1)\left( \mathcal{A}- \mathcal{B}\right) }{\left( \sigma +1\right) \left( (1-\mathcal{B})\psi _{2}^{m}+(\mathcal{A}-1)\psi _{2}^{n}\right) }r. \end{equation*}$

The equality holds if the function is $\widehat{\mathcal{G}}$ given by $(4.6)$ .

Proof. Since the proof is quite analogous with those of Theorem 5, so it will be omitted. □

The next result deals with the fact that a convex combination of functions of the class $\mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{ A}, \mathcal{B}\right)$ belongs to the same class, as follows:

Theorem 7. Let $\mathcal{G}_{i}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ given by

$\begin{equation} \mathcal{G}_{i}\left( \xi \right) = \xi +\sum\limits_{j = 2}^{\infty }a_{i, j}\xi ^{j}, \;i = 1, 2, 3, \dots , m. \end{equation}$

(4.10)

Then $H\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ , where

$\begin{equation} H\left( \xi \right) : = \sum\limits_{i = 1}^{m}c_{i}\mathcal{G}_{i}\left( \xi \right) , \;\mathit{\text{and}}\;\sum\limits_{i = 1}^{m}c_{i} = 1. \end{equation}$

(4.11)

Proof. By Theorem 4 we have

and,

$\begin{equation*} H\left( \xi \right) = \sum\limits_{i = 1}^{m}c_{i}\left( \xi +\sum\limits_{j = 2}^{\infty }a_{i, j}\xi ^{j}\right) = \xi +\sum\limits_{j = 2}^{\infty }\left( \sum\limits_{i = 1}^{m}c_{i}a_{i, j}\right) \xi ^{j}. \end{equation*}$

Therefore

$\begin{eqnarray*} &&\sum\limits_{j = 2}^{\infty }\left( (1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A} -1)\psi _{j}^{n}\right) (\tfrac{\sigma +1}{\sigma -\upsilon +1})\left\vert \sum\limits_{i = 1}^{m}c_{i}a_{i, j}\right\vert \\ &\leq &\sum\limits_{i = 1}^{m}\left[ \sum\limits_{j = 2}^{\infty }\left( (1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A}-1)\psi _{j}^{n}\right) (\tfrac{\sigma +1}{\sigma -\upsilon +1})\left\vert a_{i, j}\right\vert \right] c_{i} \\ & = &\sum\limits_{i = 1}^{m}\left( \mathcal{A}-\mathcal{B}\right) c_{i} = \left( \mathcal{ A}-\mathcal{B}\right) \sum\limits_{i = 1}^{m}c_{i} = \left( \mathcal{A}-\mathcal{B} \right) , \end{eqnarray*}$

thus $H\left(\xi \right) \in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ . □

Regarding the arithmetic means of the functions of the family $\mathcal{R} _{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ the next result holds:

Theorem 8. If $\mathcal{G}_{i}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ are given by (4.10), then

$\begin{equation} \mathcal{G}\left( \xi \right) : = \xi +\frac{1}{k}\sum\limits_{j = 2}^{\infty }\left( \sum\limits_{i = 1}^{k}a_{i, j}\xi ^{j}\right) \in \mathcal{R}_{t, \delta , \upsilon }^{m, n, \sigma }\left( \mathcal{A}, \mathcal{B}\right). \end{equation}$

(4.12)

Where $\mathcal{G}$ is the arithmetic mean of $\mathcal{G}_{i}$ , $i = 1, 2, 3, \dots, k$ .

Proof. From the definition relation (4.12) we get

$\begin{equation*} \mathcal{G}\left( \xi \right) = \frac{1}{k}\sum\limits_{i = 1}^{k}f_{i}\left( \xi \right) = \frac{1}{k}\sum\limits_{i = 1}^{k}\left( \xi +\sum\limits_{j = 2}^{\infty }a_{i, j}\xi ^{j}\right) = \xi +\sum\limits_{j = 2}^{\infty }\left( \frac{1}{k} \sum\limits_{i = 1}^{k}a_{i, j}\right) \xi ^{j}, \end{equation*}$

and to prove that $\mathcal{G}\left(\xi \right) \in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ , according to the Theorem 4.1 it is sufficient to prove that

$\begin{equation*} \sum\limits_{j = 2}^{\infty }\left( (1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A}-1)\psi _{j}^{n}\right) (\tfrac{\sigma +1}{\sigma -\upsilon +1})\left( \frac{1}{k} \sum\limits_{i = 1}^{k}\left\vert a_{i, j}\right\vert \right) \leq \left( \mathcal{A}- \mathcal{B}\right) . \end{equation*}$

A simple computation shows that

$\begin{eqnarray*} &&\sum\limits_{j = 2}^{\infty }\left( (1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A} -1)\psi _{j}^{n}\right) (\tfrac{\sigma +1}{\sigma -\upsilon +1})\left( \frac{ 1}{k}\sum\limits_{i = 1}^{k}\left\vert a_{i, j}\right\vert \right) \\ & = &\frac{1}{k}\sum\limits_{i = 1}^{k}\left( \sum\limits_{j = 2}^{\infty }\left( (1-\mathcal{B} )\psi _{j}^{m}+(\mathcal{A}-1)\psi _{j}^{n}\right) (\tfrac{\sigma +1}{\sigma -\upsilon +1})\left\vert a_{i, j}\right\vert \right) \\ &\leq &\frac{1}{k}\sum\limits_{i = 1}^{k}\left( \mathcal{A}-\mathcal{B}\right) = \left( \mathcal{A}-\mathcal{B}\right). \end{eqnarray*}$

Therefore $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ . □

Theorem 9. If $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ , then $\mathcal{G}$ is a starlike functions of order $\vartheta$ $\left(0\leq \vartheta < 1\right),$ $\left\vert \xi \right\vert < r_{1}^{\ast },$

$\begin{equation*} r_{1}^{\ast } = \inf\limits_{j\geq 2}\left( \frac{\left( 1-\vartheta \right) \left( (1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A}-1)\psi _{j}^{n}\right) \left( \sigma +1\right) }{\left( j-\vartheta \right) (\sigma -\upsilon +1)\left( \mathcal{A}-\mathcal{B}\right) }\right) ^{\frac{1}{j-1}}. \end{equation*}$

The equality holds for $\mathcal{G}$ given in $\left(4.4\right).$

Proof. Let $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ . We see that $\mathcal{G}$ is a starlike functions of order $\vartheta$ , if

$\begin{equation*} \left\vert \frac{\xi \mathcal{G}^{\prime }\left( \xi \right) }{\mathcal{G} \left( \xi \right) }-1\right\vert < 1-\vartheta . \end{equation*}$

By simple calculation, we deduce

$\begin{equation} \sum\limits_{j = 2}^{\infty }\left( \frac{j-\vartheta }{1-\vartheta }\right) \left\vert a_{j}\right\vert \left\vert \xi \right\vert ^{j-1} < 1. \end{equation}$

(4.13)

Since $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ , from $\left(4.1\right)$ we get

$\begin{equation} \sum\limits_{j = 2}^{\infty }\frac{\left( (1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A} -1)\psi _{j}^{n}\right) \left( \sigma +1\right) }{(\sigma -\upsilon +1)\left( \mathcal{A}-\mathcal{B}\right) }\left\vert a_{j}\right\vert < 1. \end{equation}$

(4.14)

The relation $\left(4.13\right)$ will holds true if

$\begin{eqnarray*} &&\sum\limits_{j = 2}^{\infty }\left( \frac{j-\vartheta }{1-\vartheta }\right) \left\vert a_{j}\right\vert \left\vert \xi \right\vert ^{j-1} \\ & < &\sum\limits_{j = 2}^{\infty }\frac{\left( (1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A} -1)\psi _{j}^{n}\right) \left( \sigma +1\right) }{(\sigma -\upsilon +1)\left( \mathcal{A}-\mathcal{B}\right) }\left\vert a_{j}\right\vert , \end{eqnarray*}$

which implies that

$\begin{equation*} \left\vert \xi \right\vert ^{j-1} < \left( \frac{\left( 1-\vartheta \right) \left( (1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A}-1)\psi _{j}^{n}\right) \left( \sigma +1\right) }{\left( j-\vartheta \right) (\sigma -\upsilon +1)\left( \mathcal{A}-\mathcal{B}\right) }\right) , \end{equation*}$

or, equivalently

$\begin{equation*} \left\vert \xi \right\vert < \left( \frac{\left( 1-\vartheta \right) \left( (1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A}-1)\psi _{j}^{n}\right) \left( \sigma +1\right) }{\left( j-\vartheta \right) (\sigma -\upsilon +1)\left( \mathcal{A}-\mathcal{B}\right) }\right) ^{\frac{1}{^{j-1}}}, \end{equation*}$

which yields the starlikeness of the family. □

Theorem 10. If $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ , then $\mathcal{G}$ is a close-to-convex function of order $\vartheta$ $\left(0\leq \vartheta < 1\right),$ $\left\vert \xi \right\vert < r_{2}^{\ast },$

$\begin{equation*} r_{2}^{\ast } = \inf\limits_{j\geq 2}\left( \frac{\left( 1-\vartheta \right) \left( \sigma +1\right) \left( (1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A}-1)\psi _{j}^{n}\right) }{j(\sigma -\upsilon +1)\left( \mathcal{A}-\mathcal{B} \right) }\right) ^{\frac{1}{j-1}}. \end{equation*}$

Proof. Let $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ . If $\mathcal{G}$ is close-to-convex function of order $\vartheta$ , then we find that

$\begin{equation*} \left\vert \mathcal{G}^{\prime }\left( \xi \right) -1\right\vert < 1-\vartheta , \end{equation*}$

that is

$\begin{equation} \sum\limits_{j = 2}^{\infty }\frac{j}{1-\vartheta }\left\vert a_{j}\right\vert \left\vert \xi \right\vert ^{j-1} < 1. \end{equation}$

(4.15)

Since $\mathcal{G}\in \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ , by $\left(4.1\right)$ we have

$\begin{equation} \sum\limits_{j = 2}^{\infty }\frac{\left( \sigma +1\right) \left( (1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A}-1)\psi _{j}^{n}\right) }{(\sigma -\upsilon +1)\left( \mathcal{A}-\mathcal{B}\right) }\left\vert a_{j}\right\vert < 1. \end{equation}$

(4.16)

The relation $\left(4.13\right)$ will holds true if

$\begin{eqnarray*} &&\sum\limits_{j = 2}^{\infty }\frac{j}{1-\vartheta }\left\vert a_{j}\right\vert \left\vert \xi \right\vert ^{j-1} \\ & < &\sum\limits_{j = 2}^{\infty }\frac{\left( \sigma +1\right) \left( (1-\mathcal{B} )\psi _{j}^{m}+(\mathcal{A}-1)\psi _{j}^{n}\right) }{(\sigma -\upsilon +1)\left( \mathcal{A}-\mathcal{B}\right) }\left\vert a_{j}\right\vert , \end{eqnarray*}$

which implies that

$\begin{equation*} \left\vert \xi \right\vert ^{j-1} < \left( \frac{\left( 1-\vartheta \right) \left( \sigma +1\right) \left( (1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A} -1)\psi _{j}^{n}\right) }{j(\sigma -\upsilon +1)\left( \mathcal{A}-\mathcal{B }\right) }\right) , \end{equation*}$

or, equivalently

$\begin{equation*} \left\vert \xi \right\vert < \left( \frac{\left( 1-\vartheta \right) \left( \sigma +1\right) \left( (1-\mathcal{B})\psi _{j}^{m}+(\mathcal{A}-1)\psi _{j}^{n}\right) }{j(\sigma -\upsilon +1)\left( \mathcal{A}-\mathcal{B} \right) }\right) ^{\frac{1}{^{j-1}}}, \end{equation*}$

which yields the desired result. □

5. Conclusions

In this paper, we introduced a new class $\mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ of holomorphic functions defined in the open unit disk, which is connected to the combination of the Binomial series and the Babalola operator. We employed differential subordination involving Janowski-type functions to investigate these properties. Utilizing well-established results, such as Carathéodory's inequality for functions with real positive parts, as well as the Keogh-Merkes and Ma-Minda inequalities, we established upper bounds for the first two initial coefficients of the Taylor-Maclaurin power series expansion. Additionally, we derived an upper bound for the Fekete-Szegő functional for functions within this family.

We also extended our findings to include similar results for the first two coefficients and for the Fekete-Szegő inequality for functions $\mathcal{G}^{-1}$ when $\mathcal{G}\in \mathcal{R} _{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right)$ . Furthermore, we determined coefficient estimates, distortion bounds, radius problems, and the radius of starlikeness and close-to-convexity for these newly defined functions.

Author contributions

Kholood M. Alsager: Conceptualization, validation, formal analysis, investigation, supervision; Sheza M. El-Deeb: Methodology, formal analysis, investigation; Ala Amourah: Methodology, validation, writing-original draft; Jongsuk Ro: Writing-original draft, writing-review. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIT) (No. NRF-2022R1A2C2004874). This work was supported by the Korea Institute of Energy Technology Evaluation and Planning(KETEP) and the Ministry of Trade, Industry & Energy(MOTIE) of the Republic of Korea (No. 20214000000280).

Conflict of interest

The authors declare no conflict of interest.

References

[1]	X. D. Yang, J. N. Zhang, S. Y. Ren, Q. Y. Ran, Can the new energy demonstration city policy reduce environmental pollution? Evidence from a quasi-natural experiment in China, J. Clean. Prod., 287 (2021), 125015. http://doi.org/10.1016/j.jclepro.2020.125015 doi: 10.1016/j.jclepro.2020.125015
[2]	M. Ellman, F. Germano, What do the papers sell? A model of advertising and media bias, The Economic Journal, 119 (2009), 680–704. https://doi.org/10.1111/j.1468-0297.2009.02218.x doi: 10.1111/j.1468-0297.2009.02218.x
[3]	H. T. Wu, Y. W. Li, Y. Hao, S. Y. Ren, P. F. Zhang, Environmental decentralization, local government competition, and regional green development: Evidence from China, Sci. Total Environ., 708 (2020), 135085. http://doi.org/10.1016/j.scitotenv.2019.135085 doi: 10.1016/j.scitotenv.2019.135085
[4]	Y. Liu, C. Ma, Z. Huang, Can the digital economy improve green total factor productivity? An empirical study based on Chinese urban data, Math. Biosci. Eng., 20 (2023), 6866–6893. http://doi.org/10.3934/mbe.2023296 doi: 10.3934/mbe.2023296
[5]	L. Katusiime, International monetary spillovers and macroeconomic stability in developing countries, National Accounting Review, 3 (2021), 310–329. http://doi.org/10.3934/NAR.2021016 doi: 10.3934/NAR.2021016
[6]	Y. Liu, L. Chen, L. Lv, P. Failler, The impact of population aging on economic growth: a case study on China, AIMS Mathematics, 8 (2023), 10468–10485. http://doi.org/10.3934/math.2023531 doi: 10.3934/math.2023531
[7]	K. C. Ho, X. X. Shen, C. Yan, X. Hu, Influence of green innovation on disclosure quality: mediating role of media attention, Technol. Forecast. Soc., 188 (2023), 122314. http://doi.org/10.1016/j.techfore.2022.122314 doi: 10.1016/j.techfore.2022.122314
[8]	S. Boulianne, J. Ohme, Pathways to environmental activism in four countries: social media, environmental concern, and political efficacy, J. Youth Stud., 25 (2022), 771–792. http://doi.org/10.1080/13676261.2021.2011845 doi: 10.1080/13676261.2021.2011845
[9]	C. C. Lee, M. L. Zeng, C. S. Wang, Environmental regulation, innovation capability, and green total factor productivity: new evidence from China, Environ. Sci. Pollut. Res., 29 (2022), 39384–39399. http://doi.org/10.1007/s11356-021-18388-0 doi: 10.1007/s11356-021-18388-0
[10]	S. Huang, K. T. Huat, Z. Zhou, The studies on Chinese traditional culture and corporate environmental responsibility: literature review and its implications, National Accounting Review, 4 (2022), 1–15. http://doi.org/10.3934/NAR.2022001 doi: 10.3934/NAR.2022001
[11]	J. M. Mazzarino, L. Turatti, S. T. Petter, Environmental governance: media approach on the united nations programme for the environment, Environ. Dev., 33 (2020), 100502. http://doi.org/10.1016/j.envdev.2020.100502 doi: 10.1016/j.envdev.2020.100502
[12]	Z. Li, L. Chen, H. Dong, What are bitcoin market reactions to its-related events?, Int. Rev. Econ. Financ., 73 (2021), 1–10. http://doi.org/10.1016/j.iref.2020.12.020 doi: 10.1016/j.iref.2020.12.020
[13]	P. Liu, Y. Zhao, J. Zhu, C. Yang, Technological industry agglomeration, green innovation efficiency, and development quality of city cluster, Green Finance, 4 (2022), 411–435. http://doi.org/10.3934/GF.2022020 doi: 10.3934/GF.2022020
[14]	C. Luo, Z. Li, L. Liu, Does investor sentiment affect stock pricing? Evidence from seasoned equity offerings in China, National Accounting Review, 3 (2021), 115–136. http://doi.org/10.3934/NAR.2021006 doi: 10.3934/NAR.2021006
[15]	M. Akhtaruzzaman, S. Boubaker, Z. Umar, COVID-19 media coverage and ESG leader indices, Financ. Res. Lett., 45 (2022), 102170. http://doi.org/10.1016/j.frl.2021.102170 doi: 10.1016/j.frl.2021.102170
[16]	G. S. Miller, D. J. Skinner, The evolving disclosure landscape: How changes in technology, the media, and capital markets are affecting disclosure, J. Account. Res., 53 (2015), 221–239. http://doi.org/10.1111/1475-679x.12075 doi: 10.1111/1475-679x.12075
[17]	E. Assifuah-Nunoo, P. O. Junior, A. M. Adam, A. Bossman, Assessing the safe haven properties of oil in African stock markets amid the COVID-19 pandemic: a quantile regression analysis, Quant. Financ. Econ., 6 (2022), 244–269. http://doi.org/10.3934/QFE.2022011 doi: 10.3934/QFE.2022011
[18]	Y. X. Chen, J. Zhang, P. R. Tadikamalla, X. T. Gao, The relationship among government, enterprise, and public in environmental governance from the perspective of multi-player evolutionary game, Int. J. Environ. Res. Public Health, 16 (2019), 3351. http://doi.org/10.3390/ijerph16183351 doi: 10.3390/ijerph16183351
[19]	P. C. Tetlock, Giving content to investor sentiment: the role of media in the stock market, J. Financ., 62 (2007), 1139–1168. http://doi.org/10.1111/j.1540-6261.2007.01232.x doi: 10.1111/j.1540-6261.2007.01232.x
[20]	G. X. Zhang, Y. Q. Jia, B. Su, J. Xiu, Environmental regulation, economic development and air pollution in the cities of China: spatial econometric analysis based on policy scoring and satellite data, J. Clean. Prod., 328 (2021), 129496. http://doi.org/10.1016/j.jclepro.2021.129496 doi: 10.1016/j.jclepro.2021.129496
[21]	M. Irfan, A. Razzaq, A. Sharif, X. Yang, Influence mechanism between green finance and green innovation: exploring regional policy intervention effects in China, Technol. Forecast. Soc., 182 (2022), 121882. http://doi.org/10.1016/j.techfore.2022.121882 doi: 10.1016/j.techfore.2022.121882
[22]	M. Pichlak, A. R. Szromek, Eco-innovation, sustainability and business model innovation by open innovation dynamics, Journal of Open Innovation: Technology, Market, and Complexity, 7 (2021), 149. http://doi.org/10.3390/joitmc7020149 doi: 10.3390/joitmc7020149
[23]	P. A. Nylund, A. Brem, N. Agarwal, Enabling technologies mitigating climate change: the role of dominant designs in environmental innovation ecosystems, Technovation, 117 (2021), 102271. http://doi.org/10.1016/j.technovation.2021.102271 doi: 10.1016/j.technovation.2021.102271
[24]	Y. Xu, W. F. Ge, G. L. Liu, X. F. Su, J. N. Zhu, C. Y. Yang, et al., The impact of local government competition and green technology innovation on economic low-carbon transition: new insights from China, Environ. Sci. Pollut. Res., 30 (2022), 23714–23735. http://doi.org/10.1007/s11356-022-23857-1 doi: 10.1007/s11356-022-23857-1
[25]	C. Fussler, P. James, Driving eco-innovation: a breakthrough discipline for innovation and sustainability, Financial Times/Prentice Hall, 1996.
[26]	J. Hartmann, Toward a more complete theory of sustainable supply chain management: the role of media attention, Supply Chain Management, 26 (2021), 532–547. http://doi.org/10.1108/scm-01-2020-0043 doi: 10.1108/scm-01-2020-0043
[27]	Z. H. Li, Z. M. Huang, Y. Y. Su, New media environment, environmental regulation and corporate green technology innovation: Evidence from China, Energ. Econ., 119 (2023), 106545. http://doi.org/10.1016/j.eneco.2023.106545 doi: 10.1016/j.eneco.2023.106545
[28]	E. R. Gray, J. M. T. Balmer, Managing corporate image and corporate reputation, Long Range Plann., 31 (1998), 695–702. https://doi.org/10.1016/S0024-6301(98)00074-0 doi: 10.1016/S0024-6301(98)00074-0
[29]	E. Blankespoor, E. deHaan, C. Zhu, Capital market effects of media synthesis and dissemination: evidence from robo-journalism, Rev. Account. Stud., 23 (2018), 1–36. http://doi.org/10.1007/s11142-017-9422-2 doi: 10.1007/s11142-017-9422-2
[30]	X. F. Jiang, C. X. Zhao, J. J. Ma, J. Q. Liu, S. H. Li, Is enterprise environmental protection investment responsibility or rent-seeking? Chinese evidence, Environ. Dev. Econ., 26 (2021), 169–187. http://doi.org/10.1017/s1355770x20000327 doi: 10.1017/s1355770x20000327
[31]	I. S. Farouq, N. U. Sambo, A. U. Ahmad, A. H. Jakada, I. A. Danmaraya, Does financial globalization uncertainty affect CO₂ emissions? Empirical evidence from some selected SSA countries, Quant. Financ. Econ., 5 (2021), 247–263. http://doi.org/10.3934/QFE.2021011 doi: 10.3934/QFE.2021011
[32]	T. C. Chiang, Geopolitical risk, economic policy uncertainty and asset returns in Chinese financial markets, China Financ. Rev. Int., 11 (2021), 474–501. http://doi.org/10.1108/CFRI-08-2020-0115 doi: 10.1108/CFRI-08-2020-0115
[33]	J. L. Guan, H. J. Xu, D. Huo, Y. C. Hua, Y. F. Wang, Economic policy uncertainty and corporate innovation: Evidence from China, Pac.-Basin Financ. J., 67 (2021), 101542. http://doi.org/10.1016/j.pacfin.2021.101542 doi: 10.1016/j.pacfin.2021.101542
[34]	Y. Liu, J. Liu, L. Zhang, Enterprise financialization and R & D innovation: a case study of listed companies in China, Electron. Res. Arch., 31 (2023), 2447–2471. http://doi.org/10.3934/era.2023124 doi: 10.3934/era.2023124
[35]	T. C. Chiang, Can gold or silver be used as a hedge against policy uncertainty and COVID-19 in the Chinese market?, China Financ. Rev. Int., 12 (2022), 571–600. http://doi.org/10.1108/CFRI-12-2021-0232 doi: 10.1108/CFRI-12-2021-0232
[36]	L. H. Yin, C. Q. Wu, Promotion incentives and air pollution: from the political promotion tournament to the environment tournament, J. Environ. Manage., 317 (2022), 115491. http://doi.org/10.1016/j.jenvman.2022.115491 doi: 10.1016/j.jenvman.2022.115491
[37]	X. Yang, H. Wu, S. Ren, Q. Ran, J. Zhang, Does the development of the internet contribute to air pollution control in China? Mechanism discussion and empirical test, Struct. Change Econ. Dyn., 56 (2021), 207–224. http://doi.org/10.1016/j.strueco.2020.12.001 doi: 10.1016/j.strueco.2020.12.001
[38]	S. Y. Ren, Y. Hao, H. T. Wu, How does green investment affect environmental pollution? Evidence from China, Environ. Resource Econ., 81 (2022), 25–51. http://doi.org/10.1007/s10640-021-00615-4 doi: 10.1007/s10640-021-00615-4
[39]	Y. Li, X. D. Yang, Q. Y. Ran, H. T. Wu, M. Irfan, M. Ahmad, Energy structure, digital economy, and carbon emissions: evidence from China, Environ. Sci. Pollut. Res., 28 (2021), 64606–64629. http://doi.org/10.1007/s11356-021-15304-4 doi: 10.1007/s11356-021-15304-4
[40]	A. Biscione, R. Caruso, A. de Felice, Environmental innovation in European transition countries, Appl. Econ., 53 (2021), 521–535. http://doi.org/10.1080/00036846.2020.1808185 doi: 10.1080/00036846.2020.1808185
[41]	S. C. Zyglidopoulos, A. P. Georgiadis, C. E. Carroll, D. S. Siegel, Does media attention drive corporate social responsibility?, J. Bus. Res., 65 (2012), 1622–1627. http://doi.org/10.1016/j.jbusres.2011.10.021 doi: 10.1016/j.jbusres.2011.10.021
[42]	C. H. Yu, X. Wu, D. Zhang, S. Chen, J. Zhao, Demand for green finance: resolving financing constraints on green innovation in China, Energ. Policy, 153 (2021), 112255. https://doi.org/10.1016/j.enpol.2021.112255 doi: 10.1016/j.enpol.2021.112255
[43]	J. von Bloh, T. Broekel, B. Özgun, R. Sternberg, New(s) data for entrepreneurship research? An innovative approach to use Big Data on media coverage, Small Bus. Econ., 55 (2020), 673–694. http://doi.org/10.1007/s11187-019-00209-x doi: 10.1007/s11187-019-00209-x
[44]	G. B. Xiong, Y. D. Luo, Smog, media attention, and corporate social responsibility-empirical evidence from Chinese polluting listed companies, Environ. Sci. Pollut. Res., 28 (2021), 46116–46129. http://doi.org/10.1007/s11356-020-11978-4 doi: 10.1007/s11356-020-11978-4
[45]	S. R. Baker, N. Bloom, S. J. Davis, Measuring economic policy uncertainty, Quarterly Journal of Economics, 131 (2016), 1593–1636. http://doi.org/10.1093/qje/qjw024 doi: 10.1093/qje/qjw024
[46]	H. T. Wu, L. N. Xu, S. Y. Ren, Y. Hao, G. Y. Yan, How do energy consumption and environmental regulation affect carbon emissions in China? New evidence from a dynamic threshold panel model, Resour. Policy, 67 (2020), 101678. http://doi.org/10.1016/j.resourpol.2020.101678 doi: 10.1016/j.resourpol.2020.101678
[47]	T. Li, X. Li, G. Liao, Business cycles and energy intensity. Evidence from emerging economies, Borsa Istanb. Rev., 22 (2022), 560–570. http://doi.org/10.1016/j.bir.2021.07.005 doi: 10.1016/j.bir.2021.07.005
[48]	C. J. Hadlock, J. R. Pierce, New evidence on measuring financial constraints: moving beyond the KZ index, Rev. Financ. Stud., 23 (2010), 1909–1940. http://doi.org/10.1093/rfs/hhq009 doi: 10.1093/rfs/hhq009
[49]	Z. X. He, C. S. Cao, C. Feng, Media attention, environmental information disclosure and corporate green technology innovations in China's heavily polluting industries, Emerg. Mark. Financ. Tr., 58 (2022), 3939–3952. http://doi.org/10.1080/1540496x.2022.2075259 doi: 10.1080/1540496x.2022.2075259
[50]	M. A. Khan, X. Z. Qin, K. Jebran, A. Rashid, The sensitivity of firms' investment to uncertainty and cash flow: evidence from listed state-owned enterprises and non-state-owned enterprises in China, Sage Open, 10 (2020), 17. http://doi.org/10.1177/2158244020903433 doi: 10.1177/2158244020903433
[51]	X. Chang, Impact of risks on forced CEO turnover, Quant. Financ. Econ., 6 (2022), 177–205. http://doi.org/10.3934/QFE.2022008 doi: 10.3934/QFE.2022008
[52]	Y. Yao, D. Hu, C. Yang, Y. Tan, The impact and mechanism of fintech on green total factor productivity, Green Finance, 3 (2021), 198–221. http://doi.org/10.3934/gf.2021011 doi: 10.3934/gf.2021011
[53]	Z. Y. Li, M. Tuerxun, J. H. Cao, M. Fan, C. Y. Yang, Does inclusive finance improve income: a study in rural areas, AIMS Mathematics, 7 (2022), 20909–20929. http://doi.org/10.3934/math.20221146 doi: 10.3934/math.20221146
[54]	Z. Li, C. Yang, Z. Huang, How does the fintech sector react to signals from central bank digital currencies?, Financ. Res. Lett., 50 (2022), 103308. http://doi.org/10.1016/j.frl.2022.103308 doi: 10.1016/j.frl.2022.103308
[55]	S. El Ghoul, O. Guedhami, R. Nash, A. Patel, New evidence on the role of the media in corporate social responsibility, J. Bus. Ethics, 154 (2019), 1051–1079. http://doi.org/10.1007/s10551-016-3354-9 doi: 10.1007/s10551-016-3354-9
[56]	T. Vanacker, D. P. Forbes, M. Knockaert, S. Manigart, Signal strength, media attention, and resource mobilization: evidence from new private equity firms, Acad. Manage. J., 63 (2020), 1082–1105. http://doi.org/10.5465/amj.2018.0356 doi: 10.5465/amj.2018.0356

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