Local well-posedness to the thermal boundary layer equations in Sobolev space

Yonghui Zou; Xin Xu; An Gao; Yonghui Zou; Xin Xu; An Gao

doi:10.3934/math.2023503

AIMS Mathematics

2023, Volume 8, Issue 4: 9933-9964. doi: 10.3934/math.2023503

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Research article

Local well-posedness to the thermal boundary layer equations in Sobolev space

Yonghui Zou ¹,
Xin Xu ¹,
An Gao ^{2
,
,}

1.
School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China
2.
Linyi Senior School of Finance and Economics, Linyi 276000, China

Received: 17 October 2022 Revised: 04 January 2023 Accepted: 18 January 2023 Published: 23 February 2023
MSC : 35Q30, 35Q35, 76D10

In this paper, we study the local well-posedness of the thermal boundary layer equations for the two-dimensional incompressible heat conducting flow with nonslip boundary condition for the velocity and Neumann boundary condition for the temperature. Under Oleinik's monotonicity assumption, we establish the local-in-time existence and uniqueness of solutions in Sobolev space for the boundary layer equations by a new weighted energy method developed by Masmoudi and Wong.

Keywords:

Citation: Yonghui Zou, Xin Xu, An Gao. Local well-posedness to the thermal boundary layer equations in Sobolev space[J]. AIMS Mathematics, 2023, 8(4): 9933-9964. doi: 10.3934/math.2023503

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Abstract

1. Introduction and definitions

Assume that $\mathcal{A}$ represents the family of analytic functions in the open unit disc

$\begin{equation*} \mathcal{U} = \{z:z\in \mathbb{C} \text{ and }\left \vert z\right \vert < 1\}. \end{equation*}$

For $f_{1}$ , $f_{2}\in \mathcal{A},$ we say that $f_{1}$ subordinate to $f_{2}$ in $\mathcal{U}$ , indicated by

$\begin{equation*} f_{1}(z)\prec f_{2}(z), \ z\in \mathcal{U}, \end{equation*}$

if there exists a Schwarz function $w$ , defined by

$\begin{equation*} w\in \mathcal{B = }\left \{ w:w\in \mathcal{A}, \text{ }\left \vert w(z)\right \vert < 1\text{ and }w(0) = 0, \text{ }z\in \mathcal{U}\right \}, \end{equation*}$

that satisfies the condition

$\begin{equation*} f_{1}(z) = f_{2}(w(z)), \ z\in \mathcal{U}. \end{equation*}$

Indeed, it is known that

$\begin{equation*} f_{1}(z)\prec f_{2}(z)\Longrightarrow f_{1}(0) = f_{2}(0)\text{ and }f_{1}( \mathcal{U})\subset f_{2}(\mathcal{U}). \end{equation*}$

Moreover, if the function $f_{2}$ is univalent in $\mathcal{U}$ then

$\begin{equation*} f_{1}(z)\prec f_{2}(z)\Leftrightarrow f_{1}(0) = f_{2}(0)\text{ and }f_{1}( \mathcal{U})\subset f_{2}(\mathcal{U}). \end{equation*}$

Let the class $\mathcal{P}$ be defined by

$\begin{equation*} \mathcal{P} = \left\{ h\in \mathcal{A}:h(0) = 1\text{ and Re}(h(z) > 0\right\} . \end{equation*}$

The class of all functions in the normalized analytic function class $\mathcal{A}$ that are univalent in $\mathcal{U}$ is also denoted by the symbol $\mathcal{S}$ . The maximization of the non-linear functional $\left \vert a_{3}-\mu a_{2}^{2}\right \vert$ or other classes and subclasses of univalent functions has been the subject of a number of established results and these results are known as Fekete-Szegö problems, see ^[1]. If $f\in \mathcal{S}$ and of the form (1.1), then

$\begin{equation*} \left \vert a_{3}-\mu a_{2}^{2}\right \vert \leq \left \{ \begin{array}{llc} 3-4\mu, & \text{ if } \mu \leq 0, \\ 1+2\exp \left( \frac{2\mu }{\mu -1}\right), & \text{ if } 0\leq \mu < 1, \\ 4\mu -3, & \text{ if } \mu \geq 1, \end{array} \right \} \end{equation*}$

and the result $\left \vert a_{3}-\mu a_{2}^{2}\right \vert$ are sharp (see ^[1]). The Fekete-Szegö problems have a rich history in literature and for complex number $\mu$ .

In the area of geometric function theory (GFT), the $q$ -calculus and fractional $q$ -calculus have been extensively employed by scholars who have developed and investigated a number of novel subclasses of analytic, univalent and bi-univalent functions. Jackson ^[2,3] first proposed the concept of the $q$ -calculus operator and gave the definition of the $q$ -difference operator $D_{q}$ in 1909. In instance, Ismail et al. were the first to define a class of $q$ -starlike functions in open unit disc $\mathcal{U}$ using $D_{q}$ in ^[4]. The most significant 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, ^[5]). See the following articles ^[6,7,8,9,10] for more information about $q$ -calculus operator theory in GFT.

Now we review some fundamental definitions and ideas of the $q$ -calculus, we utilize them to create some new subclasses in this paper.

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

$\begin{equation*} \lbrack l]_{q} = \frac{1-ql}{1-q}\; \text{ and }\; [0] = 0, \end{equation*}$

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

$\begin{eqnarray*} \lbrack l]_{q}! & = &[1]_{q}[2]_{q}[3]_{q} \cdots [l]_{q}, \\ \lbrack 0]_{q}! & = &1. \end{eqnarray*}$

For $q\rightarrow 1^{-},$ then $[l]!$ reduces to $l!$ .

The $q$ -generalized Pochhammer symbol is defined by

$\begin{equation*} \lbrack l]_{k} = \frac{\Gamma _{q}(l+k)}{\Gamma _{q}(l)}, \; \; \ \ k\in \mathbb{N} , \text{ }l\in \mathbb{C} . \end{equation*}$

The $q$ -gamma function $\Gamma _{q}$ is defined by

$\begin{equation*} \Gamma _{q}(l) = \left( 1-q\right) ^{l}\prod\limits_{j = 0}^{\infty }\frac{ 1-q^{j+1}}{1-q^{j+l}}. \end{equation*}$

The $q$ -generalized Pochhammer symbol is defined by

$\begin{equation*} \lbrack l]_{k} = \frac{\Gamma _{q}(l+k)}{\Gamma _{q}(l)}, \; \; \ \ k\in \mathbb{N} , \text{ }l\in \mathbb{C}. \end{equation*}$

Remark 1. For $q\rightarrow 1^{-},$ then $[l]_{q, k}$ reduces to $\left(l\right) _{k} = \frac{\Gamma (l+k)}{\Gamma (l)}$ .

Definition 1. Jackson ^[3] defined the $q$ -integral of function $f(z)$ as follows:

$\begin{equation*} \int f(z)d_{q}(z) = \sum \limits_{n = 0}^{\infty }z\left( 1-q\right) f\left( q^{n}(z)\right) q^{n}. \end{equation*}$

Jackson ^[2] introduced the $q$ -difference operator for analytic functions as follows:

Definition 2. ^[2] For $f\in \mathcal{A},$ the $q$ -difference operator is defined as:

$\begin{equation*} D_{q}f(z) = \frac{f(qz)-f(z)}{z(q-1)}, \ \ z\in \mathcal{U}. \end{equation*}$

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

$\begin{equation*} D_{q}(z^{n}) = [n]_{q}z^{n-1}, \; \ \ D_{q}\left( \sum \limits_{n = 1}^{\infty }a_{n}z^{n}\right) = \sum \limits_{n = 1}^{\infty }[n]_{q}a_{n}z^{n-1}. \end{equation*}$

Let $\mathcal{A}_{p}$ stand for the class of analytic functions with the form

$\begin{equation} f(z) = z^{p}+\sum \limits_{n = 1}^{\infty }a_{n+p}z^{n+p}, \; \ \ p\in \mathbb{N} , z\in \mathcal{U} \end{equation}$

(1.1)

in the open unit disk $\mathcal{U}$ . More specifically, $\mathcal{A}_{1} = \mathcal{A}$ and

$\begin{equation} f(z) = z+\sum \limits_{n = 1}^{\infty }a_{n+1}z^{n+1}, \; \ \ z\in \mathcal{U }\text{.} \end{equation}$

(1.2)

Consider the $q$ -difference operator for $f\in \mathcal{A}_{p}$ as follows:

Definition 3. ^[11] For $f\in \mathcal{A}_{p},$ the $q$ -difference operator is defined as:

$\begin{equation*} D_{q}f(z) = \frac{f(qz)-f(z)}{z(q-1)}, \ \ z\in \mathcal{U}. \end{equation*}$

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

$\begin{equation*} D_{q}(z^{n+p}) = [n+p]_{q}z^{n+p-1}, \; \ \ D_{q}\left( \sum\limits_{n = 1}^{\infty }a_{n+p}z^{n+p}\right) = \sum\limits_{n = 1}^{\infty }[n+p]_{q}a_{n+p}z^{n+p-1}. \end{equation*}$

Let $\mathcal{S}^{\ast }\left(p\right)$ represents the class of $p$ -valent starlike functions and every $f\in \mathcal{S}^{\ast }\left(p\right),$ if

$\begin{equation*} {Re}\left( \frac{zf^{^{\prime }}(z)}{pf(z)}\right) > 0, \ \ z\in \mathcal{U}, \end{equation*}$

and $\mathcal{K}\left(p\right)$ represents the class of $p$ -valent convex functions and every $f\in \mathcal{K}(p),$ if

$\begin{equation*} \frac{1}{p}\left( 1+{Re}\left( \frac{zf^{^{^{\prime \prime }}}(z)}{ f^{^{\prime }}(z)}\right) \right) > 0, \ \ z\in \mathcal{U}. \end{equation*}$

These conditions are equivalent in terms of subordination as follows:

$\begin{equation*} \mathcal{S}^{\ast }\left( p\right) = \left \{ f\in \mathcal{A}_{p}:\frac{ zf^{^{\prime }}(z)}{pf(z)}\prec \frac{1+z}{1-z}\right \} \end{equation*}$

and

$\begin{equation*} \mathcal{K}(p) = \left \{ f\in \mathcal{A}_{p}:\frac{1}{p}\left( 1+\frac{ zf^{^{^{\prime \prime }}}(z)}{f^{^{\prime }}(z)}\right) \prec \frac{1+z}{1-z} \right \}. \end{equation*}$

The aforementioned two classes can be generalized as follows:

$\begin{equation*} \mathcal{S}^{\ast }\left( p, \varphi \right) = \left \{ f\in \mathcal{A}_{p}: \frac{zf^{^{\prime }}(z)}{pf(z)}\prec \varphi \left( z\right) \right \} \end{equation*}$

and

$\begin{equation*} \mathcal{K}\left( p, \varphi \right) = \left \{ f\in \mathcal{A}_{p}:\frac{1}{p }\left( 1+\frac{zf^{^{^{\prime \prime }}}(z)}{f^{^{\prime }}(z)}\right) \prec \varphi \left( z\right) \right \} , \end{equation*}$

where $\varphi \left(z\right)$ is a real part function that is positive and is normalized by the rule

$\begin{equation*} \varphi (0) = 1\; \text{ and }\; \varphi ^{^{\prime }}(0) > 0, \end{equation*}$

and $\varphi$ maps $\mathcal{U}$ onto a space that is symmetric with regard to the real axis and starlike with respect to $1.$ If $p = 1,$ then

$\begin{equation*} \mathcal{S}^{\ast }\left( p, \varphi \right) = \mathcal{S}^{\ast }\left( \varphi \right) \end{equation*}$

and

$\begin{equation*} \mathcal{K}\left( p, \varphi \right) = \mathcal{K}\left( \varphi \right) . \end{equation*}$

These two classes $\mathcal{S}^{\ast }\left(\varphi \right)$ and $\mathcal{ K}\left(\varphi \right)$ defined by Ma ^[12].

A function $f\in \mathcal{A}_{p},$ is called $p$ -valently starlike of order $\alpha$ ( $0\leq\alpha < 1$ ) with complex order $b\in \mathbb{C} \backslash \{0\},$ if it satisfies the inequality

$\begin{equation*} {Re}\left \{ 1+\frac{1}{b}\left( \frac{zf^{^{\prime }}(z)}{pf(z)} -1\right) > \alpha , \ z\in \mathcal{U}\right \} . \end{equation*}$

The class $\mathcal{S}_{p}^{\ast}(\alpha, b)$ denotes the collection of all $f\in \mathcal{A}_{p}$ functions that satisfy the aforementioned condition.

A function $f\in \mathcal{A}_{p},$ is called $p$ -valently convex function of order $\alpha$ $(0\leq \alpha < 1)$ with complex order $b\in \mathbb{C} \backslash \{0\},$ if it satisfies the inequality

$\begin{equation*} {Re}\left \{ 1-\frac{1}{b}+\frac{1}{bp}\left( 1+\frac{zf^{^{^{\prime \prime }}}(z)}{f^{^{\prime }}(z)}\right) > \alpha , \ z\in \mathcal{U} \right \} . \end{equation*}$

The class $\mathcal{K}_{p}(\alpha, b)$ denotes the collection of all functions $f\in \mathcal{A}_{p}$ that satisfy the aforementioned condition.

Note that

$\begin{equation*} f\in \mathcal{K}_{p}(\alpha , b)\Leftrightarrow \frac{1}{p}zf^{^{\prime }}\in \mathcal{S}_{p}^{\ast }(\alpha , b). \end{equation*}$

Kargar et al. ^[13] investigated the classes $\mathcal{S}_{p}\left(\alpha, \beta \right)$ for $f\in \mathcal{A}_{p}$ and defined as follows:

$\begin{equation*} f\in \mathcal{S}_{p}\left( \alpha , \beta \right) \Leftrightarrow \alpha < {Re}\left( \frac{1}{p}\frac{zf^{^{\prime }}(z)}{f(z)}\right) < \beta , \ \ \ \left( 0\leq \alpha < 1 < \beta , \ z\in \mathcal{U} \right) . \end{equation*}$

For $0\leq \alpha < 1 < \beta$ and $b\in \mathbb{C} \backslash \{0\},$ then the function $f\in \mathcal{A}_{p}$ belongs to the class $\mathcal{K}_{b, p}\left(\alpha, \beta \right)$ if it satisfies the inequality

$\begin{equation*} \alpha < {Re}\left( 1-\frac{1}{b}+\frac{1}{bp}\left( 1+\frac{ zf^{^{^{\prime \prime }}}(z)}{f^{^{\prime }}(z)}\right) \right) < \beta , \ \ \ \left( 0\leq \alpha < 1 < \beta , \ z\in \mathcal{U} \right) . \end{equation*}$

If $p = 1,$ then $\mathcal{K}_{b, p}\left(\alpha, \beta \right) = \mathcal{K} _{b}\left(\alpha, \beta \right)$ , studied by Kargar et al. in ^[13] and if $\beta \rightarrow \infty$ in above definition, $\mathcal{K} _{b, p}\left(\alpha, \beta \right) = \mathcal{K}_{b, p}\left(\alpha, b\right)$ .

Recently, Bult ^[14] used the definition of subordination and defined new subclasses of $p$ -valent starlike and convex functions associated with vertical strip domain as follows:

$\begin{equation*} \mathcal{S}_{p, b}^{\ast }\left( \alpha , \beta \right) = \left \{ f\in \mathcal{A}_{p}:1+\frac{1}{b}\left( \frac{1}{p}\frac{zf^{^{\prime }}(z)}{f(z) }-1\right) \prec f\left( \alpha , \beta ;z\right) \right \} \end{equation*}$

and

$\begin{equation*} \mathcal{K}_{p, b}\left( \alpha , \beta \right) = \left \{ f\in \mathcal{A} _{p}:1-\frac{1}{b}+\frac{1}{bp}\left( 1+\frac{zf^{^{^{\prime \prime }}}(z)}{ f^{^{\prime }}(z)}\right) \prec f\left( \alpha , \beta ;z\right) \right \} , \end{equation*}$

where

$\begin{equation*} f\left( \alpha , \beta ;z\right) = 1+\frac{\beta -\alpha }{\pi }i\log \left( \frac{1-e^{2\pi i\frac{1-\alpha }{\beta -\alpha }}z}{1-z}\right) \end{equation*}$

and

$\begin{equation*} 0\leq \alpha < 1 < \beta, \; b\in \mathbb{C} \backslash \{0\}, \text{ }z\in \mathcal{U}. \end{equation*}$

Bult ^[14] determined the coefficient bounds for functions belonging to these new classes.

On the basis of the geometrical interpretation of their image domains, numerous subclasses of analytic functions have established using the concept of subordination. Right half plane ^[15], circular disc ^[16], oval and petal type domains ^[17], conic domain ^[18,19], leaf-like domain ^[20], generalized conic domains ^[21], and the most important one is shell-like curve ^{[22,23,24,25]} are some fascinating geometrical classes we obtain with this domain. The function

$\begin{equation} h(z) = \frac{1+\tau ^{2}z^{2}}{1-\tau z-\tau ^{2}z^{2}} \end{equation}$

(1.3)

is essential for the shell-like shape, where

$\begin{equation*} \tau = \frac{1-\sqrt{5}}{2}. \end{equation*}$

The image of unit circle under the function $h$ gives the conchoid of Maclaurin's, due to the function

$\begin{equation*} h\left( e^{i\varphi }\right) = \frac{\sqrt{5}}{2\left( 3-2\cos \varphi \right) }+i\frac{\sin \varphi \left( 4\cos \varphi -1\right) }{2\left( 3-2\cos \varphi \right) \left( 1+\cos \varphi \right) }, \; \ 0\leq \varphi < 2\pi . \end{equation*}$

The function given in (1.3) has the following series representation:

$\begin{equation*} h(z) = 1+\sum \limits_{n = 1}^{\infty }\left( u_{n-1}+u_{n+1}\right) \tau ^{n}z^{n}, \end{equation*}$

where

$\begin{equation*} u_{n} = \frac{\left( 1-\tau \right) ^{n}-\tau ^{n}}{\sqrt{5}}, \end{equation*}$

and $u_{n}$ produces a Fibonacci series of coefficient constants that are more closely related to the Fibonacci numbers.

Taking motivation from the idea of circular disc and shell-like curves, Malik et al. ^[26] defined new domain for analytic functions which is named as cardioid domain. A new class of analytic functions is defined associated with cardioid domain, for more detail, see ^[26].

Definition 4. ^[26] Assume that $\mathcal{CP}\left(L, N\right)$ represents the class of functions $p$ that are defined as

$\begin{equation*} p(z)\prec \overline{p}\left( L, N, z\right) , \end{equation*}$

where $\overline{p}\left(L, N, z\right)$ is defined by

$\begin{equation} \overline{p}\left( L, N, z\right) = \frac{2L\tau ^{2}z^{2}+\left( L-1\right) \tau z+2}{2N\tau ^{2}z^{2}+\left( N-1\right) \tau z+2} \end{equation}$

(1.4)

with $-1 < N < L\leq 1$ , $\tau = \frac{1-\sqrt{5}}{2}$ and $z\in \mathcal{U}$ .

To understand the class $\mathcal{CP}\left(L, N\right)$ , an explanation of the function $\overline{p}\left(L, N, z\right)$ in geometric terms might be helpful in this instance. If we denote

$\begin{equation*} R_{\overline{p}}\left( L, N;e^{i\theta }\right) = u \end{equation*}$

and

$\begin{equation*} I_{\overline{p}}\left( L, N;e^{i\theta }\right) = v, \end{equation*}$

then the image $\overline{p}\left(L, N, e^{i\theta }\right)$ of the unit circle is a cardioid like curve defined by

$\begin{equation} \left\{ \begin{array}{c} u = \frac{4+(L-1)(N-1)\tau ^{2}+4LN\tau ^{4}+2\lambda \cos \theta +4\left( L+N\right) \tau ^{2}\cos 2\theta }{4+(N-1)^{2}\tau ^{2}+4N^{2}\tau ^{4}+4\left( N-1\right) \left( \tau +N\tau ^{3}\right) \cos \theta +8N\tau ^{2}\cos 2\theta }, \\ v = \frac{(L-N)\left( \tau -\tau ^{3}\right) \sin \theta +2\tau ^{2}\sin 2\theta }{4+(N-1)^{2}\tau ^{2}+4N^{2}\tau ^{4}+4\left( N-1\right) \left( \tau +N\tau ^{3}\right) \cos \theta +8N\tau ^{2}\cos 2\theta}, \end{array} \right\} \end{equation}$

(1.5)

where

$\begin{equation*} \lambda = (L+N-2)\tau +(2LN-L-N)\tau ^{3}, {\ }-1 < N < L\leq 1, {\ }\tau = \frac{1-\sqrt{5}}{2}, \end{equation*}$

and

$0\leq \theta \leq 2\pi.$

Moreover, we observe that

$\overline{p}\left( L, N, 0\right) = 1,$

and

$\overline{p}\left( L, N, 1\right) = \frac{LN+9\left( L+N\right) +1+4\left( N-L\right) \sqrt{5}}{ N^{2}+18N+1}.$

According to (1.5), the cusp of the cardioid-like curve is provided by

$\begin{equation*} \gamma \left( L, N\right) = \overline{p}\left( L, N;e^{\pm i\arccos \left( \frac{1}{4}\right) }\right) = \frac{2LN-3\left( L+N\right) +2+(L-N)\sqrt{5}}{ 2\left( N^{2}-3N+1\right) }. \end{equation*}$

The image of each inner circle is a nested cardioid-like curve if the open unit disc $\mathcal{U}$ is considered a collection of concentric circles with origin at the center. As a result, the open unit disc $\mathcal{U}$ is mapped onto a cardioid region by the function $\overline{p}(L, N, z)$ . This means that $\overline{p}\left(L, N;\mathcal{U}\right)$ is a cardioid domain. The above discussed cardioid like curve with different values of parameters can be seen in Figures 1 and 2.

Figure 1. The curve (1.5) with L = 0.8, N = 0.6 and the curve (1.5) with L = 0.5, N = -0.5.

DownLoad: Full-Size Img PowerPoint

Figure 2. The curve (1.5) with L = 0.6, N = 0.8 and the curve (1.5) with L = -0.5, N = 0.5.

DownLoad: Full-Size Img PowerPoint

The relationship $N < L$ links the parameters $L$ and $N$ . The cardioid-like curve is flipped by its voilation, as seen in the figures below.

See , if collection of concentric circles having origin as center. Thus, the function $\overline{p}(L, N, z)$ maps the open unit disk $\mathcal{U}$ onto a cardioid region. See ^[26] for more details about cardioid region.

Figure 3. The open unit disk

$\mathcal{U}$ .

DownLoad: Full-Size Img PowerPoint

The operator theory of quantum calculus is the primary result of this research. Using standard uses in quantum calculus operator theory and the $q$ -difference operator, we develop numerous novel $q$ -analogous of the differential and integral operators. We construct a large number of new classes of $q$ -starlike and $q$ -convex functions using these operators and study some interesting characteristics of the corresponding analytic functions. In this paper, we gain inspiration from recent research by ^[14,26,27] and define two new classes of $p$ -valent starlike, convex functions connected with the cardioid domain using the $q$ -difference operator.

Influenced by recent studies ^[14,26,27], we defined two new classes of $p$ -valent starlike, convex functions related with cardioid domain.

Definition 5. The function $f$ of the form (1.1) related with cardioid domain, represented by $\mathcal{S}_{p}^{\ast }\left(L, N, q, b\right),$ is defined to be the functions $f$ such that

$\begin{equation*} 1+\frac{1}{b}\left( \frac{1}{\left[ p\right] _{q}}\frac{zD_{q}f(z)}{f(z)} -1\right) \prec \overline{p}\left( L, N;z\right) , \end{equation*}$

where, $b\in \mathbb{C} \backslash \{0\}$ and $\overline{p}\left(L, N;z\right)$ is given by (1.4).

Definition 6. The function $f$ of the form (1.1) related with cardioid domain, represented by $\mathcal{K}_{p}\left(L, N, q, b\right),$ is defined to be the functions $f$ such that

$\begin{equation*} 1-\frac{1}{b}+\frac{1}{b\left[ p\right] _{q}}\left( 1+\frac{zD_{q}^{2}f(z)}{ D_{q}f(z)}\right) \prec \overline{p}\left( L, N;z\right), \end{equation*}$

where $b\in \mathbb{C} \backslash \{0\}$ and $\overline{p}\left(L, N;z\right)$ is given by (1.4) and $\mathcal{K}_{p}\left(L, N, q, b\right)$ is the class of convex functions of order $b$ related with cardioid domain.

Special cases:

(i) For $q\rightarrow 1^{-}, b = 1$ and $p = 1$ , in Definition 5, we have known class $\mathcal{S}^{\ast }\left(L, N\right)$ of starlike functions associated with cardioid domain proved by Zainab et al. in ^[27].

(ii) For $q\rightarrow 1^{-}, L = 1, N = -1$ , $b = 1$ and $p = 1$ in Definition 5, then class $\mathcal{S}_{p}^{\ast }\left(L, N, q, b\right) = S\mathcal{L}$ and this class is defined on starlike functions associated with Fibonacci numbers, introduced and studied by Sokół in ^[28].

(iii) For $q\rightarrow 1^{-},$ $L = 1,$ $N = -1$ , $b = 1$ and $p = 1$ in Definition 6 then class $\mathcal{K}_{p}\left(L, N, q, b\right) = \mathcal{K},$ and this family is referred to as a class of convex functions connected with Fibonacci numbers.

There are four parts to this article. In Section $1$ , we briefly reviewed some basic concepts from geometric function theory, quantum calculus, and cardioid domain, studied the $q$ -difference operator, and finally discussed this operator to define two new subclasses of multivalent $q$ -starlike and $q$ -convex functions. The established lemmas are presented in Section $2$ . Our main results and some known corollaries will be presented in Section $3$ , then some concluding remarks in Section $4$ .

2. A set of lemmas

By utilizing the following lemmas, we will determine our main results.

Lemma 1. ^[26] Let the function $\overline{p}\left(L, N;z\right),$ defined by (1.4). Then,

(i) For the disc $\left \vert z\right \vert < \tau ^{2}$ , the function $\overline{p}\left(L, N;z\right)$ is univalent.

(ii) If $h(z)\prec \overline{p}\left(L, N;z\right),$ then ${Re} h(z) > \alpha,$ where

$\begin{equation*} \alpha = \frac{2\left( L+N-2\right) \tau +2\left( 2LN-L-N\right) \tau ^{3}+16\left( L+N\right) \tau ^{2}\eta }{4\left( N-1\right) \left( \tau +N\tau ^{3}\right) +32N\tau ^{2}\eta }, \end{equation*}$

where

$\begin{equation*} \eta = \frac{4+\tau ^{2}-N^{2}\tau ^{2}-4N^{2}\tau ^{4}-\left( 1-N\tau ^{2}\right) \chi (N)}{4\tau \left( 1+N^{2}\tau ^{2}\right) }, \end{equation*}$

$\begin{equation*} \quad\quad \chi (N) = \sqrt{5\left( 2N\tau ^{2}-\left( N-1\right) \tau +2\right) \left( 2N\tau ^{2}+\left( N-1\right) \tau +2\right), } \end{equation*}$

$\begin{equation*} -1 < N < L\leq 1 \end{equation*}$

and

$\begin{equation*} \tau = \frac{1-\sqrt{5}}{2}. \end{equation*}$

(iii) If

$\overline{p}\left( L, N;z\right) = 1+\sum \limits_{n = 1}^{\infty } \overline{Q}_{n}z^{n},$

then

$\begin{equation} \overline{Q}_{n} = \left \{ \begin{array}{lll} \left( L-N\right) \frac{\tau }{2}, &\text{for }n = 1, \\ \left( L-N\right) \left( 5-N\right) \frac{\tau ^{2}}{2^{2}}, &\text{for}\ n = 2, \\ \frac{1-N}{2}\tau p_{n-1}-N\tau ^{2}p_{n-2}, &\text{for}\ n = 3, 4, 5, \cdots, \end{array} \right. \end{equation}$

(2.1)

where

$\begin{equation*} -1 < N < L\leq 1. \end{equation*}$

(iv) Let $h(z)\prec \overline{p}\left(L, N;z\right)$ and of the form $h(z) = 1+\sum \limits_{n = 1}^{\infty }h_{n}z^{n}.$ Then

$\begin{equation*} \left \vert h_{2}-vh_{1}^{2}\right \vert \leq \frac{\left( L-N\right) \left \vert \tau \right \vert }{4}\max \left \{ 2, \left \vert \tau \left( v\left( L-N\right) +N-5\right) \right \vert \right \} {, \ \ }v\in \mathbb{C} . \end{equation*}$

Lemma 2. ^[29] Let $h\in \mathcal{P},$ such that $h(z) = 1+\sum \limits_{n = 1}^{\infty }c_{n}z^{n}.$ Then

$\begin{equation} \left \vert c_{2}-\frac{v}{2}c_{1}^{2}\right \vert \leq \max \left \{ 2, 2\left \vert v-1\right \vert \right \} = \left \{ \begin{array}{ll} &2, \quad \quad \quad \mathit{\text{if}}~~ 0\leq v\leq 2, \\ &2\left \vert v-1\right \vert, \ \ \ \mathit{\text{elsewhere}}, \end{array} \right \} \end{equation}$

(2.2)

and

$\begin{equation} \left \vert c_{n}\right \vert \leq 2\ \mathit{{for}} \ n\geq 1. \end{equation}$

(2.3)

Lemma 3. ^[30] Let $h\in \mathcal{P},$ such that

$h(z) = 1+\sum \limits_{n = 1}^{\infty }c_{n}z^{n}.$

Then for any complex number $v$

$\begin{equation*} \left \vert c_{2}-vc_{1}^{2}\right \vert \leq 2\max \left \{ 1, \left \vert 2v-1\right \vert \right \} \end{equation*}$

and the result is sharp for

$\begin{equation*} h(z) = \frac{1+z^{2}}{1-z^{2}}\; \; \mathit{\text{and}}\; \; h(z) = \frac{1+z}{1-z}. \end{equation*}$

Lemma 4. ^[31] Let the function $g$ given by

$\begin{equation*} g(z) = \sum \limits_{k = 1}^{\infty }b_{k}z^{k} \end{equation*}$

be convex in $\mathcal{U}$ . Also let the function $f$ given by

$\begin{equation*} f(z) = \sum \limits_{k = 1}^{\infty }a_{k}z^{k} \end{equation*}$

be analytic in $\mathcal{U}.$ If

$\begin{equation*} f(z)\prec g(z), \end{equation*}$

then

$\begin{equation*} \left \vert a_{k}\right \vert < \left \vert b_{1}\right \vert ,~~ k = 1, 2, 3, \cdots. \end{equation*}$

For the recently described classes of multivalent $q$ -starlike ( $\mathcal{S} _{p}^{\ast }\left(L, N, q, b\right))$ and multivalent $q$ -convex ( $\mathcal{K} _{p}\left(L, N, q, b\right)$ ) functions, we get sharp estimates for the coefficients of Taylor series, Fekete-Szegő problems and coefficient inequalities.

3. Main results

In the following theorems, we investigate the functions $f(z)$ which can be used to find the sharpness of the results of this article.

Theorem 5. A function $f\in \mathcal{A}_{p}$ given by (1.1) is in the class $\mathcal{S}_{p}^{\ast }\left(L, N, q, b\right)$ if and only if there exists an analytic function $S$ ,

$\begin{equation*} S(z)\prec \overline{p}\left( L, N, z\right) = \frac{2L\tau ^{2}z^{2}+\left( L-1\right) \tau z+2}{2N\tau ^{2}z^{2}+\left( N-1\right) \tau z+2}, \end{equation*}$

where, $\overline{p}\left(L, N, 0\right) = 1,$ such that

$\begin{equation} f(z) = z^{p}\exp \left\{ b\left[ p\right] _{q}\int\limits_{0}^{z}\frac{S(t)-1}{ t}dt\right\} , ~~ z\in \mathcal{U}. \end{equation}$

(3.1)

Proof. Let $f\in \mathcal{S}_{p}^{\ast }\left(L, N, q, b\right)$ and

$\begin{equation*} 1+\frac{1}{b}\left( \frac{1}{\left[ p\right] _{q}}\frac{zD_{q}f^{^{}}(z)}{ f(z)}-1\right) = h(z)\prec \overline{p}\left(L, N, z\right.). \end{equation*}$

Then by integrating this equation we obtain (3.1). Conversely, if given by (3.1) with an analytic function $S(z)$ such that $S(z)\prec \overline{p}\left(L, N, z\right),$ then by logarithmic differentiation of (3.1) we obtain

$\begin{equation*} 1+\frac{1}{b}\left( \frac{1}{\left[ p\right] _{q}}\frac{zD_{q}f^{^{}}(z)}{ f(z)}-1\right) = S(z). \end{equation*}$

Therefore we have

$\begin{equation*} 1+\frac{1}{b}\left( \frac{1}{\left[ p\right] _{q}}\frac{zD_{q}f^{^{}}(z)}{ f(z)}-1\right) \prec \overline{p}\left( L, N, z\right) \end{equation*}$

and $f\in \mathcal{S}_{p}^{\ast }\left(L, N, q, b\right)$ . □

The initial coefficient bounds $\left\vert a_{p+1}\right\vert$ and $\left\vert a_{p+2}\right\vert$ for the functions $f\in \mathcal{S} _{p}^{\ast }\left(L, N, b\right)$ are investigated in Theorem 6 using the Lemma 2.

Theorem 6. Let $f\in \mathcal{S}_{p}^{\ast }\left(L, N, q, b\right)$ be given by (1.1) $,$ $-1\leq N < L\leq 1.$ Then

$\begin{eqnarray*} \left \vert a_{p+1}\right \vert &\leq &\frac{\left[ p\right] _{q}\left \vert b\right \vert \left( L-N\right) \tau }{2}, \\ \left \vert a_{p+2}\right \vert &\leq &\frac{\left[ p\right] _{q}\left \vert b\right \vert \left( L-N\right) \left \vert \tau \right \vert ^{2}}{8}\left( 5-N+\left[ p\right] _{q}b(L-N)\right.). \end{eqnarray*}$

These bounds are sharp.

Proof. Let $f\in \mathcal{S}_{p}^{\ast }\left(L, N, q, b\right)$ , and of the form (1.1). Then

$\begin{equation} 1+\frac{1}{b}\left( \frac{1}{\left[ p\right] _{q}}\frac{zD_{q}f^{^{}}(z)}{ f(z)}-1\right) \prec \overline{p}\left( L, N;z\right.), \end{equation}$

(3.2)

where

$\begin{equation*} \overline{p}\left( L, N, z\right) = \frac{2L\tau ^{2}z^{2}+\left( L-1\right) \tau z+2}{2N\tau ^{2}z^{2}+\left( N-1\right) \tau z+2}. \end{equation*}$

By applying the concept of subordination, there exists a function $w$ with

$\begin{equation*} w(0) = 0\; \text{ and }\; |w(z)| < 1, \end{equation*}$

such that

$\begin{equation} 1+\frac{1}{b}\left( \frac{1}{\left[ p\right] _{q}}\frac{zD_{q}f^{^{}}(z)}{ f(z)}-1\right) = \overline{p}\left( L, N;w(z)\right). \end{equation}$

(3.3)

Let

$\begin{eqnarray} w\left( z\right) & = &\frac{h(z)-1}{h(z)+1} \\ & = &\frac{c_{1}z+c_{2}z^{2}+c_{3}z^{3}+\cdots}{2+c_{1}z+c_{2}z^{2}+\cdots} \\ & = &\frac{1}{2}c_{1}z+\frac{1}{2}\left( c_{2}-\frac{1}{2}c_{1}^{2}\right) z^{2}+\frac{1}{2}\left( c_{3}-c_{1}c_{2}+\frac{1}{4}c_{1}^{3}\right) z^{3}+\cdots . \end{eqnarray}$

(3.4)

Since

$\overline{p}\left( L, N;z\right) = 1+\sum \limits_{n = 1}^{\infty } \overline{Q}_{n}z^{n},$

then

$\begin{eqnarray} \overline{p}\left( L, N;w(z)\right) & = &1+\overline{Q}_{1}\left \{ \frac{1}{2}c_{1}z+\frac{1}{2}\left( c_{2}- \frac{1}{2}c_{1}^{2}\right) z^{2}\cdots\right \} +\overline{Q}_{2}\left \{ \frac{1}{2}c_{1}z+\frac{1}{2}\left( c_{2}-\frac{1}{2}c_{1}^{2}\right) z^{2}\cdots\right \}^{2} +\cdots \\ & = &1+\frac{\overline{Q}_{1}c_{1}}{2}z+\left( \frac{1}{2}\left( c_{2}-\frac{1 }{2}c_{1}^{2}\right) \overline{Q}_{1}+\frac{\overline{Q}_{2}c_{1}^{2}}{4} \right) z^{2}+\cdots. \end{eqnarray}$

(3.5)

Also consider the function

$\begin{equation*} \overline{p}\left( L, N;z\right) = \frac{2L\tau ^{2}z^{2}+\left( L-1\right) \tau z+2}{2N\tau ^{2}z^{2}+\left( N-1\right) \tau z+2}. \end{equation*}$

Let $\tau z = \alpha _{0},$ then

$\begin{eqnarray*} \overline{p}\left( L, N, z\right) & = &\frac{2L\alpha _{0}^{2}+\left( L-1\right) \alpha _{0}+2}{2N\alpha _{0}^{2}+\left( N-1\right) \alpha _{0}+2} \\ & = &\frac{L\alpha _{0}^{2}+\frac{\left( L-1\right) }{2}\alpha _{0}+1}{N\alpha _{0}^{2}+\frac{\left( N-1\right) }{2}\alpha _{0}+1} \\ & = &\left( L\alpha _{0}^{2}+\frac{\left( L-1\right) }{2}\alpha _{0}+1\right) \left[ 1+\frac{1}{2}\left( 1-N\right) \alpha _{0}+\left( \frac{N^{2}-6N+1}{4} \right) \alpha _{0}^{2}+\cdots\right] \\ & = &1+\frac{1}{2}\left( L-N\right) \alpha _{0}+\frac{1}{4}\left( L-N\right) \left( 5-N\right) \alpha _{0}^{2}+\cdots. \end{eqnarray*}$

This implies that

$\begin{equation} \overline{p}\left( L, N;z\right) = 1+\frac{1}{2}\left( L-N\right) \tau z+\frac{ 1}{4}\left( L-N\right) \left( 5-N\right) \tau ^{2}z^{2}+\cdots. \end{equation}$

(3.6)

It is simple to observe from (3.5) that

$\begin{eqnarray} \overline{p}\left( L, N;w(z)\right) = 1+\frac{1}{4}\left( L-N\right) \tau c_{1}z+\left( \frac{1}{4}\left( L-N\right) \tau \left( c_{2}-\frac{1}{2}c_{1}^{2}\right) +\frac{\left( L-N\right) \left( 5-N\right) \tau ^{2}c_{1}^{2}}{16}\right) z^{2}+\cdots. \end{eqnarray}$

(3.7)

Since $f\in \mathcal{S}_{p}^{\ast }\left(L, N, b\right),$ then

$\begin{equation} 1+\frac{1}{b}\left( \frac{1}{[p]_{q}}\frac{zD_{q}f^{^{}}(z)}{f(z)} -1\right) = 1+\frac{1}{b[p]_{q}}a_{p+1}z+\frac{1}{b[p]_{q}}\left( 2a_{p+2}-a_{p+1}^{2}\right) z^{2}+\cdots. \end{equation}$

(3.8)

It is simple to show that by utilizing (3.3) and comparing the coefficients from (3.7) and (3.8), we get

$\begin{equation} a_{p+1} = \frac{b[p]_{q}\left( L-N\right) \tau c_{1}}{4}. \end{equation}$

(3.9)

Applying modulus on both side, we have

$\begin{equation*} \left \vert a_{p+1}\right \vert \leq \frac{\lbrack p]_{q}\left \vert b\right \vert \left( L-N\right) \tau }{2}. \end{equation*}$

Now again comparing the coefficients from (3.7) and (3.8), we have

$\begin{eqnarray*} \frac{2}{b[p]_{q}}a_{p+2} = \frac{1}{4}\left( L-N\right) \tau \left( c_{2}- \frac{1}{2}c_{1}^{2}\right) +\frac{\left( L-N\right) \left( 5-N\right) \tau ^{2}c_{1}^{2}}{16}+\frac{1}{[p]_{q}b}a_{p+1}^{2}, \end{eqnarray*}$

$\begin{eqnarray} \left \vert a_{p+2}\right \vert = \frac{b[p]_{q}\left( L-N\right) \tau }{8} \left \vert c_{2}-\frac{v}{2}c_{1}^{2}\right \vert, \qquad\qquad\qquad\qquad\qquad\quad \end{eqnarray}$

(3.10)

where

$\begin{equation*} v = 1-\frac{\tau }{2}\left( 5-N+[p]_{q}b(L-N)\right) . \end{equation*}$

It shows that $v > 2$ which is satisfied by the relation $L > N.$ Hence, by applying Lemma 2, we obtain the required result.

Result is sharp for the function

$\begin{eqnarray} f_{\ast }(z) & = &z^{p}\exp \left( b[p]_{q}\int \limits_{0}^{z}\frac{\overline{ p}\left( L, N, t\right) -1}{t}dt\right) \\ & = &z^{p}+\frac{b[p]_{q}\left( L-N\right) \tau }{2}z^{p+1}+\frac{ b[p]_{q}\left( L-N\right) \left( 5-N\right) \tau ^{2}}{8}z^{p+2}+\cdots, \end{eqnarray}$

(3.11)

where $\overline{p}\left(L, N, \right)$ defined in (1.4). □

Letting $q\rightarrow 1^{-},$ $b = 1$ and $p = 1$ in Theorem 6, we get the known corollary proved in ^[32] for starlike functions connected with cardioid domain.

Corollary 1. ^[32] Let $f\in \mathcal{S}^{\ast }\left(L, N\right)$ be given by (1.2) $,$ $-1\leq N < L\leq 1.$ Then

$\begin{eqnarray*} \left \vert a_{2}\right \vert &\leq &\frac{\left( L-N\right) \left \vert \tau \right \vert }{2}, \\ \left \vert a_{3}\right \vert &\leq &\frac{\left( L-N\right) \left \vert \tau \right \vert ^{2}}{8}\left \{ L-2N+5\right \} . \end{eqnarray*}$

Fekete-Szegö problem $\left\vert a_{p+2}-\mu a_{p+1}^{2}\right\vert$ for the functions $f\in \mathcal{S}_{p}^{\ast }\left(L, N, b\right)$ are investigated in Theorem 7.

Theorem 7. Let $f\in \mathcal{S}_{p}^{\ast }\left(L, N, q, b\right)$ and of the form (1.1). Then

$\begin{eqnarray*} \quad\quad \left \vert a_{p+2}-\mu a_{p+1}^{2}\right \vert \leq\frac{[p]_{q}\left \vert b\right \vert \left( L-N\right) \left \vert \tau \right \vert }{8}\max \left \{ 2, \left \vert \tau \left( -\left( L-N\right) [p]_{q}b+N-5+2[p]_{q}b\left( L-N\right) \mu \right) \right \vert \right\}. \end{eqnarray*}$

This result is sharp.

Proof. Since $f\in \mathcal{S}_{p}^{\ast }\left(L, N, q, b\right),$ we have

$\begin{equation*} 1+\frac{1}{b}\left( \frac{1}{[p]_{q}}\frac{zD_{q}f^{^{}}(z)}{f(z)} -1\right) = \overline{p}\left( L, N;w(z)\right), \ \ z\in \mathcal{U}, \end{equation*}$

where $w$ is Schwarz function such that $w(0)$ and $|w(z)| < 1$ in $\mathcal{U}$ . Therefore

$\begin{eqnarray*} 1+\frac{1}{b}\left( \frac{1}{[p]_{q}}\frac{zD_{q}f^{^{}}(z)}{f(z)} -1\right) & = &h(z), \\ 1+\frac{1}{[p]_{q}b}\frac{zD_{q}f^{^{}}(z)}{f(z)} & = &\frac{1}{b}+\left( 1+h_{1}z+h_{2}z^{2}+\cdots\right) , \\ zD_{q}f^{^{}}(z) & = &[p]_{q}bf(z)\left( \frac{1}{b}+h_{1}z+h_{2}z^{2}+\cdots \right), \end{eqnarray*}$

and after some simple calculation, we have

$\begin{eqnarray*} [p]_{q}z^{p}&+&[p+1]_{q}a_{p+1}z^{p+1}+\left[ p+2\right] _{q}a_{p+2}z^{p+2}+\cdots \\ & = &[p]_{q}b\left \{ z^{p}+a_{p+1}z^{p+1}+a_{p+2}z^{p+2}+\cdots\right \} \left( \frac{1}{b}+h_{1}z+h_{2}z^{2}+\cdots\right) \\ & = &[p]_{q}\left \{ z^{p}+a_{p+1}z^{p+1}+a_{p+2}z^{p+2}+\cdots\right \} \left( 1+bh_{1}z+bh_{2}z^{2}+\cdots\right) . \end{eqnarray*}$

Comparing the coefficients of both sides, we get

$\begin{equation*} a_{p+1} = [p]_{q}bh_{1}, { \ \ \ }2a_{p+2} = [p]_{q}b\left( h_{1}a_{p+1}+h_{2}\right) . \end{equation*}$

This implies that

$\begin{eqnarray*} \left \vert a_{p+2}-\mu a_{p+1}^{2}\right \vert & = &\frac{[p]_{q}\left \vert b\right \vert }{2}\left \vert h_{2}+\left( 1-2\mu \right) [p]_{q}bh_{1}^{2}\right \vert \\ & = &\frac{[p]_{q}\left \vert b\right \vert }{2}\left \vert h_{2}-vh_{1}^{2}\right \vert , \end{eqnarray*}$

where

$\begin{equation*} v = \left( 2\mu -1\right) \left[ p\right] _{q}b. \end{equation*}$

By using (iv) of Lemma 1 for

$v = \left( 2\mu -1\right) [p]_{q}b,$

we have the required result. The equality

$\begin{equation*} \left \vert a_{p+2}-\mu a_{p+1}^{2}\right \vert = \frac{[p]_{q}\left \vert b\right \vert \left( L-N\right) \left \vert \tau \right \vert ^{2}}{8}\left \vert \left( L-N\right) [p]_{q}b-N+5-2[p]_{q}b\left( L-N\right) \mu \right \vert \end{equation*}$

holds for $f_{\ast }$ given in (3.11). Consider $f_{0}$ : $\mathcal{U} \rightarrow \mathbb{C}$ defined as:

$\begin{equation} f_{0}(z) = z^{p}\exp \left( [p]_{q}b\int \limits_{0}^{z}\frac{\overline{p} \left( L, N;t^{2}\right) -1}{t}dt\right) = z^{p}+\frac{[p]_{q}b\tau }{2}\left( L-N\right) z^{p+2}+\cdots, \end{equation}$

(3.12)

where, $\overline{p}\left(L, N;z\right)$ is defined in ( $1.4$ ). Hence

$\begin{equation*} 1+\frac{1}{b}\left( \frac{1}{[p]_{q}}\frac{zD_{q}f^{^{}}(z)}{f(z)} -1\right) = \overline{p}\left( L, N;z^{2}\right) . \end{equation*}$

This demonstrates $f_{0}\in \mathcal{S}_{p}^{\ast }\left(L, N, q, b\right)$ . Hence the equality

$\begin{equation*} \left \vert a_{p+2}-\mu a_{p+1}^{2}\right \vert = \frac{[p]_{q}\left \vert b\right \vert \left( L-N\right) \left \vert \tau \right \vert }{2} \end{equation*}$

holds for the function $f_{0}$ given in (3.12). □

Letting $q\rightarrow 1^{-},$ $b = 1$ and $p = 1$ in Theorem 7, we get the known corollary proved in ^[32] for starlike functions associated with cardioid domain.

Corollary 2. ^[32] Let $f\in \mathcal{S}^{\ast }\left(L, N\right)$ and of the form (1.2). Then

$\begin{equation*} \left \vert a_{3}-\mu a_{2}^{2}\right \vert \leq \frac{\left( L-N\right) \left \vert \tau \right \vert }{8}\max \left \{ 2, \left \vert \tau \left( -\left( L-2N+5\right) +2\left( L-N\right) \mu \right) \right \vert \right \} . \end{equation*}$

This result is sharp.

Coefficient inequality for the class $\mathcal{S}_{p}^{\ast }\left(L, N, b\right)$ :

Theorem 8. For function $f\in \mathcal{A}_{p},$ given by (1.1), if $f\in \mathcal{S}_{p}^{\ast }\left(L, N, q, b\right),$ then

$\begin{equation*} \left \vert a_{p+n}\right \vert \leq \frac{\prod\limits_{k = 2}^{n+1}\left( \left[ k-2\right] _{q}+\frac{\left[ p\right] _{q}}{q^{p}}\left \vert b\left( \left( L-N\right) \frac{\tau }{2}\right) \right \vert \right) }{\left[ n \right] _{q}!}, ~~ \left( ~~ p, n\in \mathbb{N} \right) . \end{equation*}$

Proof. Suppose $f\in \mathcal{S}_{p}^{\ast }\left(L, N, q, b\right)$ and the function $S(z)$ define by

$\begin{equation} S(z) = 1+\frac{1}{b}\left( \frac{1}{[p]_{q}}\frac{zD_{q}f^{^{}}(z)}{f(z)} -1\right) . \end{equation}$

(3.13)

Then by Definition 5, we have

$\begin{equation*} S(z)\prec \overline{p}\left( L, N;z\right) , \end{equation*}$

where, $b\in \mathbb{C} \backslash \{0\}$ and $\overline{p}\left(L, N;z\right)$ is given by (1.4). Hence, applying the Lemma 4, we get

$\begin{equation} \left \vert \frac{S^{(m)}(0)}{m!}\right \vert = \left \vert c_{m}\right \vert \leq \left \vert \overline{Q}_{1}\right \vert , \ \ m\in \mathbb{N} , \end{equation}$

(3.14)

where

$\begin{equation*} S(z) = 1+c_{1}z+c_{2}z^{2}+\cdots, \end{equation*}$

and by (2.1), we have

$\begin{equation} \left \vert \overline{Q}_{1}\right \vert = \left \vert \left( L-N\right) \frac{\tau }{2}\right \vert . \end{equation}$

(3.15)

Also from (3.13), we find

$\begin{equation} zD_{q}f(z) = [p]_{q}\left \{ b\left[ q(z)-1\right] +1\right \} f(z). \end{equation}$

(3.16)

Since $a_{p} = 1,$ in view of (3.16), we obtain

$\begin{align} \left[ n+p\right] _{q}-\left[ p\right] _{q}a_{p+n}& = [p]_{q}b\left \{ c_{n}+c_{n-1}a_{p+1}+\cdots+c_{1}a_{p+n-1}\right \}\\& = b[p]_{q}\sum \limits_{i = 1}^{n}c_{i}a_{p+n-i}. \end{align}$

(3.17)

Applying (3.14) into (3.17), we get

$\begin{equation*} q^{p}\left[ n\right] _{q}\left \vert a_{p+n}\right \vert \leq \left[ p\right] _{q}\left \vert b\right \vert \left \vert \overline{Q}_{1}\right \vert \sum \limits_{i = 1}^{n}\left \vert a_{p+n-i}\right \vert , {\ \ \ }p, n\in \mathbb{N} . \end{equation*}$

For $n = 1, 2, 3$ , we have

$\begin{eqnarray*} \left \vert a_{p+1}\right \vert &\leq &\frac{\left[ p\right] _{q}}{q^{p}} \left \vert b\overline{Q}_{1}\right \vert , \\ \left \vert a_{p+2}\right \vert &\leq &\frac{\left[ p\right] _{q}\left \vert b\overline{Q}_{1}\right \vert }{q^{p}\left[ 2\right] _{q}}\left( 1+\left \vert a_{p+1}\right \vert \right) \\ &\leq &\frac{\left[ p\right] _{q}\left \vert b\overline{Q}_{1}\right \vert }{ q^{p}\left[ 2\right] _{q}}\left( 1+\frac{\left[ p\right] _{q}}{q^{p}}\left \vert b\overline{Q}_{1}\right \vert \right) \end{eqnarray*}$

and

$\begin{eqnarray*} \left \vert a_{p+3}\right \vert &\leq &\frac{\left[ p\right] _{q}\left \vert b\overline{Q}_{1}\right \vert }{q^{p}\left[ 3\right] _{q}}\left( 1+\left \vert a_{p+1}\right \vert +\left \vert a_{p+2}\right \vert \right) \\ &\leq &\frac{\left[ p\right] _{q}\left \vert b\overline{Q}_{1}\right \vert }{ q^{p}\left[ 3\right] _{q}}\left( 1+\frac{\left[ p\right] _{q}}{q^{p}}\left \vert b\overline{Q}_{1}\right \vert +\frac{\left[ p\right] _{q}\left \vert b \overline{Q}_{1}\right \vert }{q^{p}\left[ 2\right] _{q}}\left( 1+\frac{ \left[ p\right] _{q}}{q^{p}}\left \vert b\overline{Q}_{1}\right \vert \right) \right) \\ & = &\frac{\left[ p\right] _{q}}{q^{p}}\left \vert b\overline{Q}_{1}\right \vert \left( \frac{\left( 1+\frac{\left[ p\right] _{q}}{q^{p}}\left \vert b \overline{Q}_{1}\right \vert \right) \left( \left[ 2\right] _{q}+\frac{\left[ p\right] _{q}}{q^{p}}\left \vert b\overline{Q}_{1}\right \vert \right) }{ \left[ 3\right] _{q}\left[ 2\right] _{q}}\right) , \end{eqnarray*}$

respectively. Applying the equality (3.15) and using the mathematical induction principle, we obtain

$\begin{eqnarray*} \left \vert a_{p+n}\right \vert &\leq &\frac{\prod \limits_{k = 2}^{n+1}\left( \left[ k-2\right] _{q}+\frac{\left[ p\right] _{q}}{ q^{p}}\left \vert b\overline{Q}_{1}\right \vert \right) }{\left[ n\right] _{q}!} \\ & = &\frac{\prod \limits_{k = 2}^{n+1}\left( \left[ k-2\right] _{q}+\frac{\left[ p\right] _{q}}{q^{p}}\left \vert b\left( \left( L-N\right) \frac{\tau }{2} \right) \right \vert \right) }{\left[ n\right] _{q}!}. \end{eqnarray*}$

This evidently completes the proof of Theorem 8. □

The initial coefficient bounds $\left\vert a_{p+1}\right\vert$ and $\left\vert a_{p+2}\right\vert$ for the functions $f\in \mathcal{K} _{p}\left(L, N, q, b\right)$ are investigated in Theorem 9 using the Lemma 2.

Theorem 9. Let $f\in \mathcal{K}_{p}\left(L, N, q, b\right)$ be given by (1.1) $,$ $-1\leq N < L\leq 1.$ Then

$\begin{eqnarray*} \left \vert a_{p+1}\right \vert &\leq &\frac{\left[ p\right] _{q}^{2}\left \vert b\right \vert \left( L-N\right) \tau }{2\left[ p+1\right] _{q}}, \\ \left \vert a_{p+2}\right \vert &\leq &\frac{b^{2}\left[ p\right] _{q}\left( L-N\right) \tau }{8\left[ p+2\right] _{q}}\left( 5-N+\left[ p\right] _{q}b^{2}(L-N)\right) . \end{eqnarray*}$

These bounds are sharp.

Proof. Let $f\in \mathcal{K}_{p}\left(L, N, q, b\right)$ , and be of the form (1.1). Then

$\begin{equation} 1-\frac{1}{b}+\frac{1}{b\left[ p\right] _{q}}\left( 1+\frac{zD_{q}^{2}f(z)}{ D_{q}f(z)}\right) \prec \overline{p}\left( L, N;z\right) , \end{equation}$

(3.18)

where

$\begin{equation*} \overline{p}\left( L, N, z\right) = \frac{2L\tau ^{2}z^{2}+\left( L-1\right) \tau z+2}{2N\tau ^{2}z^{2}+\left( N-1\right) \tau z+2}. \end{equation*}$

By applying the concepts of subordination, there exists a function $w$ with

$\begin{equation*} w(0) = 0\; \text{ and }\; |w(z)| < 1, \end{equation*}$

such that

$\begin{equation} 1-\frac{1}{b}+\frac{1}{b\left[ p\right] _{q}}\left( 1+\frac{zD_{q}^{2}f(z)}{ D_{q}f(z)}\right) = \overline{p}\left( L, N;w(z)\right). \end{equation}$

(3.19)

Let

(3.20)

Since

$\overline{p}\left( L, N;z\right) = 1+\sum \limits_{n = 1}^{\infty } \overline{Q}_{n}z^{n},$

then

(3.21)

Also consider the function

$\begin{equation*} \overline{p}\left( L, N;z\right) = \frac{2L\tau ^{2}z^{2}+\left( L-1\right) \tau z+2}{2N\tau ^{2}z^{2}+\left( N-1\right) \tau z+2}. \end{equation*}$

Let $\tau z = \alpha _{0}.$ Then

This implies that

$\begin{equation} \overline{p}\left( L, N;z\right) = 1+\frac{1}{2}\left( L-N\right) \tau z+\frac{ 1}{4}\left( L-N\right) \left( 5-N\right) \tau ^{2}z^{2}+\cdots. \end{equation}$

(3.22)

It is simple to observe from (3.21) that

$\begin{aligned} \overline{p}\left( L, N;w(z)\right) = 1+\frac{1}{4}\left( L-N\right)\tau c_{1}z+\left(\frac{1}{4}\left( L-N\right)\tau \left( c_{2}-\frac{1}{2}c_{1}^{2}\right) +\frac{\left( L-N\right)\left( 5-N\right)\tau ^{2}c_{1}^{2}}{16}\right) z^{2}+\cdots. \end{aligned}$

(3.23)

Since $f\in \mathcal{K}_{p}\left(L, N, q, b\right),$ then

$\begin{aligned} 1-\frac{1}{b}+\frac{1}{b\left[ p\right] _{q}}\left( 1+\frac{zD_{q}^{2}f(z) }{D_{q}f(z)}\right) & = 1+\frac{\left[ p+1\right] _{q}}{b\left[ p\right] _{q}}a_{p+1}z+\frac{1}{b \left[ p\right] _{q}}\left( 2\left[ p+2\right] _{q}a_{p+2}-\frac{\left[ p+1 \right] _{q}^{2}}{\left[ p\right] _{q}}a_{p+1}^{2}\right) z^{2} +\cdots.\end{aligned}$

(3.24)

It is simple to show that by utilizing (3.19) and comparing the coefficients from (3.23) and (3.24), we get

$\begin{equation*} a_{p+1} = \frac{b\left[ p\right] _{q}^{2}\left( L-N\right) \tau c_{1}}{4\left[ p+1\right] _{q}}. \end{equation*}$

Applying modulus on both side, we have

$\begin{equation*} \left \vert a_{p+1}\right \vert \leq \frac{\left[ p\right] _{q}^{2}\left \vert b\right \vert \left( L-N\right) \tau }{2\left[ p+1\right] _{q}}. \end{equation*}$

Now again comparing the coefficients from (3.7) and (3.8), we have

$\begin{eqnarray*} &&\frac{2\left[ p+2\right] _{q}}{b\left[ p\right] _{q}}a_{p+2} = \frac{1}{4} \left( L-N\right) \tau \left( c_{2}-\frac{1}{2}c_{1}^{2}\right) +\frac{ \left( L-N\right) \left( 5-N\right) \tau ^{2}c_{1}^{2}}{16}+\frac{\left( p+1\right) ^{2}}{bp^{2}}a_{p+1}^{2}, \end{eqnarray*}$

$\begin{eqnarray*} \left \vert a_{p+2}\right \vert = \frac{b\left[ p\right] _{q}\left( L-N\right) \tau }{8\left[ p+2\right] _{q}}\left \vert c_{2}-\frac{v}{2} c_{1}^{2}\right \vert, \qquad\qquad\qquad\ \ \ \end{eqnarray*}$

where

$\begin{equation*} v = 1-\frac{\tau }{2}\left( 5-N+\left[ p\right] _{q}b^{2}(L-N)\right) , \end{equation*}$

it shows that $v > 2$ which is satisfied by the relation $L > N.$ Hence, by applying Lemma 2, we obtain the required result.

Result is sharp for the function

$\begin{eqnarray} f_{\ast }(z) & = &z^{p}\exp \left( b\left[ p\right] _{q}^{2}\int \limits_{0}^{z}\frac{\overline{p}\left( L, N, t\right) -1}{t}dt\right) \\ & = &z^{p}+\frac{b\left[ p\right] _{q}^{2}\left( L-N\right) \tau }{2}z^{p+1}+ \frac{b\left[ p\right] _{q}^{2}\left( L-N\right) \left( 5-N\right) \tau ^{2} }{8}z^{p+2}+\cdots, \end{eqnarray}$

(3.25)

where $\overline{p}\left(L, N, \right)$ defined in (1.4). □

Fekete-Szegö problem $\left\vert a_{p+2}-\mu a_{p+1}^{2}\right\vert$ for the functions $f\in \mathcal{K}_{p}\left(L, N, q, b\right)$ are investigated in Theorem 10.

Theorem 10. Let $f\in \mathcal{K}_{p}\left(L, N, q, b\right)$ and be of the form (1.1). Then

$\begin{eqnarray*} \left \vert a_{p+2}-\mu a_{p+1}^{2}\right \vert &\leq &\frac{\left[ p\right] _{q}\left \vert b\right \vert \left( L-N\right) \left \vert \tau \right \vert }{4(\left[ p+1\right] _{q}-\left[ p\right] _{q}+1)\left[ p+2\right] _{q}} \\ &\times & \max \left \{ 2, \left \vert \tau \left( \begin{array}{c} -\left( L-N\right) \left[ p\right] _{q}^{2}b+N-5 \\ \\ +\frac{\left[ p\right] _{q}^{3}b\left[ p+2\right] _{q}\left( \left[ p+1 \right] _{q}-\left[ p\right] _{q}+1\right) \left( L-N\right) }{\left[ p+1 \right] _{q}^{2}}\mu \end{array} \right) \right \vert \right \} . \end{eqnarray*}$

This result is sharp.

Proof. Since $f\in \mathcal{K}_{p}\left(L, N, b\right),$ we have

$\begin{equation*} 1-\frac{1}{b}+\frac{1}{b\left[ p\right] _{q}}\left( 1+\frac{zD_{q}^{2}f(z)}{ D_{q}f(z)}\right) = \overline{p}\left( L, N;w(z)\right), {\ \ \ }z\in \mathcal{ U}, \end{equation*}$

where $w$ is Schwarz function such that $w(0)$ and $|w(z)| < 1$ in $\mathcal{U}$ . Therefore

$\begin{eqnarray*} 1-\frac{1}{b}+\frac{1}{b\left[ p\right] _{q}}\left( 1+\frac{zD_{q}^{2}f(z)}{ D_{q}f(z)}\right) & = &h(z), \\ 1+\frac{zD_{q}^{2}f(z)}{D_{q}f(z)} & = &\left( \frac{1}{b}-1\right) b\left[ p \right] _{q}+b\left[ p\right] _{q}\left( 1+h_{1}z+h_{2}z^{2}+\cdots\right) , \\ 1+\frac{zD_{q}^{2}f(z)}{D_{q}f(z)} & = &\left( \left[ p\right] _{q}+b\left[ p \right] _{q}h_{1}z+b\left[ p\right] _{q}h_{2}z^{2}+\cdots\right), \\ 1+\frac{zD_{q}^{2}f(z)}{D_{q}f(z)} & = &\left[ p\right] _{q}\left[ 1+bh_{1}z+bh_{2}z^{2}+\cdots\right], \end{eqnarray*}$

and after some simple calculation, we have

$\begin{eqnarray*} &&\left[ p\right] _{q}\left( 1+\left[ p-1\right] _{q}\right) z^{p-1}+(\left[ p+1\right] _{q}\left( \left[ p\right] _{q}+1\right) a_{p+1}z^{p} +\left[ p+2\right] _{q}\left( \left[ p+1\right] _{q}+1\right) a_{p+2}z^{p+1}+\cdots \\ && = \left[ p\right] _{q}\left \{ \left[ p\right] _{q}z^{p-1}+\left[ p+1\right] _{q}a_{p+1}z^{p}+\left[ p+2\right] _{q}a_{p+2}z^{p+1}+\cdots\right \} \left( 1+bh_{1}z+bh_{2}z^{2}+\cdots\right) \\ && = \left[ p\right] _{q}^{2}z^{p-1}+\left( \left[ p\right] _{q}(\left[ p+1 \right] _{q}a_{p+1}+\left[ p\right] _{q}^{2}bh_{1}\right) z^{p} \\ &&+\left \{ \left[ p\right] _{q}\left[ p+2\right] _{q}a_{p+2}+\left[ p\right] _{q}\left[ p+1\right] _{q}bh_{1}a_{p+1}+\left[ p\right] _{q}^{2}bh_{2}\right \} z^{p+1}. \end{eqnarray*}$

Comparing the coefficients of both sides, we get

$\begin{equation*} a_{p+1} = \frac{\left[ p\right] _{q}^{2}bh_{1}}{\left[ p+1\right] _{q}}, \ \ \ a_{p+2} = \frac{\left[ p\right] _{q}b}{\left[ p+2\right] _{q}\left( \left[ p+1\right] _{q}-\left[ p\right] _{q}+1\right) }\left( \left[ p+1 \right] _{q}h_{1}a_{p+1}+h_{2}\right) . \end{equation*}$

This implies that

$\begin{eqnarray*} a_{p+2}-\mu a_{p+1}^{2} & = &\frac{\left[ p\right] _{q}\left \vert b\right \vert }{(\left[ p+1\right] _{q}-\left[ p\right] _{q}+1)\left[ p+2\right] _{q}}\\ &\times& \left( h_{2}+\left( 1-\frac{\left[ p+2\right] _{q}\left[ p\right] _{q}\left( \left[ p+1\right] _{q}-\left[ p\right] _{q}+1\right) }{\left[ p+1 \right] _{q}^{2}}\mu \right) \left[ p\right] _{q}^{2}bh_{1}^{2}\right) \\ & = &\frac{\left[ p\right] _{q}\left \vert b\right \vert }{(\left[ p+1\right] _{q}-\left[ p\right] _{q}+1)\left[ p+2\right] _{q}}\left( h_{2}-vh_{1}^{2}\right) , \end{eqnarray*}$

where

$\begin{equation*} v = \left( \frac{\left[ p+2\right] _{q}\left[ p\right] _{q}\left( \left[ p+1 \right] _{q}-\left[ p\right] _{q}+1\right) }{\left[ p+1\right] _{q}^{2}}\mu -1\right) \left[ p\right] _{q}^{2}b. \end{equation*}$

By using (iv) of Lemma 1 for

$v = \left( \frac{\left[ p+2\right] _{q} \left[ p\right] _{q}\left( \left[ p+1\right] _{q}-\left[ p\right] _{q}+1\right) }{\left[ p+1\right] _{q}^{2}}\mu -1\right) \left[ p\right] _{q}^{2}b,$

we have the required result. The equality

$\begin{eqnarray*} \left \vert a_{p+2}-\mu a_{p+1}^{2}\right \vert & = &\frac{\left[ p\right] _{q}\left \vert b\right \vert \left( L-N\right) \left \vert \tau \right \vert ^{2}}{4(\left[ p+1\right] _{q}-\left[ p\right] _{q}+1)\left[ p+2\right] _{q}}\\&\times& \left \vert \left( L-N\right) \left[ p\right] _{q}^{2}b-N+5-\frac{\left[ p \right] _{q}^{3}b\left( \left[ p+1\right] _{q}-\left[ p\right] _{q}+1\right) \left[ p+2\right] _{q}\left( L-N\right) }{\left[ p+1\right] _{q}^{2}}\mu \right \vert \end{eqnarray*}$

holds for $f_{\ast }$ given in (3.25). Consider the function $f_{0}$ : $\mathcal{U}\rightarrow \mathbb{C}$ be defined as:

$\begin{eqnarray*} f_{0}(z)& = &z^{p}\exp \left( \left[ p\right] _{q}^{2}b\int \limits_{0}^{z}\frac{ \overline{p}\left( L, N;t^{2}\right) -1}{t}dt\right)\\ & = &z^{p}+\frac{\tau \left[ p\right] _{q}^{2}b}{2}\left( L-N\right) z^{p+2}+\cdots, \end{eqnarray*}$

where, $\overline{p}\left(L, N;z\right)$ is defined in ( $1.4$ ). Hence and

$\begin{equation*} 1-\frac{1}{b}+\frac{1}{b\left[ p\right] _{q}}\left( 1+\frac{zD_{q}^{2}f(z)}{ D_{q}f(z)}\right) = \overline{p}\left( L, N;z^{2}\right) . \end{equation*}$

□

Theorem 11. Let $f\in \mathcal{A}_{p},$ be given by (1.1). If $f\in \mathcal{K}_{p}\left(L, N, b\right),$ then

$\begin{equation*} \left \vert a_{p+n}\right \vert \leq \frac{\left[ p\right] _{q}\prod \limits_{k = 2}^{n+1}\left( \left[ k-2\right] _{q}+\left[ 2\right] _{q}\frac{ \left[ p\right] _{q}}{q^{p}}\left \vert b\left( \left( L-N\right) \frac{\tau }{2}\right) \right \vert \right) }{\left[ n\right] _{q}!\left( \left[ p+n \right] _{q}\right) }, \ \ \ p, n\in \mathbb{N} . \end{equation*}$

Proof. We can obtain Theorem 11, by using the same technique of Theorem 8. □

4. Conclusions

In this article, we have used the ideas of cardioid domain, multivalent analytic functions, and $q$ -calculus operator theory to define the new subfamilies of multivalent $q$ -starlike and $q$ -convex functions. In Section $1$ , we discussed some basic concepts from geometric functions, analytic functions, multivalent functions, $q$ -calculus operator theory, and the idea of the cardioid domain. We also define two new classes of $p$ -valent starlike, convex functions connected with the cardioid domain using the $q$ -difference operator. The already known lemmas are presented in Section $2$ . In Section $3$ , for the class $\mathcal{S}_{p}^{\ast }\left(L, N, q, b\right)$ , we investigated sharp coefficient bounds, Fekete-Szegö functional, and coefficient inequalities. Same type of results also studied for the class $\mathcal{S}_{p}^{\ast }\left(L, N, q, b\right)$ . The research also demonstrated how the parameters, including some new discoveries, expand and enhance the results.

For future studies, researchers can use a number of ordinary differential and $q$ -analogous of difference and integral operators and can define a number of new subclasses of multivalent functions. By applying the ideas of this article, many new results can be found. The idea presented in this article can be implemented on papers ^[33,34,35], and researchers can discuss the new properties of multivalent functions associated with the cardioid domain.

Use of AI tools declaration

The authors declare that they did not employ any artificial intelligence in the execution of this work.

Acknowledgments

The authors extend their appreciation to the Arab Open University for funding this work through research fund No. (AOURG-2023-007).

Conflict of interest

All the authors claim to have no conflicts of interest.

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AIMS Mathematics

Local well-posedness to the thermal boundary layer equations in Sobolev space

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3. Main results

4. Conclusions

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Local well-posedness to the thermal boundary layer equations in Sobolev space

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