Extended suprametric spaces and Stone-type theorem

Sumati Kumari Panda; Ravi P Agarwal; Erdal Karapínar; Sumati Kumari Panda; Ravi P Agarwal; Erdal Karapínar

doi:10.3934/math.20231179

AIMS Mathematics

2023, Volume 8, Issue 10: 23183-23199. doi: 10.3934/math.20231179

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Research article Special Issues

Extended suprametric spaces and Stone-type theorem

1.
Department of Mathematics, GMR Institute of Technology, Rajam-532127, Andhra Pradesh, India
2.
Department of Mathematics, Texas A & M University-Kingsville 700 University Blvd., MSC 172 Kingsville, Texas 78363-8202
3.
Department of Medical Research, China Medical University Hospital, China Medical University, 40402, Taichung, Taiwan
4.
Department of Mathematics, Çankaya University, 06790, Etimesgut, Ankara, Turkey

Received: 28 March 2023 Revised: 01 July 2023 Accepted: 13 July 2023 Published: 20 July 2023
MSC : 47H10, 54E35, 54E99

Extended suprametric spaces are defined, and the contraction principle is established using elementary properties of the greatest lower bound instead of the usual iteration procedure. Thereafter, some topological results and the Stone-type theorem are derived in terms of suprametric spaces. Also, we have shown that every suprametric space is metrizable. Further, we prove the existence of a solution of Ito-Doob type stochastic integral equations using our main fixed point theorem in extended suprametric spaces.

Keywords:

an extended suprametric space,
metrization,
fixed point and Ito-Doob type stochastic integral equations

Citation: Sumati Kumari Panda, Ravi P Agarwal, Erdal Karapínar. Extended suprametric spaces and Stone-type theorem[J]. AIMS Mathematics, 2023, 8(10): 23183-23199. doi: 10.3934/math.20231179

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Abstract

1. Introduction and definitions

Assume that $\mathfrak{A}$ denotes the set of all analytic functions $\mathfrak{h}(z)$ in the open symmetric unit disk

$\mathcal{U} = \left\{z:|z| < 1\right\},$

which are normalized by

$\mathfrak{h}(0) = 0\; \; and\; \; \mathfrak{h'}(0) = 1.$

Thus, every function $\mathfrak{h} \in \mathfrak{A}$ can be expressed in the form given in (1.1)

$\begin{align} \mathfrak{h}(z) = z+\sum\limits_{k = 2}^{\infty}a_kz^k. \end{align}$

(1.1)

Let an analytic function $\mathfrak{h}$ is said to be univalent if it satisfy the following condition:

$\begin{align*} \mathfrak{h}(z_1) \ne \mathfrak{h}(z_2)\Rightarrow z_1\ne z_2,\; \forall\; z_1, z_2 \in \mathcal{U}. \end{align*}$

Furthermore, $\mathcal{S}$ is the subclass of $\mathfrak{A}$ whose members are univalent in $\mathcal{U}$ . The idea of subordination was initiated by Lindelof ^[30] and Little-wood and Rogosinski have further improved this idea, see ^[31,35,36]. For $\mathfrak{h}$ , $y \in \mathfrak{A}$ , and $\mathfrak{h}$ subordinate to $y$ in $\mathcal{U}$ , denoted by

$\mathfrak{h}(z)\prec y(z),\qquad z\in \mathcal{U},$

if we have a function $u$ , such that

$u \in \mathcal{B} = \left\{u : u \in \mathfrak{A}, |u(z)| < 1,\; and\; u(0) = 0,\; z \in \mathcal{U} \right\}$

and

$\begin{align*} \mathfrak{h}(z) = y(u(z)), \; \; z \in \mathcal{U}. \end{align*}$

According to the Koebe one-quarter theorem (see ^[13]), the image of $\mathcal{U}$ under $\mathfrak{h} \in \mathcal{S}$ contains a disk of radius one-quarter centered at origin. Thus, every function $\mathfrak{h} \in \mathcal{S}$ has an inverse $\mathfrak{h}^{-1} = g$ , defined as:

$g(\mathfrak{h}(z)) = z, \qquad z \in \mathcal{U}$

and

$\mathfrak{h}(g(w)) = w,\; \; |w| < r_0(\mathfrak{h}), \; \; r_0(\mathfrak{h}) \geq \frac{1}{4}.$

The power series for the inverse function $g(w)$ is given by

$\begin{align} g(w) = w-a_2w^2+(2a_2^2-a_3)w^3- \mathcal{Q}(a)w^4+,\cdots, \end{align}$

(1.2)

Where

$\mathcal{Q}(a) = (5a_2^3-5a_2a_3+a_4).$

An analytic function $\mathfrak{h}$ is called bi-univalent in $\mathcal{U}$ if $\mathfrak{h}$ and $\mathfrak{h}^{-1}$ are univalent in $\mathcal{U}$ and class of all bi-univalent functions are denoted by $\Sigma$ . In $1967$ , for $\mathfrak{h} \in \Sigma$ , Levin ^[32] showed that $|a_2| < 1.51$ and after twelve years Branan and Clunie ^[8] gave the improvement of $|a_2|$ and proved that $|a_2|\leq \sqrt{2}$ . Furthermore, for $\mathfrak{h} \in \Sigma$ , Netanyahu ^[34] proved that $\max |a_2| = \frac{4}{3}$ and an intriguing subclass of analytic and bi-univalent functions was proposed and studied by Branan and Taha ^[9], who also discovered estimates for the coefficients of the functions in this subclass. Recently, the investigation of numerous subclasses of the analytic and bi-univalent function class $\Sigma$ was basically revitalized by the pioneering work of Srivastava et al. ^[41]. In 2012, Xu et al. ^[44] defined a general subclass of class $\Sigma$ and investigated coefficient estimates for the functions belonging to the new subclass of class $\Sigma$ . Recently, several different subclasses of class $\Sigma$ were introduced and investigated by a number of authors (see for details (^[23,29,38]). In these recent papers only non-sharp estimates on the initial coefficients were obtained.

Faber polynomials was introduced by Faber ^[15] and first time he used it to determine the general coefficient bounds $|a_k|$ for $k \geq 4$ . Gong ^[16] interpreted significance of Faber polynomials in mathematical sciences, particularly in Geometric Function Theory. In 1913, Hamidi et al. ^[18] first time used the Faber polynomials expansion technique on meromorphic bi-starlike functions and determined the coefficient estimates. The Faber polynomials expansion method for analytic bi-close-to-convex functions was examined by Hamidi and Jahangiri ^[21,22], who also discovered some new coefficient bounds for new subclasses of close-to-convex functions. Furthermore, many authors ^{[3,4,7,11,12,14,20]} used the same technique and determined some interesting and useful properties for analytic bi-univalent functions. For $\mathfrak{h} \in \Sigma$ , by using the Faber polynomial expansions methods, only a few works have been done so far and we recognized very little over the bounds of Maclaurin's series coefficient $|a_k|$ for $k \geq 4$ in the literature. Recently only a few authors, used the Faber polynomials expansion technique and determined the general coefficient bounds $|a_k|$ for $k \geq 4$ , (see for detail ^{[6,11,24,39,40,42]}).

A domain $\mathcal{U}$ is said to be the $\upsilon$ -fold symmetric if

$\begin{align*} \mathfrak{h}_\upsilon\bigg(e^{k(\frac{2\pi}{\upsilon})}(z) \bigg) = e^{k(\frac{2\pi}{\upsilon})}\mathfrak{h}_\upsilon(z),\; \; \; z \in \mathcal{U}, \upsilon \in \mathbb{Z^+}, \mathfrak{h} \in \mathfrak{A} \end{align*}$

and every $\mathfrak{h}_\upsilon$ has the series of the form

$\begin{align} \mathfrak{h}_\upsilon = z+ \sum\limits_{k = 1}^{\infty}a_{\upsilon k+1}z^{\upsilon k+1}. \end{align}$

(1.3)

The class $\mathcal{S}^\upsilon$ represents the set of all $\upsilon$ -fold symmetric univalent functions. For $\upsilon = 1$ , then $\mathcal{S}^\upsilon$ reduce to the class $\mathcal{S}$ of univalent functions. If the inverse $g_\upsilon$ of univalent $\mathfrak{h}$ is univalent then $\mathfrak{h}$ is called $\upsilon$ -fold symmetric bi-univalent functions in $\mathcal{U}$ and denoted by $\Sigma_\upsilon$ . The series expansion of inverse function $g_\upsilon$ investigated by Srivastava et al. in ^[43]:

$\begin{align} g_\upsilon(w) & = w-a_{\upsilon+1}w^{\upsilon+1} +((\upsilon+1)a^2_{\upsilon+1}-a_{2\upsilon+1} )w^{2\upsilon+1}-\bigg\{\frac{1}{2}(\upsilon+1)(3\upsilon+2)a^3_{\upsilon+1}\\ &\qquad{}-\dfrac{1}{2}(\upsilon+1)(\upsilon+2)a_{\upsilon+1}^3-((3\upsilon+2)a_{\upsilon+1}a_{2\upsilon+1}+a_{3\upsilon+1})\bigg\}w^{3\upsilon+1}. \end{align}$

(1.4)

For $\upsilon = 1$ , the series in (1.4) reduces to the (1.2) of the class $\Sigma$ . In ^[43] Srivastava et al. defined a subclass of $\upsilon$ -fold symmetric bi-univalent functions and investigated coeffiients problem for $\upsilon$ -fold symmetric bi-univalent functions. Hamidi and Jahangiri ^[19] defined $\upsilon$ -fold symmetric bi-starlike functions and discussed the unpredictability of the coefficients of $\upsilon$ -fold symmetric bi-starlike functions.

Many researchers have used the $q$ -calculus and fractional $q$ -calculus in the field of Geometric Function Theory (GFT) and they defined and studied several new subclasses of analytic, univalent and bi-univalent functions. In 1909, Jackson (^[26,27]), gave the idea of $q$ -calculus operator and defined the q-difference operator ( $D_q$ ) while in ^[25], Ismail et al. was the first who used $D_q$ in order to define a class of q-starlike functions in open unit disk $\mathcal{U}$ . The most signifcant usages of $q$ -calculus in the perspective of GFT was basically furnished and the basic $(or\; q-)$ hypergeometric functions were first used in GFT in a book chapter by Srivastava (see, for details, ^[37]). For more study about $q$ -calculus operator theory in GFT, see the following articles ^[5,28,33].

Now we recall, some basic definitions and concepts of the $q$ -calculus which will be used to define some new subclasses of the this paper.

For a non-negative integer $t$ , the $q$ -number $[t, q],$ $(0 < q < 1),$ is defined by

$[t, q] = \frac{1-q^t}{1-q},\; and\; [0, q] = 0$

and the $q$ -number shift factorial is given by

$[t, q]! = [1, q][2, q][3, q]\cdots[t, q],$

$[0, q]! = 1.$

For $q \rightarrow1-$ , then $[t, q]!$ reduces to $t!$ .

The q-generalized Pochhammer symbol is defined by

$[t, q]_k = \frac{\Gamma_q(t+k)}{\Gamma_q(t)},\; k \in \mathbb{N},\; t \in \mathbb{C}.$

Remark 1.1. For $q \rightarrow1-$ , then $[t, q]_k$ reduces to $(t)_k = \frac{\Gamma(t+k)}{\Gamma(t)}.$

Definition 1.2. Jackson ^[27] defined the $q$ -integral of function $\mathfrak{h}(z)$ as follows:

$\int \mathfrak{h}(z)d_q(z) = \sum\limits_{k = 0}^{\infty}z(1-q)\mathfrak{h}(q^k(z))q^k.$

Jackson ^[26] introduced the $q$ -difference operator for analytic functions as follows:

Definition 1.3. ^[26]. For $\mathfrak{h} \in \mathcal{A}$ , the $q$ -difference operator is defined as:

$D_q\mathfrak{h}(z) = \frac{\mathfrak{h}(qz)-\mathfrak{h}(z)}{z(q-1)}, \qquad z \in \mathcal{U}.$

Note that, for $k \in \mathbb{N}$ and $z \in \mathcal{U}$ and

$D_q(z^k) = [k, q]z^{k-1},\; \; D_q\bigg(\sum\limits_{k = 1}^{\infty}a_kz^k\bigg) = \sum\limits_{k = 1}^{\infty}[k,q]a_kz^{k-1}.$

Here, we introduce the $q$ -difference operator for $\upsilon$ -fold symmetric functions related to the $q$ -calculus as follows:

Definition 1.4. Let $\mathfrak{h}_{\upsilon} \in \Sigma_\upsilon$ , of the form (1.3). Then $q$ -difference operator will be defined as

$\begin{align} D_q\mathfrak{h}_{\upsilon}(z) = \frac{\mathfrak{h}_\upsilon(qz)-\mathfrak{h}_\upsilon(z)}{(q-1)z}, \qquad z \in \mathcal{U}, \end{align}$

(1.5)

$= 1+ \sum\limits_{k = 1}^{\infty}[\upsilon k+1, q]a_{\upsilon k+1}z^{\upsilon k}$

and

$D_q\bigg(\sum\limits_{k = 1}^{\infty}a_{\upsilon k+1}z^{\upsilon k+1} \bigg) = \sum\limits_{k = 1}^{\infty}[\upsilon k+1, q]a_{\upsilon k+1}z^{\upsilon k},$

$D_q(z)^{\upsilon k+1} = [\upsilon k+1, q]z^{\upsilon k}.$

Now we define Salagean $q$ -differential operator for $\upsilon$ -fold symmetric functions as follows:

Definition 1.5. For $m \in \mathbb{N}$ , the Salagean $q$ -differential operator for $\mathfrak{h}_{\upsilon} \in \Sigma_\upsilon$ is defined by

$\nabla^0_q\mathfrak{h}_\upsilon(z) = \mathfrak{h}_\upsilon(z),\; \; \; \nabla^1_q\mathfrak{h}_\upsilon(z) = zD_q\mathfrak{h}_\upsilon(z) = \frac{\mathfrak{h}_\upsilon(qz)-\mathfrak{h}_\upsilon(z)}{(q-1)},\cdots,$

$\nabla^m_q\mathfrak{h}_\upsilon(z) = zD_q( \nabla^{m-1}_q \mathfrak{h}_\upsilon(z)) = \bigg(z+ \sum\limits_{k = 1}^{\infty}([\upsilon k+1, q])^m z^{\upsilon k+1} \bigg),$

$\begin{align} \nabla^m_q\mathfrak{h}_\upsilon(z) = z+ \sum\limits_{k = 1}^{\infty}([\upsilon k+1, q])^m a_{\upsilon k+1} z^{\upsilon k+1}. \end{align}$

(1.6)

Remark 1.6. For $\upsilon = 1$ , we have Salagean $q$ -differential operator for analytic functions proved in [17].

Motivated by the following articles ^[1,10,25] and using the $q$ -analysis in order to define new subclasses of class $\Sigma_\upsilon$ , we apply Faber polynomial expansions technique in order to determine the estimates for the general coefficient bounds $|a_{\upsilon k+1}|$ . We also derive initial coefficients $|a_{\upsilon +1}|$ and $|a_{2\upsilon +1}|$ and obtain Feketo-Sezego coefficient bounds for the functions belonging to the new subclasses of $\Sigma_\upsilon$ .

Definition 1.7. A function $\mathfrak{h}_\upsilon \in \Sigma_\upsilon$ is in the class $\mathcal{R}^{\upsilon, \gamma}_{b, q}(\varphi)$ if and only if

$1+\frac{1}{b} \biggl\{(D_q\mathfrak{h}_{\upsilon}(z)+\gamma zD^2_q \mathfrak{h}_\upsilon(z) )-1\biggr\} \prec \varphi(z)$

and

$1+\frac{1}{b} \biggl\{(D_qg_\upsilon(w)+\gamma w D^2_q g_\upsilon(w) )-1\biggr\} \prec \varphi(w),$

where, $\varphi \in \mathcal{P}$ , $\gamma \geq 0$ , $b \in \mathbb{C} \setminus \{0\}$ , $z$ , $w \in \mathcal{U}$ , and $g_\upsilon(w)$ is defined by (1.4).

Remark 1.8. For $q \rightarrow 1-$ , $\upsilon = 1$ , and $\gamma = 0$ , then $\mathcal{R}^{\upsilon, \gamma}_{b, q}(\varphi) = \mathcal{R}_b(\varphi)$ introduced in ^[22].

Definition 1.9. A function $\mathfrak{h}_\upsilon \in \Sigma_\upsilon$ , is in the class $\mathcal{R}^\upsilon_b(b, \alpha, \gamma)$ if and only if

$\bigg| \Bigg( 1+\frac{1}{b} \biggl\{(D_q\mathfrak{h}_{\upsilon}(z)+\gamma zD^2_q \mathfrak{h}_\upsilon(z))-1 \Biggr\} \Bigg) -\frac{1-\alpha q}{1-q} \bigg| < \frac{1-\alpha }{1-q}$

and

$\bigg| \bigg( 1+\frac{1}{b} \biggl\{(D_qg_\upsilon(w)+\gamma zD^2_q g_\upsilon(w) )-1\biggr\} \bigg) -\frac{1-\alpha q}{1-q} \bigg| < \frac{1-\alpha }{1-q}.$

Or equivalently by using subordination, we can write the above conditions as:

$1+\frac{1}{b} \biggl\{(D_q\mathfrak{h}_{\upsilon}(z)+\gamma zD^2_q \mathfrak{h}_\upsilon(z) )-1\biggr\} \prec \frac{1+[1-\alpha(1+q)]z}{1-qz}$

and

$1+\frac{1}{b} \biggl\{(D_qg_\upsilon(w)+\gamma w D^2_q g_\upsilon(w) )-1 \biggr\} \prec \frac{1+[1-\alpha(1+q)]w}{1-qw},$

where, $0\leq \alpha < 1$ , $\gamma \geq 0$ , $b \in \mathbb{C} \setminus \left\{0\right\}$ , $z$ , $w \in \mathcal{U}$ , $g_\upsilon(w)$ is defined by (1.4).

Remark 1.10. For $q \rightarrow1-$ , $\upsilon = 1$ , $\alpha = 0$ and $\gamma = 0$ , then $\mathcal{R}^\upsilon_b(b, \alpha, \gamma) = \mathcal{R}_b(\varphi)$ introduced in [22].

Definition 1.11. A function $\mathfrak{h}_\upsilon \in \Sigma_\upsilon$ , is in the class $\mathcal{R}^{\upsilon, \gamma, m}_{b, q}(\varphi)$ if and only if

$1+\frac{1}{b}\biggl\{\bigg( \frac{\nabla^m_q\mathfrak{h}_\upsilon(z)}{z} +\gamma zD_q\bigg( \frac{\nabla^m_q\mathfrak{h}_\upsilon(z)}{z}\bigg) \bigg) -1 \biggr\} \prec \varphi(z)$

and

$1+\frac{1}{b}\biggl\{\bigg( \frac{\nabla^m_qg_\upsilon(w)}{w} +\gamma wD_q\bigg( \frac{\nabla^m_qg_\upsilon(w)}{w}\bigg) \bigg) -1 \biggr\} \prec \varphi(w),$

where, $\varphi\in \mathcal{P}$ , $\gamma \geq 0, m \in \mathbb{N}$ , $b \in \mathbb{C} \setminus \left\{0\right\}$ , $z$ , $w \in \mathcal{U}$ , $g_\upsilon(w)$ is defined by (1.4).

2. The faber polynomial expansion method and application

Using the Faber polynomial technique for the analytic function $\mathfrak{h},$ then the coefficient of its inverse map $g$ can be written as follows (see ^[2,4]):

$\begin{eqnarray*} g_{\upsilon}w) = w+ \sum\limits_{k = 2}^{\infty} \dfrac{1}{k}\Re^{k}_{k-1}(a_{2},a_{3},...)w^{k}, \end{eqnarray*}$

where

$\begin{align*} \Re^{-k}_{k-1} = &\dfrac{(-k)!}{(-2k+1)!(k-1)!} a^{k-1}_{2}+\dfrac{(-k)!}{[2(-k+1)]!(k-3)!} a^{k-3}_{2}a_{3} \\ & +\dfrac{(-k)!}{(-2k+3)!(k-4)!} a^{k-4}_{2}a_{4} \\ & +\dfrac{(-k)!}{[2(-k+2)]!(k-5)!} a^{k-5}_{2}[a_{5}+(-k+2)a^{2}_{3}] \\ & +\dfrac{(-k)!}{(-2k+5)!(k-6)!} a^{k-6}_{2}[a_{6}+(-2k+5)a_{3}a_{4}] \\ & +\sum\limits_{i \geq 7}a^{k-i}_{2}\mathcal{Q}_{i}, \end{align*}$

and $\mathcal{Q}_{i}$ is a homogeneous polynomial in the variables $a_{2}, a_{3}, ...a_{k},$ for $7 \leq i \leq k.$ Particularly, the first three term of $\Re^{-k}_{k-1}$ are

$\begin{eqnarray*} \dfrac{1}{2}\Re^{-2}_{1} = -a_{2},\dfrac{1}{3}\Re^{-3}_{2} = 2a^{2}_{2}-a_{3},\\ \dfrac{1}{4}\Re^{-4}_{3} = -(5a^{3}_{2}-5a_{2}a_{3}+a_{4}). \end{eqnarray*}$

In general, for $r\in N \ \ and \ \ k\geq 2,$ an expansion of $\Re^{r}_{k}$ of the form:

$\begin{eqnarray*} \Re^{r}_{k} = ra_{k}+\dfrac{r(r-1)}{2}E^{2}_{k}+\dfrac{r!}{(r-3)!3!}E^{3}_{k}+...+\dfrac{r!}{(r-k)!k!}E^{k}_{k}, \end{eqnarray*}$

where,

$\begin{eqnarray*} E^{r}_{k} = E^{r}_{k}(a_{2},a_{3},...) \end{eqnarray*}$

and by ^[2], we have

$\begin{eqnarray*} E^{\upsilon}_{k}(a_{2},a_{3},...a_{k}) = \sum\limits_{k = 1}^{\infty} \dfrac{\upsilon!(a_{2})^{\mu_{1}}...(a_{k})^{\mu_{k}}}{\mu_{1!},...,\mu_{k!}}, \ \ for \ \ a_{1} = 1 \ \ and \ \ \upsilon \leq k. \end{eqnarray*}$

The sum is taken over all non negative integer $\mu_{1}, ..., \mu_{k}$ which is satisfying

$\begin{eqnarray*} \mu_{1}+\mu_{2}+...+\mu_{k} = \upsilon,\\ \mu_{1}+2\mu_{2}+...+(k)\mu_{k} = k. \end{eqnarray*}$

Clearly,

$\begin{eqnarray*} E^{k}_{k}(a_{1},...,a_{k}) = E^{k}_{1} \end{eqnarray*}$

and

$\begin{eqnarray*} E^{k}_{k} = a^{k}_{1} \ \ and \ \ E^{1}_{k} = a_{k} \end{eqnarray*}$

are first and last polynomials.

Now, using the Faber polynomial expansion for $\mathfrak{h}_{\upsilon}$ of the form (1.3) we have

$\begin{eqnarray*} \mathfrak{h}_{\upsilon}(z) = z+\sum\limits_{k = 1}^{\infty}a_{\upsilon k+1} z^{\upsilon k+1}. \end{eqnarray*}$

The coefficient of inverse map $g_{\upsilon}$ can be expressed of the form:

$\begin{eqnarray*} {g_{\upsilon}(z)} = w+\sum\limits_{k = 1}^{\infty}\dfrac{1}{(\upsilon k+1)}\Re^{-(\upsilon k+1)}_{k}(a_{\upsilon+1},a_{2\upsilon+1},...a_{\upsilon k+1})w^{\upsilon k+1}. \end{eqnarray*}$

Theorem 2.1. For $b \in \mathbb{C}\setminus \{0\}.$ Let $\mathfrak{h}_{\upsilon} \in \mathcal{R}^{\upsilon, \gamma}_{b, q}(\varphi)$ by given by (1.3). If $a_{\upsilon i+1} = 0, \ \ 1\leq i \leq k-1$ , then

$\begin{eqnarray*} |a_{\upsilon k+1}| \leq \dfrac{2|b|}{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]} ,\ \ for \ \ k \geq 2. \end{eqnarray*}$

Proof. For $\mathfrak{h}_{\upsilon} \in \mathcal{R}^{\upsilon, \gamma}_{b, q}(\varphi)$ we have

$\begin{eqnarray*} 1+\dfrac{1}{b}\left\{(D_{q}\mathfrak{h}_{\upsilon}(z)+\gamma zD^{2}_{q}\mathfrak{h}_{\upsilon}(z))-1 \right\} \end{eqnarray*}$

$\begin{eqnarray} = 1+ \sum\limits_{k = 1}^{\infty}\dfrac{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]}{b} a_{\upsilon k+1}z^{\upsilon k} \end{eqnarray}$

(2.1)

and

$\begin{eqnarray*} 1+\dfrac{1}{b}\left\{ (D_{q}g_{\upsilon}(w)+\gamma w D^{2}_{q}g_{\upsilon}(w))-1\right\} \end{eqnarray*}$

$\begin{eqnarray} = 1+ \sum\limits_{k = 1}^{\infty}\dfrac{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]}{b} A_{\upsilon k+1}w^{\upsilon k}, \end{eqnarray}$

(2.2)

where,

$\begin{eqnarray*} A_{\upsilon k+1} = \dfrac{1}{(\upsilon k+1)}\Re^{-(\upsilon k+1)}_{k}(a_{\upsilon+1},a_{2\upsilon+1},...a_{\upsilon k+1}), \ \ for \ \ k \geq 1. \end{eqnarray*}$

Since $\mathfrak{h}_{\upsilon} \in \mathcal{R}^{\upsilon, \gamma}_{b, q}(\varphi)$ and $g_{\upsilon} \in \mathcal{R}^{\upsilon, \gamma}_{b, q}(\varphi)$ by definition, we have

$\begin{eqnarray} p(z) = \sum\limits_{k = 1}^{\infty} c_{k}z^{\upsilon k} \end{eqnarray}$

(2.3)

and

$\begin{eqnarray} r(w) = \sum\limits_{k = 1}^{\infty} d_{k}w^{\upsilon k} \end{eqnarray}$

(2.4)

where

$\begin{eqnarray} \varphi (p(z)) = 1+\sum\limits_{k = 1}^{\infty}\sum\limits_{l = 1}^{\infty} \varphi_{l} \Re^{l}_{k}(c_{1},c_{2},...,c_{k})z^{\upsilon k}, \end{eqnarray}$

(2.5)

$\begin{eqnarray} \varphi(r(w)) = 1+\sum\limits_{k = 1}^{\infty}\sum\limits_{l = 1}^{\infty} \varphi_{l} \Re^{l}_{k}(d_{1},d_{2},...,d_{k})w^{\upsilon k}. \end{eqnarray}$

(2.6)

Equating the coefficient of (2.1) and (2.5) we obtain

$\begin{eqnarray} \Big(\dfrac{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]}{b}\Bigg) a_{\upsilon k+1} = \sum\limits_{l = 1}^{k-1} \varphi_{l} \Re^{l}_{k}(c_{1},c_{2},...,c_{k}). \end{eqnarray}$

(2.7)

Similarly, corresponding coefficient of (2.2) and (2.6), we have

$\begin{eqnarray} \Big(\dfrac{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]}{b}\Big) A_{\upsilon k+1} = \sum\limits_{l = 1}^{k-1} \varphi_{l} \Re^{l}_{k}(d_{1},d_{2}...,d_{k}). \end{eqnarray}$

(2.8)

Since, $1\leq i \leq k-1, \ \ and \ \ a_{\upsilon i+1} = 0$ ; we have

$\begin{eqnarray*} A_{\upsilon k+1} = -a_{\upsilon k+1} \end{eqnarray*}$

and

$\begin{eqnarray} \dfrac{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]}{b} a_{\upsilon k+1} = \varphi_{1}c_{k}, \end{eqnarray}$

(2.9)

$\begin{eqnarray} \dfrac{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]}{b} A_{\upsilon k+1} = \varphi_{1}d_{k}. \end{eqnarray}$

(2.10)

Taking the modulus on both sides of (2.9) and (2.10), we have

$\begin{eqnarray*} \Big|\dfrac{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]}{b} a_{\upsilon k+1}\Big| = |\varphi_{1}c_{k}|, \end{eqnarray*}$

$\begin{eqnarray*} \Big|\dfrac{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]}{b} A_{\upsilon k+1}\Big| = |\varphi_{1}d_{k}|. \end{eqnarray*}$

Now using the fact $|\varphi_{1}|\leq 2, |c_{k}|\leq 1,$ and $|d_{k}|\leq 1,$ we have

$\begin{eqnarray*} | a_{\upsilon k+1} | \leq \dfrac{|b|}{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]}|\varphi_{1}c_{k}| \end{eqnarray*}$

$\begin{eqnarray*} = \dfrac{|b|}{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]} )|\varphi_{1}d_{k}|, \end{eqnarray*}$

$\begin{eqnarray*} |a_{\upsilon k+1} | \leq \dfrac{2|b|}{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]}. \end{eqnarray*}$

Hence, Theorem 2.1 is completed.

For $\upsilon = 0, \gamma = 0, q \rightarrow 1-, k = n-1,$ in Theorem 2.1, we obtain known corollary proved in ^[22].

Corollary 2.2. For $b\in \mathbb{C}\setminus\{0\},$ Let $\mathfrak{h}_{\upsilon} \in \mathcal{R}_{b}(\varphi)$ , If $a_{\upsilon i+1} = 0, 1\leq i \leq n.$ Then

$\begin{align*} |a_{n}| \leq \dfrac{2|b|}{n}, \ \ for \ \ n \geq 3. \end{align*}$

Theorem 2.3. For $b \in \mathbb{C}\setminus\{0\}.$ Let $\mathfrak{h}_{\upsilon} \in \mathcal{R}^{\upsilon, \gamma}_{b, q}(\varphi)$ be given by (1.3). Then

$\begin{eqnarray*} | a_{\upsilon+1}| \leq \left\{ \begin{array}{rcl} \frac{2|b|}{(1+\gamma[\upsilon k,q])[\upsilon +1,q]} ,\ \ if \ \ |b| < \psi_{1}(\upsilon,q), \\ \\ \sqrt{|b|\; \psi_{1}(\upsilon,q)}, \ \ if \ \ |b| \geq \psi_{1}(v,q), \end{array}\right. \end{eqnarray*}$

$\begin{align*} |a_{2\upsilon + 1}| \leq \left\{ \begin{array}{rcl} &|b|\psi_2(\upsilon,q)+\frac{2|b|^2}{(1+\gamma[\upsilon,q])[\upsilon+1,q]} , \ \ if \ \ |b| < \psi_2(\upsilon,q), \\ \\ & 2|b|\psi_2(\upsilon,q), \ \ if \ \ |b|\geq\psi_2(\upsilon,q),\end{array}\right. \end{align*}$

$|a_{2\upsilon+1}-(1+\gamma[\upsilon,q])[\upsilon+1,q]a^2_{\upsilon+1}| \leq 2|b|\psi_2(\upsilon,q),$

$\Big| a_{2\upsilon+1}-\frac{1}{\psi_2(\upsilon,q)} a^2_{\upsilon+1} \Big| \leq |b|\psi_2(\upsilon,q),$

where,

$\psi_1(\upsilon,q) = \frac{8}{((1+\gamma[2\upsilon,q])[2\upsilon+1,q])((1+\gamma[\upsilon,q])[\upsilon+1,q])},$

$\psi_2(\upsilon,q) = \frac{2}{((1+\gamma[2\upsilon,q])[2\upsilon+1,q]}.$

Proof. Taking $k = 1$ and $k = 2$ in (2.7) and (2.8), then, we have

$\begin{align} \frac{(1+\gamma[\upsilon,q])[\upsilon+1,q]}{b} a_{\upsilon+1} & = \varphi_1 c_1, \end{align}$

(2.11)

$\begin{align} \frac{(1+\gamma[2\upsilon,q])[2\upsilon+1,q]}{b} a_{2\upsilon+1} & = \varphi_1 c_2+ \varphi_2 c^2_1, \end{align}$

(2.12)

$\begin{align} -\frac{(1+\gamma[\upsilon,q])[\upsilon+1,q]}{b} a_{\upsilon+1} & = \varphi_1 d_1, \end{align}$

(2.13)

$\begin{align} \{(1+\gamma[\upsilon,q])[\upsilon+1,q]a^2_{\upsilon+1}-a_{2\upsilon+1}\} & = \frac{b(\varphi_1d_2+\varphi_2d^2_1)}{(1+\gamma[2\upsilon,q])[2\upsilon+1,q]}. \end{align}$

(2.14)

From (2.11) and (2.13) and using the fact $|\varphi_{1}|\leq 2, |c_{k}|\leq 1$ and $|d_{k}|\leq 1,$ we have

$\begin{align} |a_{\upsilon+1}| & \leq \frac{|b|}{(1+\gamma[\upsilon,q])[\upsilon+1,q]}|\varphi_{1}c_{1}| = \frac{|b|}{(1+\gamma[\upsilon,q])[\upsilon+1,q]}|\varphi_{1}d_{1}| \\ & \leq \frac{2|b|}{1+\gamma[\upsilon,q])[\upsilon+1,q]}. \end{align}$

(2.15)

Adding (2.12) and (2.14) we have

$\begin{equation} a^{2}_{\upsilon+1} = \frac{b\{\varphi_{1}(c_{2}+d_{2})+\varphi_{2}(c^{2}_{1}+d^{2}_{1})\}}{((1+\gamma[2\upsilon,q])[2\upsilon+1,q])((1+\gamma[\upsilon,q])[\upsilon+1,q])}. \end{equation}$

(2.16)

Taking absolute value of (2.16), we have

$\begin{eqnarray*} |a_{\upsilon+1}|\leq\sqrt{\frac{8|b|}{((1+\gamma[2\upsilon,q])[2\upsilon+1,q])((1+\gamma[\upsilon,q])[\upsilon+1,q])}}. \end{eqnarray*}$

Now the bounds given for $|a_{\upsilon+1}|$ can be justified since

$\begin{eqnarray*} |b| < \sqrt{\frac{8}{((1+\gamma[2\upsilon,q])[2\upsilon+1,q])((1+\gamma[\upsilon,q])[\upsilon+1,q])}} \end{eqnarray*}$

for

$|b| < \dfrac{8}{((1+\gamma[2\upsilon,q])[2\upsilon+q])((1+\gamma[\upsilon,q])[\upsilon+1,q])}.$

From (2.12), we get

$\begin{equation} |a_{2\upsilon+1}|\; = \; \frac{|b||\varphi_1 c_2 +\varphi_2 c^2_1|}{(1+\gamma[2\upsilon,q])[2\upsilon+1,q]}\; \; \leq\; \; \frac{4\mid b \mid}{(1+ \gamma[2\upsilon,q])[2\upsilon+1,q]}. \end{equation}$

(2.17)

Subtract (2.14) from (2.12), we have

$\begin{align} &\frac{2 (1+\gamma [2\upsilon,q])[2\upsilon+1,q]}{b} \left\{ a_{2\upsilon+1} -\; \; \frac{(1+\gamma[\upsilon,q])[\upsilon+1,q]}{2}\; a^2_{\upsilon+1} \right\} \\ & = \varphi_1(c_2-d_2) + \varphi_2(c_1^2 - d_1^2) = \varphi_1(c_2-d_2), \end{align}$

(2.18)

$\begin{equation} a_{2\upsilon+1} = \frac{(1+\gamma[\upsilon,q])[\upsilon+1,q]}{2}a_{\upsilon+1}^2 + \frac{\varphi_1 b(c_2 -d_2)}{2(1+\gamma [2v,q])[2v+1,q]}. \end{equation}$

(2.19)

Taking the absolute, we have

$\begin{equation} \mid a_{2v+1} | \leq \frac{ \mid \varphi_1 \mid \mid b \mid \mid c_2-d_2\mid}{2(1+ \gamma[2\upsilon,q])[2\upsilon+1,q]}\; \; + \frac{(1+\gamma [\upsilon,q])[\upsilon +1,q]}{2} \mid a^2_{\upsilon +1} | . \end{equation}$

(2.20)

Using the assertion (2.15) on (2.20), we have

$\begin{equation} \mid a_{2\upsilon +1} \mid \leq \frac{2 \mid b \mid }{(1+\gamma[2\upsilon ,q])[2\upsilon +1,q]} + \frac{2\mid b \mid^2 }{(1+\gamma[\upsilon ,q])[\upsilon +1,q]}. \end{equation}$

(2.21)

Follows from (2.17) and (2.21) upon nothing that

$\begin{align*} &\frac{2\mid b \mid }{(1+\gamma [2\upsilon ,q])[2\upsilon +1,q]} + \frac{2 \mid b\mid^2}{(1+\gamma[\upsilon ,q])[\upsilon +1,q]} \\ &\leq \frac{2 \mid b \mid }{(1+ \gamma [2\upsilon ,q])[2\upsilon +1,q]} \ \ if \ \ \mid b \mid < \frac{2}{(1+ \gamma [2\upsilon ,q])[2\upsilon +1,q]}. \end{align*}$

Now, rewrite (2.14) as follows:

$\begin{equation*} (1+ \gamma [\upsilon ,q])[\upsilon +1,q] a^2_{\upsilon +1} - a_{2\upsilon +1} = \frac{b(\varphi_1d_2+\varphi_2 d_1^2)}{(1+ \gamma[2\upsilon ,q])[2\upsilon +1,q]}. \end{equation*}$

Using the fact $\mid \varphi_1 \mid \leq 2$ , $\mid c_k\mid \leq 1$ and $\mid d_k \mid \leq 1$ , we have

$\begin{equation*} \mid a_{2\upsilon +1}-(1+\gamma[\upsilon ,q])[\upsilon +1,q]a^2_{\upsilon +1} \mid \leq \frac{4 \mid b \mid }{(1+ \gamma[2\upsilon ,q])[2\upsilon +1,q]}. \end{equation*}$

From (2.18), we have

$\begin{equation*} \frac{2(1+ \gamma[2\upsilon ,q])[2\upsilon +1,q]}{b} \Big\{a_{2\upsilon +1} - \frac{(1+\gamma[2\upsilon ,q])[2\upsilon +1,q]}{2} a^2_{\upsilon +1}\Big\} = \varphi_1(c_2-d_2). \end{equation*}$

Again using the fact $\mid \varphi_1 \mid \leq 2$ , $\mid c_k \mid \leq 1$ and $\mid d_k \mid \leq 1$ , we have

$\begin{equation*} \Bigg| a_{2\upsilon +1} -\frac{(1+ \gamma[2\upsilon ,q])[2\upsilon +1,q]}{2}a^2_{\upsilon +1} \Bigg|\leq \frac{2 \mid b \mid}{(1+ \gamma [2\upsilon ,q])[2\upsilon +1,q]}. \end{equation*}$

Take $q\rightarrow 1-, \gamma = 0, \upsilon = 1,$ and $k = n-1$ in the Theorem 2.3, we get known corollary.

Corollary 2.4. ^[22]. For $b \in \mathbb{C} \setminus \{0\},$ let $\mathfrak{h} \in \mathcal{R}_b (\varphi)$ be given by (1.1), then

$\begin{eqnarray*} |a_{2}| \leq \left \{ \begin{array} {rcl} &|b|, \ \ if \ \ |b| < \frac{4}{3}, \\ \\ &\sqrt{\frac{4|b|}{3}}, \ \ if \ \ |b| \geq \frac{4}{3}, \end{array}\right. \end{eqnarray*}$

$\begin{eqnarray*} |a_{3}| \leq \left \{ \begin{array} {rcl} &\frac{2|b|}{3} +|b|^{2}, \ \ if \ \ |b| < \frac{2}{3},\\ \\ &\frac{4|b|}{3}, \ \ if \ \ |b| \geq \frac{2}{3}, \end{array}\right. \end{eqnarray*}$

$\begin{eqnarray*} |a_{3}-2a^2_{2}|\leq \dfrac{4|b|}{3}, \end{eqnarray*}$

$\begin{eqnarray*} |a_{3}-a^2_{2}|\leq \dfrac{2|b|}{3}. \end{eqnarray*}$

Theorem 2.5. For $b \in \mathbb{C} \setminus\{0\}.$ Let $\mathfrak{h}_\upsilon \in \mathcal{R}^{\upsilon }_{q}(b, \alpha, \gamma)$ by given by (1.3). If $a_{\upsilon i+1} = 0, 1\leq i \leq k-1.$ Then

$\begin{eqnarray*} | a_{\upsilon k+1}| \leq \dfrac{(\mathfrak{B}_0 - \mathfrak{B}_1)|b|}{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]}, & for\; \; k \geq 2. \end{eqnarray*}$

where, $\mathfrak{B}_0 = 1 - \alpha(1+q)$ and $\mathfrak{B}_1 = -q$ .

Proof. Let $\mathfrak{h}_\upsilon \in \mathcal{R}^{\upsilon }_{q}(b, \alpha, \gamma)$ . Then

$\begin{align} &1 + \frac{1}{b}\{ (D_q\mathfrak{h}_\upsilon (z) + \gamma z D^2_{q} \mathfrak{h}_\upsilon (z)) - 1 \} \\ = & 1 + \sum\limits_{k = 1}^{\infty} \dfrac{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]}{b} a_{\upsilon k+1} z^{\upsilon k} \end{align}$

(2.22)

and

$\begin{align} &1 + \frac{1}{b}\{(D_q g_\upsilon (w) + \gamma w D^2_{q} g_\upsilon (w)) - 1 \} \\ = & 1 + \sum\limits_{k = 1}^{\infty} \dfrac{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]}{b} A_{\upsilon k+1} w^{\upsilon k}. \end{align}$

(2.23)

where,

$\begin{align*} A_{\upsilon k+1} = \dfrac{1}{(\upsilon k+1)} \Re^{-(\upsilon k+1)} (a_{\upsilon +1}, a_{2\upsilon +1},...,a_{\upsilon k+1}), & \ \ k \geq 1. \end{align*}$

Since $h_\upsilon \in \mathcal{R}^{\upsilon }_{q}(b, \alpha, \gamma)$ and $g_\upsilon \in \mathcal{R}^{\upsilon }_{q}(b, \alpha, \gamma)$ by definition, there exist two positive real functions $p(z)$ and $r(w)$ given in (2.3) and (2.4), then we have

$\begin{align} & = \dfrac{1+\mathfrak{B}_0(p(z))}{1+\mathfrak{B}_1(p(z))} = 1 - \sum\limits_{k = 1}^{\infty} \sum\limits_{l = 1}^{k} (\mathfrak{B}_0 - \mathfrak{B}_1)\Re^{-1}_k (c_1,c_2,...,c_k,\mathfrak{B}_1) z^{\upsilon k} \end{align}$

(2.24)

$\begin{align} & = \dfrac{1+\mathfrak{B}_0(r(w))}{1+\mathfrak{B}_1(r(w))} = 1 - \sum\limits_{k = 1}^{\infty} \sum\limits_{l = 1}^{k} (\mathfrak{B}_0 - \mathfrak{B}_1)\Re^{-1}_k (d_1,d_2,...,d_k,\mathfrak{B}_1) w^{\upsilon k}. \end{align}$

(2.25)

Equating the corresponding coefficients of (2.22) and (2.24), we have

$\begin{align} \dfrac{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]}{b} a_{\upsilon k+1} = (\mathfrak{B}_0 - \mathfrak{B}_1)\Re^{-1}_k (c_1,c_2,...,c_k,\mathfrak{B}_1) z^{\upsilon k}. \end{align}$

(2.26)

Similarly, corresponding coefficient of (2.23)and (2.25), we have

$\begin{align} \dfrac{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]}{b}A_{\upsilon k+1} = (\mathfrak{B}_0 - \mathfrak{B}_1)\Re^{-1}_k(d_1,d_2,...,d_k,\mathfrak{B}_1)w^{\upsilon k}. \end{align}$

(2.27)

For $a_{\upsilon i+1} = 0;\; 1 \leq i \leq k-1,$ we get

$\begin{align*} A_{\upsilon k+1} = -a_{\upsilon k+1} \end{align*}$

and we have

$\begin{align} \dfrac{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]}{b}a_{\upsilon k+1} = (\mathfrak{B}_0 - \mathfrak{B}_1)c_k, \end{align}$

(2.28)

and

$\begin{align} -\dfrac{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]}{b}A_{\upsilon k+1} = (\mathfrak{B}_0 - \mathfrak{B}_1)d_k. \end{align}$

(2.29)

Taking modulus on (2.28) and (2.29), we have

$\begin{align*} &\Bigg|\dfrac{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]}{b}a_{\upsilon k+1} \Bigg| = | (\mathfrak{B}_0 - \mathfrak{B}_1)c_k|,\\ &\Bigg| -\dfrac{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]}{b}A_{\upsilon k+1} \Bigg| = | (\mathfrak{B}_0 - \mathfrak{B}_1)d_k |. \end{align*}$

Since

$\begin{align*} |c_k|\leqq 1 \; \; \; {\rm{and}} \; \; \; |d_k| \leqq 1 (see[14]), \end{align*}$

we have

$\begin{align*} |a_{\upsilon k+1}| & \leq \dfrac{|b|}{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]} | (\mathfrak{B}_0 - \mathfrak{B}_1)c_k|\\ & = \dfrac{|b|}{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]} | (\mathfrak{B}_0 - \mathfrak{B}_1)d_k,|\\ |a_{\upsilon k+1}| & \leq \dfrac{(\mathfrak{B}_0 - \mathfrak{B}_1)|b|}{(1+\gamma[\upsilon k,q])[\upsilon k+1,q]}, \end{align*}$

which complete the proof of Theorem.

For $b = 1, k = 1, \upsilon = n-1, q \rightarrow 1-, and \gamma \geqq 0$ in the above Theorem 2.5, we obtain the following result given in ^[40].

Corollary 2.6. Let $\mathfrak{h}_\upsilon \in \mathcal{R}(n, \alpha, \gamma)$ be given by (1.3). If $a_{n-1} = 0$ , and $1 \leq i \leq k-1$ , then

$\begin{align*} | a_n | \leqq \dfrac{2(1-\alpha)}{n(1+\gamma(n-1))}, &\; \; \; n \in \mathbb{N} \setminus \{1,2\}. \end{align*}$

Theorem 2.7. For $b \in \mathbb{C} \setminus \{0\},$ let $\mathfrak{h}_\upsilon \in \mathcal{R}_q^\upsilon (b, \alpha, \gamma)$ be given by (1.3), then

$\begin{align*} |a_{\upsilon +1} | \leq \left\{ \begin{array}{lcl} \dfrac{(\mathfrak{B}_0 - \mathfrak{B}_1)|b|}{(1+\gamma[\upsilon ,q])[\upsilon +1,q]}, & if\; \; |b| < \psi_3(\upsilon ,q), \\ \; \; \; \\ \sqrt{2|b|\psi_3(\upsilon ,q)} & if \; \; |b| \geq \psi_3(\upsilon ,q), \end{array}\right. \end{align*}$

$\begin{align*} |a_{2\upsilon +1} | \leq \left\{ \begin{array}{lcl} |b| \psi_4(\upsilon ,q) + \psi_4 (\upsilon ,q)|(\mathfrak{B}_0 - \mathfrak{B}_1)|\; |b|^2, & if\; \; |b| < \psi_4(\upsilon ,q), \\ \; \; \; \\ |b|(|\mathfrak{B}_1|+1)\psi_4(\upsilon ,q) & if \; \; |b| \geq \psi_4(\upsilon ,q), \end{array}\right. \end{align*}$

$\begin{align*} |a_{2\upsilon +1} - (1+\gamma[\upsilon ,q])[\upsilon +1,q] a^2_{\upsilon +1}| \leq |b|(|\mathfrak{B}_1|+1|)\psi_4(\upsilon ,q) \end{align*}$

and

$\begin{align*} \Bigg| a_{2\upsilon +1} - \dfrac{(1+\gamma[2\upsilon ,q])[2\upsilon +1,q]}{2} a^2_{\upsilon +1}\Bigg| \leq |b|\psi_4(\upsilon ,q), \end{align*}$

where

$\begin{align*} &\psi_3(\upsilon ,q) = \dfrac{| \mathfrak{B}_0 - \mathfrak{B}_1 |\{|\mathfrak{B}_1|+1\}}{((1+\gamma[2\upsilon ,q])[2\upsilon +1,q])( (1+\gamma[\upsilon ,q])[\upsilon +1,q])}\\ &\psi_4(\upsilon ,q) = \dfrac{| \mathfrak{B}_0 - \mathfrak{B}_1 |}{(1+\gamma[2\upsilon ,q])[2\upsilon +1,q]}. \end{align*}$

Proof. Take $k = 1$ and $k = 2$ in (2.26) and (2.27). Then we have

$\begin{align} &\dfrac{(1+\gamma[\upsilon ,q])[\upsilon +1,q]}{b} a_{\upsilon +1} = (\mathfrak{B}_0 - \mathfrak{B}_1)c_1, \end{align}$

(2.30)

$\begin{align} &\dfrac{(1+\gamma[2\upsilon ,q])[2\upsilon +1,q]}{b} a_{2\upsilon +1} = (\mathfrak{B}_0 - \mathfrak{B}_1)(-\mathfrak{B}_1 c^2_1 + c_2), \end{align}$

(2.31)

$\begin{align} &-\dfrac{(1+\gamma[\upsilon ,q])[\upsilon +1,q]}{b} a_{\upsilon +1} = -(\mathfrak{B}_0 - \mathfrak{B}_1)d_1, \end{align}$

(2.32)

$\begin{align} &(1+\gamma[\upsilon ,q])[\upsilon +1,q] a^2_{\upsilon +1} - a_{2\upsilon +1} = \dfrac{b(\mathfrak{B}_0 - \mathfrak{B}_1)(-\mathfrak{B}_1 d^2_1 + d_2)}{(1+\gamma[2\upsilon ,q])[2\upsilon +1,q]}. \end{align}$

(2.33)

From (2.30) and (2.32) and using the fact $|\varphi_1 | \leq 2, | c_k|\leq 1$ and $| d_k|\leq 1,$ we have

$\begin{align} |a_{\upsilon+1}| & \leq \dfrac{|b|}{(1+\gamma[\upsilon ,q])[\upsilon +1,q]} |(\mathfrak{B}_0 - \mathfrak{B}_1)c_1 | \\ & = \dfrac{|b|}{(1+\gamma[\upsilon ,q])[\upsilon +1,q]} |(\mathfrak{B}_0 - \mathfrak{B}_1)d_1 | \\ & \leq \dfrac{ (\mathfrak{B}_0 - \mathfrak{B}_1) |b|}{(1+\gamma[\upsilon ,q])[\upsilon +1,q]}. \end{align}$

(2.34)

Adding (2.31) and (2.33) we have

$\begin{align*} a^2_{\upsilon +1} = \dfrac{b(\mathfrak{B}_0 - \mathfrak{B}_1)\{(c_2 + d_2) + \mathfrak{B}_1(c^2_1 + d^2_1)\}}{((1+\gamma[2\upsilon ,q])[2\upsilon +1,q])((1+\gamma[\upsilon ,q])[\upsilon +1,q])} \end{align*}$

and

$\begin{align} |a_{\upsilon +1} |^2 \leq \dfrac{2|b|\; | \mathfrak{B}_0 - \mathfrak{B}_1|\{|\mathfrak{B}_1 | + 1 \}}{((1+\gamma[2\upsilon ,q])[2\upsilon +1,q])((1+\gamma[\upsilon ,q])[\upsilon +1,q])}. \end{align}$

(2.35)

Taking the square-root of (2.35), we have

$\begin{align*} |a_{\upsilon +1} | \leq \sqrt{ \dfrac{2|b||\mathfrak{B}_0 - \mathfrak{B}_1|\{|\mathfrak{B}_1| + 1 \}}{((1+\gamma[2\upsilon ,q])[2\upsilon +1,q])((1+\gamma[\upsilon ,q])[\upsilon +1,q])}}. \end{align*}$

Now the bounds given for $|a_{\upsilon +1} |$ can be justified since

$\begin{align*} &|b| < \sqrt{ \dfrac{2|b||\mathfrak{B}_0 - \mathfrak{B}_1|\{|\mathfrak{B}_1| + 1 \}}{((1+\gamma[2\upsilon ,q])[2\upsilon +1,q])((1+\gamma[\upsilon ,q])[\upsilon +1,q])}} \\ {\rm{for}} \; \; \; & |b| < \dfrac{2|b||\mathfrak{B}_0 - \mathfrak{B}_1|\{|\mathfrak{B}_1| + 1 \}}{((1+\gamma[2\upsilon ,q])[2\upsilon +1,q])((1+\gamma[\upsilon ,q])[\upsilon +1,q])}. \end{align*}$

From (2.31), we have

$\begin{align} | a_{2\upsilon +1}| & = \dfrac{|b||(\mathfrak{B}_0 - \mathfrak{B}_1)(\mathfrak{B}_1 c^2_1 + c_2)|}{(1+\gamma[2\upsilon ,q])[2\upsilon +1,q]} \\ & \leq \dfrac{|b||\mathfrak{B}_0 - \mathfrak{B}_1|(|\mathfrak{B}_1| + 1)}{(1+\gamma[2\upsilon ,q])[2\upsilon +1,q]}. \end{align}$

(2.36)

Next we subtract (2.33) from (2.31), we get

$\begin{align} &\dfrac{2(1+\gamma[2\upsilon ,q])[2\upsilon +1,q]}{b}\Bigl\{a_{2\upsilon +1} - \dfrac{(1+\gamma[\upsilon ,q])[\upsilon +1,q]}{2} a^2_{\upsilon +1}\Bigr\} \\ & = (\mathfrak{B}_0 - \mathfrak{B}_1)\{\mathfrak{B}_1(d^2_1 - c^2_1) -(c_2 -d_2)\} = (\mathfrak{B}_0 - \mathfrak{B}_1)(c_2 - d_2), \end{align}$

(2.37)

$\begin{align} &a_{2\upsilon +1} = \dfrac{(1+\gamma[\upsilon ,q])[\upsilon +1,q]}{2} a^2_{\upsilon +1} + \dfrac{(\mathfrak{B}_0 - \mathfrak{B}_1)b(c_2 -d_2)}{2(1+\gamma[2\upsilon ,q])[2\upsilon +1,q]}. \end{align}$

(2.38)

Taking the absolute values yield

$\begin{align} &| a_{2\upsilon +1} | \leq \dfrac{|(\mathfrak{B}_0 - \mathfrak{B}_1)|\; |b|\; |c_2 -d_2|}{2(1+\gamma[2\upsilon ,q])[2\upsilon +1,q]} + \dfrac{(1+\gamma[\upsilon ,q])[\upsilon +1,q]}{2}|a^2_{\upsilon +1}|. \end{align}$

(2.39)

Using the assertion (2.34) on (2.39), we have

$\begin{align} &| a_{2\upsilon +1} | \leq \dfrac{|(\mathfrak{B}_0 - \mathfrak{B}_1)|\; |b|}{(1+\gamma[2\upsilon ,q])[2\upsilon +1,q]} + \dfrac{|(\mathfrak{B}_0 - \mathfrak{B}_1)|^2\; |b|^2}{2(1+\gamma[\upsilon ,q])[\upsilon +1,q]}. \end{align}$

(2.40)

It follows from (2.36) and (2.40) upon noting that

$\begin{align*} & \dfrac{|(\mathfrak{B}_0 - \mathfrak{B}_1)|\; |b|}{(1+\gamma[2\upsilon ,q])[2\upsilon +1,q]} + \dfrac{|(\mathfrak{B}_0 - \mathfrak{B}_1)|^2\; |b|^2}{2(1+\gamma[\upsilon ,q])[\upsilon +1,q]}. \nonumber \\ & \leq \dfrac{|(\mathfrak{B}_0 - \mathfrak{B}_1)|\; |b|}{(1+\gamma[2\upsilon ,q])[2\upsilon +1,q]} \; \; {\rm{if}} \; \; |b| < \dfrac{|(\mathfrak{B}_0 - \mathfrak{B}_1)|}{(1+\gamma[2\upsilon ,q])[2\upsilon +1,q]}. \end{align*}$

Now, we rewrite (2.33) as follows:

$\begin{align*} \{ (1+\gamma[\upsilon ,q])[\upsilon +1,q] a^2_{\upsilon +1} - a_{2\upsilon +1}\} = \dfrac{b(\mathfrak{B}_0 - \mathfrak{B}_1)(-\mathfrak{B}_1 d^2_1 + d_2)}{(1+\gamma[2\upsilon ,q])[2\upsilon +1,q]}. \end{align*}$

Taking the modulus and using $|\varphi_1| \leq 2, \ |c_k| \leq 1$ and $|d_k| \leq 1,$ we have

$\begin{align*} | a_{2\upsilon +1} - (1+\gamma[\upsilon ,q])[\upsilon +1,q] a^2_{\upsilon +1} | \leq \dfrac{(\mathfrak{B}_0 - \mathfrak{B}_1)(|\mathfrak{B}_1| + 1 )\; |b|}{(1+\gamma[2\upsilon ,q])[2\upsilon +1,q]}. \end{align*}$

Finally, from (2.37), we have

$\begin{align*} \Big\{ a_{2\upsilon +1} - \dfrac{(1+\gamma[2\upsilon ,q])[2\upsilon +1,q]}{2} a^2_{\upsilon +1} \Big\} = \dfrac{b(\mathfrak{B}_0 - \mathfrak{B}_1)(c_2 - d_2)}{2(1+\gamma[2\upsilon ,q])[2\upsilon +1,q]}. \end{align*}$

Taking the modulus and using $|c_k| \leq 1$ and $|d_k| \leq 1,$ we have

$\begin{align*} \Bigg| a_{2\upsilon +1} - \dfrac{(1+\gamma[2\upsilon ,q])[2\upsilon +1,q]}{2} a^2_{\upsilon +1} \Bigg| \leq \dfrac{(\mathfrak{B}_0 - \mathfrak{B}_1)|b|}{(1+\gamma[2\upsilon ,q])[2\upsilon +1,q]}. \end{align*}$

For $\upsilon = 1, \gamma = 0, q \rightarrow 1-, k = n-1$ in Theorem 2.7, then we obtain result proved in ^[22].

Corollary 2.8. ^[22]. For $b \in \mathbb{C} \setminus \{0\},$ let $\mathfrak{h}_\upsilon \in \mathcal{R}_b (\varphi)$ be given by (1.1), then

$\begin{eqnarray*} |a_{2}| \leq \left \{ \begin{array} {rcl} &|b|, \ \ if \ \ |b| < \dfrac{4}{3}, \\ \\ &\sqrt{\dfrac{4|b|}{3}}, \ \ if \ \ |b| \geq \dfrac{4}{3}, \end{array}\right. \end{eqnarray*}$

$\begin{eqnarray*} |a_{3}| \leq \left \{ \begin{array} {rcl} &\frac{2|b|}{3} +|b|^{2}, \ \ if \ \ |b| < \dfrac{2}{3},\\ \\ &\frac{4|b|}{3}, \ \ if \ \ |b| \geq \dfrac{2}{3}, \end{array}\right. \end{eqnarray*}$

$\begin{eqnarray*} |a_{3}-2a^2_{2}|\leq \dfrac{4|b|}{3}, \end{eqnarray*}$

$\begin{eqnarray*} |a_{3}-a^2_{2}|\leq \dfrac{2|b|}{3}. \end{eqnarray*}$

2.1. Applications of our main results

Here, in this section, we consider the newly defined Salagean $q-$ differential operator for subclass of $R^{\upsilon, \gamma, m}_{b, q}(\varphi)$ of class of $\sum_{\upsilon }$ and investigate some new application in the form of results

Theorem 2.9. For $b \in \mathbb{C} \setminus\{0\}.$ Let $\mathfrak{h}_\upsilon \in R^{\upsilon, \gamma, m}_{b, q}(\varphi)$ by given by (1.3). If $a_{\upsilon i+1} = 0,$ and $1\leq i \leq k-1,$ then

$\begin{eqnarray*} | a_{\upsilon k+1}| \leq \dfrac{2|b|}{(1+\gamma[\upsilon k,q])(\upsilon k+1,q)^{m}}, & for\; \; k \geq 2. \end{eqnarray*}$

Proof. We can prove Theorem 2.9 by using the similar method of Theorem 2.1.

Theorem 2.10. For $b \in \mathbb{C} \setminus \{0\}.$ Let $\mathfrak{h}_\upsilon \in R^{\upsilon, \gamma, m}_{b, q}(\varphi)$ by given by (1.3). Then

$\begin{eqnarray*} | a_{\upsilon +1}| \leq \left \{ \begin{array} {rcl} & \frac{2|b|}{(1+\gamma[\upsilon ,q])(\upsilon +1,q)^{m}} , \; \; if \; \; |b| < \psi_{3}(\upsilon ,q), \\ \\ &\sqrt{|b| \ \ \psi_{1}(\upsilon ,q)}, \; \; if \; \; |b| \geq \psi_{3}(\upsilon ,q), \end{array}\right. \end{eqnarray*}$

$\begin{eqnarray*} | a_{2\upsilon +1}| \leq \left \{ \begin{array} {rcl} &|b| \psi_{2}(\upsilon ,q)+ \frac{2|b|^2}{(1+\gamma[\upsilon ,q])[\upsilon +1,q]^{m}}, \; if\; |b| < \psi_{4}(\upsilon ,q), \\ \\ & 2|b|\psi_{2}(\upsilon ,q) \; \; if \; \; |b| \geq \psi_{4}(\upsilon ,q), \end{array}\right. \end{eqnarray*}$

$\begin{eqnarray*} |a_{2\upsilon +1}- (1+\gamma[\upsilon ,q])[\upsilon +1,q]^{m} a^{2}_{\upsilon +1} | \leq 2|b|\psi_{4}(\upsilon ,q), \end{eqnarray*}$

$\begin{eqnarray*} \Big| a_{2\upsilon +1}-\dfrac{1}{\psi_{2}(\upsilon ,q)} a^{2}_{\upsilon +1}\Big| \leq |b|\psi_{4}(\upsilon ,q), \end{eqnarray*}$

where

$\begin{eqnarray*} \psi_{3}(\upsilon ,q) = \dfrac{8}{((1+\gamma[2\upsilon ,q])[2\upsilon +1,q]^{m})((1+\gamma[\upsilon ,q])[\upsilon +1,q]^{m})}, \end{eqnarray*}$

$\begin{eqnarray*} \psi_{4}(\upsilon ,q) = \dfrac{2}{(1+\gamma[2\upsilon ,q])[2\upsilon +1,q]^{m}}. \end{eqnarray*}$

Proof. We can prove Theorem 2.10 by using the similar method of Theorem 2.3.

3. Conclusions

In this article, first of all, we used the q-difference operator for $\upsilon$ -fold symmetric functions in order to define some new subclasses of the $\upsilon$ -fold symmetric bi-univalent functions in the open symmetric unit disk $\mathcal{U}$ . We also used the basic concepts of q-calculus and defined the Salagean q-differential operator for $\upsilon$ -fold symmetric functions. We considered this operator and investigated a new subclass of $\upsilon$ -fold symmetric bi-univalent functions. Faber Polynomial expansion method and q-analysis are used in order to determined general coefficient bounds $|a_{\upsilon+1}|$ for functions in each of these newly defined $\upsilon$ -fold symmetric bi-univalent functions classes. Feketo-Sezego problems and initial coefficient bounds $|a_{\upsilon+1}|$ and $|a_{2\upsilon+1}|$ for the function belonging to the subclasses of $\upsilon$ -fold symmetric bi-univalent functions in open symmetric unit disk $\mathcal{U}$ are also investigated.

Acknowledgments

I would like to thank to the editor and referees for their valuable comments and suggestions.

Conflict of interest

The author declares no conflict of interest.

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AIMS Mathematics

Extended suprametric spaces and Stone-type theorem

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Abstract

1. Introduction and definitions

2. The faber polynomial expansion method and application

2.1. Applications of our main results

3. Conclusions

Acknowledgments

Conflict of interest

References

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AIMS Mathematics

Extended suprametric spaces and Stone-type theorem

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Abstract

1. Introduction and definitions

2. The faber polynomial expansion method and application

2.1. Applications of our main results

3. Conclusions

Acknowledgments

Conflict of interest

References

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