Applications of a q-Differential Operator to a Class of Harmonic Mappings Defined by q-Mittag–Leffler Functions

Abstract

Many diverse subclasses of analytic functions, q-starlike functions, and symmetric q-starlike functions have been studied and analyzed by using q-analogous values of integral and derivative operators. In this paper, we define a q-analogous value of differential operators for harmonic functions with the help of basic concepts of quantum (q-) calculus operator theory; and introduce a new subclass of harmonic functions associated with the Janowski and q-Mittag–Leffler functions. We obtain several useful properties of the new class, such as necessary and sufficient conditions, criteria for analyticity, compactness, convexity, extreme points, radii of starlikeness, radii of convexity, distortion bounds, and integral mean inequality. Furthermore, we discuss some applications of this study in the form of some results and examples and highlight some known corollaries for verifying our main results presented in this investigation. Finally, the conclusion section summarizes the fact about the $(p,q)$-variations.

1. Introduction

Every complex-valued harmonic function g can be expressed as the following if it is harmonic in a domain $D\subset \mathbb{C}$ containing the origin,

$$g\left(\xi \right)=k\left(\xi \right)+\overline{l\left(\xi \right)},$$

(1)

where the function’s k is called the analytic part and l is called the co-analytic part of the harmonic function $g,$ respectively. In particular, if the co-analytic part $l=0$ then the harmonic function g reduces to an analytic function.

The Jacobian value of a harmonic function $g=u+iv$ is given by

$${\mathcal{J}}_g\left(\xi \right)=\left|\begin{array}{cc}{u}_x& v_x\\ u_y& v_y\end{array}\right|=u_x{v}_y-v_x{u}_y,$$

which can be also defined in the form of derivatives with respect to $\xi$ and the conjugate $\overline{\xi }$,

$${\mathcal{J}}_g\left(\xi \right)={\left|g_{\xi }\right|}^2-{\left|g_{\overline{\xi }}\right|}^2={\left|k^{{}^{\prime }}\left(\xi \right)\right|}^2-{\left|l^{{}^{\prime }}\left(\xi \right)\right|}^2,\xi \in D.$$

(2)

It is well known that if g is analytic in E, then

$$g_{\overline{\xi }}=0\mathrm{and}{g}_{\xi }=g^{{}^{\prime }}\left(\xi \right).$$

Ponnusamy and Silverman [1] showed that an analytic function g is locally univalent at a point ${\xi }_0$ if and only if ${\mathcal{J}}_g\left(\xi \right)\ne 0$ in D. The converse of this theorem was proved by Lewy [2] in 1936 and this theorem is also true for harmonic mappings. Therefore, a harmonic function g is sense-preserving and locally univalent if and only if

$$\left|k^{{}^{\prime }}\left(\xi \right)\right|>\left|l^{{}^{\prime }}\left(\xi \right)\right|,\xi \in D.$$

The class of harmonic functions in the open unit disk $E=E\left(1\right),$ where

$$E\left(r\right):=\left\{\xi \in \mathbb{C}:\left|\xi \right|<r\right\},$$

is denoted by $\mathcal{H},$ and ${\mathcal{H}}_0$ denotes the class of functions $g\in \mathcal{H}$ satisfying the normalization conditions

$$g\left(0\right)=g_{\xi }^{{}^{\prime }}\left(0\right)=g^{{}^{\prime }}\left(0\right)-1=0.$$

Therefore the analytic functions k and l given by (1) can be written in the form

$$k\left(\xi \right)=\xi +\sum _{n=2}^{\infty }{a}_n{\xi }^n\mathrm{and}l\left(\xi \right)=\sum _{n=1}^{\infty }{b}_n{\xi }^n,(\xi \in E).$$

Thus,

$$g\left(\xi \right)=\xi +\sum _{n=2}^{\infty }{a}_n{\xi }^n+\overline{\sum _{n=1}^{\infty }{b}_n{\xi }^n},\left(\left|b_1\right|<1,\xi \in E\right).$$

(3)

Let ${\mathcal{S}}_{\mathcal{H}}$ denote the class of harmonic functions $g\in {\mathcal{H}}_0$, which are univalent and sense-preserving in E and can be defined by

$${\mathcal{S}}_{\mathcal{H}}=\left\{g\in {\mathcal{H}}_0:g\mathrm{is}\mathrm{univalent}\mathrm{and}\text{sense-preserving}\mathrm{in}E\right\}.$$

Obviously, if the co-analytic part $l\left(\xi \right)=0$ in $E,$ then ${\mathcal{S}}_{\mathcal{H}}$ can be reduced to the class $\mathcal{S}$ of normalized univalent functions (see [3]). In [4,5], Clunie and Sheil-Small investigated the class ${\mathcal{S}}_{\mathcal{H}}$ and some of its subclasses. A harmonic function $g\in {\mathcal{H}}_0$ is said to be harmonically starlike in E if it satisfies

$$Re\frac{D_{\mathcal{H}}\left(g\left(\xi \right)\right)}{g\left(\xi \right)}>0,$$

where

$$D_{\mathcal{H}}\left(g\left(\xi \right)\right)=k^{{}^{\prime }}\left(\xi \right)-\overline{l^{{}^{\prime }}\left(\xi \right)},(\xi \in E).$$

A function $g\in {\mathcal{H}}_0$ is said to be starlike of order ${\alpha }_0$ in $E\left(r\right)$ if

$$\frac{\partial }{\partial t}\left(argg\left(\rho e^{it}\right)\right)>{\alpha }_0,0\le t\le 2\pi ,0<\rho <r<1.$$

(4)

A function $g\in {\mathcal{H}}_0$ is said to be convex of order ${\alpha }_0$ in $E\left(r\right)$ if

$$\frac{\partial }{\partial t}\left(\frac{\partial }{\partial t}\left(argg\left(\rho e^{it}\right)\right)\right)>{\alpha }_0,0\le t\le 2\pi ,0<\rho <r<1.$$

A function g is said to be subordinated to a function h (denoted by $g\prec h)$ if there is a complex-valued function w with $\left|w\left(\xi \right)\right|\le 1$ and $h\left(0\right)=0$ such that

$$g\left(\xi \right)=h\left(w\left(\xi \right)\right),(\xi \in E).$$

If h is univalent in E, we have

$$g\prec h\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}g\left(0\right)=h\left(0\right)$$

and

$$g\left(E\right)\subset h\left(E\right).$$

The convolution of two harmonic functions $g_1$ and $g_2$ in ${\mathcal{H}}_0$ (denoted by $g_1\ast$ $g_2$) is defined by

$$(g_1\ast g_2)\left(\xi \right)=\xi +\sum _{n=2}^{\infty }\left(a_{1n}{a}_{2n}{\xi }^n+\overline{b_{1n}{b}_{2n}{\xi }^n}\right),(\xi \in E)$$

where

$$g_1\left(\xi \right)=\xi +\sum _{n=2}^{\infty }{a}_{1n}{\xi }^n+\overline{\sum _{n=1}^{\infty }{b}_{1n}{\xi }^n}\mathrm{and}{g}_2\left(\xi \right)=\xi +\sum _{n=2}^{\infty }{a}_{2n}{\xi }^n+\overline{\sum _{n=1}^{\infty }{b}_{2n}{\xi }^n}.$$

The idea of circular domain was given by Janowski [6] and defined Janowski function as follows.

Definition 1

([6]). An analytic function $p\left(\xi \right)$ in E with $p\left(0\right)=1$ is called a Janowski function if there exist $-1\le L<M\le 1$ such that

$$p\left(\xi \right)\prec \frac{1+M\xi }{1+L\xi }.$$

Denote $\mathcal{T}\left(M,L\right)$ the class of all related Janowski functions.

Janowski showed that each function $p\in \mathcal{T}\left(M,L\right)$ maps E onto the circular domain $E(M,L)$ with a centre on real axis and the following diameters

$$E_1=\frac{1-M}{1-L}\mathrm{and}{E}_2=\frac{1+M}{1+L},$$

where $0<E_1<1<E_2.$

The Mittag–Leffler function is defined by

$$E_{\alpha }\left(\xi \right)=\sum _{n=0}^{\infty }\frac{{\xi }^n}{\mathrm{\Gamma }\left(\alpha n+1\right)},$$

(5)

where $\alpha \in \mathbb{C},$ $Re\left(\alpha \right)>0,$ and $\mathrm{\Gamma }\left(\xi \right)$ is gamma function.

By introducing a parameter $\beta \in \mathbb{C}$ with $Re\left(\beta \right)>0,$ Wiman [7] defined the generalized Mittag–Leffler functions $E_{\alpha ,\beta }\left(\xi \right):$

$$E_{\alpha ,\beta }\left(\xi \right)=\sum _{n=0}^{\infty }\frac{{\xi }^n}{\mathrm{\Gamma }\left(\alpha n+\beta \right)}.$$

The normalization of $E_{\alpha ,\beta }\left(\xi \right)$ can be performed as follows, denoted by $Q_{\alpha ,\beta }\left(\xi \right),$

$$\begin{array}{ccc}\hfill Q_{\alpha ,\beta }\left(\xi \right)& =& \xi \mathrm{\Gamma }\left(\beta \right)E_{\alpha ,\beta }\left(\xi \right)\hfill \\ & =& \xi +\sum _{n=2}^{\infty }\frac{\mathrm{\Gamma }\left(\beta \right)}{\mathrm{\Gamma }\left(\alpha \left(n-1\right)+\beta \right)}{\xi }^n.\hfill \end{array}$$

(6)

Elhaddad et al. [8] used the normalized Mittag–Leffler function $Q_{\alpha ,\beta }\left(\xi \right)$ and then defined the following differential operators ${\mathfrak{D}}_{\delta }^m\left(\alpha ,\beta \right),$ $m\in {\mathbb{N}}_0=\left\{0,1,2,\dots \right\}$ for the class of analytic functions $g:$

$$\begin{array}{ccc}\hfill {\mathfrak{D}}_{\delta }^m\left(\alpha ,\beta \right)g\left(\xi \right)& =& g\left(\xi \right)\ast Q_{\alpha ,\beta }\left(\xi \right),\hfill \\ \hfill {\mathfrak{D}}_{\delta }^m\left(\alpha ,\beta \right)g\left(\xi \right)& =& \xi +\sum _{n=2}^{\infty }\left({\left(1+\left(n-1\right)\delta \right)}^m\frac{\mathrm{\Gamma }\left(\beta \right)}{\mathrm{\Gamma }\left(\alpha \left(n-1\right)+\beta \right)}\right)a_n{\xi }^n,\hfill \end{array}$$

where $\delta >0.$ In particular, the operator ${\mathfrak{D}}_{\delta }^m\left(\alpha ,\beta \right)$ for harmonic functions $g\in {\mathcal{H}}_0$ was defined by Khan et al. [9] as follows:

$${\mathfrak{D}}_{\delta }^m\left(\alpha ,\beta \right)g\left(\xi \right)={\mathfrak{D}}_{\delta }^m\left(\alpha ,\beta \right)k\left(\xi \right)+\overline{{\mathfrak{D}}_{\delta }^m\left(\alpha ,\beta \right)l\left(\xi \right)},\xi \in E$$

where

$${\mathfrak{D}}_{\delta }^m\left(\alpha ,\beta \right)k\left(\xi \right)=\xi +\sum _{n=2}^{\infty }{\mathrm{\Psi }}_{n,\delta }^{\alpha ,\beta }{a}_n{\xi }^n,$$

$${\mathfrak{D}}_{\delta }^m\left(\alpha ,\beta \right)l\left(\xi \right)=\xi +\sum _{n=2}^{\infty }{\mathrm{\Psi }}_{n,\delta }^{\alpha ,\beta }{b}_n{\xi }^n,$$

and

$${\mathrm{\Psi }}_{n,\delta }^{\alpha ,\beta }={\left(1+\left(n-1\right)\delta \right)}^m\frac{\mathrm{\Gamma }\left(\beta \right)}{\mathrm{\Gamma }\left(\alpha \left(n-1\right)+\beta \right)}.$$

Recently, q-calculus has been extensively used in various areas of mathematics and physics such as fractional calculus, q-difference equation, q-integral equations, as well as in geometric function theory. Initially, Jackson [10] was the first to apply the basic concepts of q-calculus operator theory and introduced the q-derivative and q-integral operator $\left({\partial }_q\right)$ and after that, in the year 1990, Ismail et al. [11] used the q-derivative operator and starlike functions and defined a new class ${\mathcal{S}}^{\ast }$ of q-starlike functions. However, in the article [12] published in 1989, Srivastava provided a firm footing for use of the q-calculus and the basic (or q-) hypergeometric functions $rFs$ $\left(r,s\in {\mathbb{N}}_0\right)$ in the study of geometric function theory (for details, see (p. 347 et seq., [12])).

After the invention of q-derivative and q-integral operator, many researchers investigated the q-analogous of differential operators. In 1914, Kanas and Raducanu [13] defined the q-analogous of the Ruscheweyh differential operator and used it to examine a new class of analytic functions in the conic domain. Aouf and Seoudy [14] defined a new subclass of analytic functions associated with q-analogue of Ruscheweyh operator and found Fekete-Szegö problem. Later on, Arif et al. [15] provided generalizations for the multivalent functions. Zang et al. [16] used the basic concepts of q-calculus operator theory defining a generalized conic domain and then studied a new subclass of q-starlike functions in this conic domain. Many mathematicians have done a lot of work so far in the field of geometric function theory along with q-calculus theory (see [17,18,19,20,21,22]).

Jahangiri [23] utilized the idea of the q-difference operator (${\partial }_q$) on harmonic functions and defined a new subclass of harmonic functions and explored some valuable results. Later, Porwal and Gupta [24] discussed some significant applications of the q-difference operator (${\partial }_q$) for harmonic univalent functions. More recently, Srivastava et al. [25] systematically used the q-analogous of the difference operator along with the technique of convolution and derived a q-derivative operator for meromorphically harmonic functions. Khan et al. [26] established new subclasses of meromorphic functions using the principles of q-analogous value of the difference operator. For more detail about harmonic univalent functions along with the q-calculus operator theory (see [27,28]).

Now we provide some basic definitions and concepts of the q-calculus which is used in this paper. Let $q\in (0,1)$ and $\eta \in \mathbb{C},$ Gasper and Rahman [29] defined q-number,

$$\begin{array}{ccc}\hfill {\left[\eta \right]}_q& =& \frac{1-q^{\eta }}{1-q},{\left[0\right]}_q=0,\hfill \\ & & \\ \hfill {\left[n\right]}_q& =& 1+q+q^2+\dots +q^{n-1},\eta =n\in \mathbb{N}.\hfill \end{array}$$

Let $q\in \left(0,1\right),$ Jackson [10] defined the q-derivative operator $\left({\partial }_q\right)$ for analytic functions g as

$$\begin{array}{ccc}\hfill {\partial }_{q}g\left(\xi \right)& =& \frac{g\left(\xi \right)-g\left(q\xi \right)}{(1-q)\xi },\xi \ne 0,0<q<1,\hfill \\ & =& 1+\sum _{n=2}^{\infty }{\left[n\right]}_q{a}_n{\xi }^{n-1}.\hfill \end{array}$$

(7)

Let the q-derivative operator $\left({\partial }_q\right)$ for a harmonic function $g\in {\mathcal{S}}_{\mathcal{H}}$ be defined by

$$\begin{array}{ccc}\hfill {\partial }_{q}g\left(\xi \right)& =& {\partial }_{q}k\left(\xi \right)+\overline{{\partial }_{q}l\left(\xi \right)},\hfill \\ & & \\ & =& 1+\sum _{n=2}^{\infty }{\left[n\right]}_q{a}_n{\xi }^{n-1}+\sum _{n=2}^{\infty }{\left[n\right]}_q\overline{b_n{\xi }^{n-1}}.\hfill \end{array}$$

We observe that

$${\partial }_{q}k\left(\xi \right)=k\left(\xi \right)\ast {\partial }_q\left(\frac{\xi }{1-\xi }\right),$$

and

$$\begin{array}{ccc}\hfill {\partial }_q\left(\frac{\xi }{1-\xi }\right)& =& \xi +\sum _{n=2}^{\infty }{\left[n\right]}_q{a}_n{\xi }^{n-1}\hfill \\ & =& \frac{\xi }{\left(1-\xi \right)\left(1-q\xi \right)}.\hfill \end{array}$$

The q-Mittag–Leffler function is defined by Sharma and Jain (see [30]) as

$${\mathcal{H}}_{\alpha ,\beta }(\xi ,q)=\sum _{n=0}^{\infty }\frac{1}{{\mathrm{\Gamma }}_q(\alpha n+\beta )}{\xi }^n,\left(\alpha ,\beta \in \mathbb{C},Re\left(\alpha \right),Re\left(\beta \right)\right)>0.$$

(8)

(see, for more details, [31,32]). The normalization of q-Mittag–Leffler function $\mathcal{Q}(\alpha ,\beta ,q)$ can be defined by

$$\begin{array}{ccc}\hfill \mathcal{Q}(\alpha ,\beta ,q)& =& \xi {\mathrm{\Gamma }}_q\left(\beta \right){\mathcal{H}}_{\alpha ,\beta }(\xi ,q)\hfill \\ & & \\ \hfill \mathcal{Q}(\alpha ,\beta ,q)& =& \xi +\sum _{n=2}^{\infty }\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\alpha \left(n-1\right)+\beta )}{\xi }^n.\hfill \end{array}$$

Now we define the following q-analogous of differential operators ${\mathcal{D}}^m(\alpha ,\beta ,q),$$m\in {\mathbb{N}}_0,$ for analytic functions $g,$ by applying for the definition of convolution along with normalized q-Mittag–Leffler function.

$$\begin{array}{ccc}\hfill {\mathcal{D}}^0(\alpha ,\beta ,\delta ,q)g\left(\xi \right)& =& g\left(\xi \right)\ast \mathcal{Q}(\alpha ,\beta ,q), \hfill \\ & & \hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\mathcal{D}}^1(\alpha ,\beta ,\delta ,q)g\left(\xi \right)& =& \left(1-\delta \right)\left(g\left(\xi \right)\ast \mathcal{Q}(\alpha ,\beta ,q)\right)\hfill \\ & & +\delta {\partial }_q\left(g\left(\xi \right)\ast \mathcal{Q}(\alpha ,\beta ,q)\right)\hfill \end{array}$$

(9)

$$\begin{array}{ccc}& & ⋮\hfill \\ \hfill {\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right)& =& {\mathcal{D}}^1\left({\mathcal{D}}^{m-1}(\alpha ,\beta ,\delta ,q)g\left(\xi \right)\right).\hfill \end{array}$$

(10)

After some simple calculation we get the following series form of q-differential operator ${\mathcal{D}}^m(\alpha ,\beta ,q)$,

$${\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right)=\xi +\sum _{n=2}^{\infty }{\mathrm{{\rm Y}}}_{\delta }^q\left(n,\alpha ,\beta \right)a_n{\xi }^n,$$

where

$${\mathrm{{\rm Y}}}_{\delta }^q\left(n,\alpha ,\beta \right)={\left(1+\left({\left[n\right]}_q-1\right)\delta \right)}^m\frac{{\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q(\alpha \left(n-1\right)+\beta )}.$$

(11)

In particular, above newly defined q-analogous of differential operators becomes the following known operators by taking the specific values of parameter $\alpha ,$ $\beta ,\delta ,m,$ and $q.$

(1)

For $q\to 1^{-},$ we get differential operator defined by Elhaddad et al. [8].

(2)

For $q\to 1^{-},$ $\alpha =0,$ and $\beta =1,$ we get Al-Oboudi operator [33].

(3)

For $\alpha =0,$ $\beta =1,$ and $\delta =1,,$ we get q-Salagean operator [34].

(4)

For $q\to 1^{-},$ $\alpha =0,$ $\beta =1,$ and $\delta =1,$ we get Salagean operator [35].

(5)

For $q\to 1^{-},$ and $m=0,$ we get Mittag–Leffler function defined in [36].

Next, we define the following new q-differential operators ${\mathcal{D}}^m(\alpha ,\beta ,\delta ,q),$ $m\in {\mathbb{N}}_0,$ for harmonic functions $g\in \mathcal{H}$, by combining the concepts of q-calculus operator theory and normalized q-Mittag–Leffler function.

$${\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right)={\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)k\left(\xi \right)+\overline{{\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)l\left(\xi \right)},\xi \in E,$$

where

$$\begin{array}{ccc}\hfill {\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)k\left(\xi \right)& =& \xi +\sum _{n=2}^{\infty }{\mathrm{{\rm Y}}}_{\delta }^q\left(n,\alpha ,\beta \right)a_n{\xi }^n,\hfill \\ \hfill {\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)l\left(\xi \right)& =& \xi +\sum _{n=2}^{\infty }{\mathrm{{\rm Y}}}_{\delta }^q\left(n,\alpha ,\beta \right)b_n{\xi }^n,\hfill \end{array}$$

and ${\mathrm{{\rm Y}}}_{\delta }^q\left(n,\alpha ,\beta \right)$ is given by (11).

Remark 1.

For $q\to 1^{-},$ we get differential operator for a function $g\in \mathcal{H}$, defined by Khan et al. in [9].

Definition 2.

Let $G\subset \mathcal{H}.$ A function $g\in G$ is called an extreme point of G if the condition

$$g=\gamma g_1+\left(1-\gamma \right)g_2,\left(g_1,g_2\in G,0<\gamma <1\right)$$

implies $g_1=g_2=g.$ The set of all extreme points of G is denoted by $EG,$ and hence $EG\subset G.$

Definition 3.

Let $0<r<1,$ and there is a real constant $M=M\left(r\right)$ such that

$$\left|g\left(\xi \right)\right|\le M\left(g\in G,\left|\xi \right|\le r\right),$$

then G is locally uniformly bounded and class G is convex if

$$\gamma g+\left(1-\gamma \right)l\in G,\left(g,l\in G,0\le \gamma \le 1\right).$$

The intersection of all closed convex subsets of $\mathcal{H}$ that contain G is called the closed convex hull of G and it is denoted by $\overline{co}G$.

Definition 4.

A real-valued function $\mathcal{J}:\mathcal{H}\to R$ is called convex on a convex class $G\subset \mathcal{H}$ if

$$\mathcal{J}\left(\gamma g+\left(1-\gamma \right)l\right)\le \gamma \mathcal{J}\left(g\right)+\left(1-\gamma \right)\mathcal{J}\left(l\right),\left(g,l\in G,0\le \gamma \le 1\right).$$

On the count of principle of subordination in conjunction with the q-differential operator ${\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)$ for $g\in \mathcal{H}$ and Janowski functions, we define a new class of harmonic univalent functions below.

Definition 5.

Let $-L\le M<L\le 1$ and ${\mathcal{S}}_{\mathcal{H}}^{\alpha ,\beta }(m,\delta ,q,M,L)$ denote the class of functions $g\in {\mathcal{S}}_{\mathcal{H}}$ such that

$$\frac{{\mathcal{D}}_{\mathcal{H}}\left({\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right)\right)}{{\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right)}\prec \frac{1+M\xi }{1+L\xi }.$$

(12)

Equivalently,

$$\left|\frac{{\mathcal{D}}_{\mathcal{H}}\left({\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right)\right)-{\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right)}{L{\mathcal{D}}_{\mathcal{H}}\left({\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right)\right)-M{\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right)}\right|<1,$$

(13)

with

$${\mathcal{D}}_{\mathcal{H}}\left({\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right)\right)={\mathcal{D}}_{\mathcal{H}}\left({\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)k\left(\xi \right)\right)-\overline{{\mathcal{D}}_{\mathcal{H}}\left({\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)l\left(\xi \right)\right)}.$$

This new class gives the following known classes of harmonic univalent functions by taking $q\to 1^{-}$ and the specific values of parameter $m,\delta ,M,$ and $L.$

1.

${\mathcal{S}}_{\mathcal{H}}^{\alpha ,1}(m,\delta ,q\to 1^{-},M,L)=$${\mathcal{S}}_{\mathcal{H}}^{\alpha ,1}(m,\delta ,M,L),$ defined by Khan et al. in [9].

2.

${\mathcal{S}}_{\mathcal{H}}^{0,1}(0,\delta ,q\to 1^{-},M,L)=$${\mathcal{S}}_{\mathcal{H}}^{\ast }(M,L),$ studied by Dziok [37].

3.

${\mathcal{S}}_{\mathcal{H}}^{0,1}(0,\delta ,q\to 1^{-},2a-1,1)=$${\mathcal{S}}_{\mathcal{H}}^{\ast }\left(a\right),$$(0\le a<1),$ defined by Jahangiri in [38].

4.

${\mathcal{S}}_{\mathcal{H}}^{0,1}(1,1,q\to 1^{-},2a-1,1)=$${\mathcal{S}}_{\mathcal{H}}^c\left(a\right),$$(0\le a<1),$ introduced by Jahangiri in [39].

Definition 6.

Using the approach by Ruscheweyh [40], we define the dual set of V by

$$V^{\ast }=\left\{g\in {\mathcal{H}}_0:\left(g\ast h\right)\left(\xi \right)\ne 0forallh\in Vand\xi \in E_0\right\},$$

where

$$V\subset {\mathcal{H}}_{0}and{E}_0=E\backslash \left\{0\right\}.$$

Taking motivation from Silverman [41], we define a class $\tau$ and ${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L)$ for functions $g\in {\mathcal{H}}_0$ of the form (3) as follows:

Definition 7.

Let τ denote the class of a functions $g=k+\overline{l}\in \mathcal{H}$ such that $a_n=-\left|a_n\right|,$ $b_n=\left|b_n\right|,$ $(n\ge 2),$ where

$$\begin{array}{ccc}\hfill k\left(\xi \right)& =& \xi -\sum _{n=2}^{\infty }\left|a_n\right|{\xi }^n,\hfill \\ \hfill l\left(\xi \right)& =& \sum _{n=2}^{\infty }\left|b_n\right|{\overline{\xi }}^n,(\xi \in E).\hfill \end{array}$$

(14)

Further, class ${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L)$ define by

$${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L)=\tau \cap {\mathcal{S}}_{\mathcal{H}}^{\alpha ,\beta }(m,\delta ,q,M,L).$$

Remark 2.

In particular, if $q\to 1^{-},$$\alpha =0$, $\beta =1,$ and $m=0,$ then the class $\mathcal{S}$${}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L)$$=\mathcal{S}$${}_{\mathcal{H},\tau }(m,\delta ,M,L),$ introduced and investigated by Dziok [37].

It is easy to verify that for function $g\in \tau$ the condition (4) is equivalent to the following inequality,

$$Re\frac{{\mathcal{D}}_{\mathcal{H}}g\left(\xi \right)}{g\left(\xi \right)}>{\alpha }_0,\xi \in E\left(r\right).$$

Equivalently,

$$\left|\frac{{\mathcal{D}}_{\mathcal{H}}g\left(\xi \right)-\left(1+{\alpha }_0\right)g\left(\xi \right)}{{\mathcal{D}}_{\mathcal{H}}g\left(\xi \right)+\left(1+{\alpha }_0\right)g\left(\xi \right)}\right|<1\xi \in E\left(r\right).$$

(15)

Definition 8.

Let B be a subclass of the class ${\mathcal{H}}_0$. We define the radius of starlikeness and convexity as follows

$$R_a^{\ast }\left(B\right)=\underset{g\in B}{inf}\left(sup\{r\in (0,1]:g\mathit{is}\mathit{starlike}\mathit{of}\mathit{order}{\alpha }_0\mathit{in}E\left(r\right)\right),$$

and

$$R_a^c\left(B\right)=\underset{g\in B}{inf}\left(sup\{r\in (0,1]:g\mathit{is}\mathit{convex}\mathit{of}\mathit{order}{\alpha }_0\mathit{in}E\left(r\right)\right).$$

2. A Set of Lemmas

We give some lemmas here to investigate the main results in this paper.

Lemma 1

([42]). Let G be a non-empty convex compact subclass of the class $\mathcal{H},$ and let $\mathcal{J}$$:\mathcal{H}\to R$ be a real-valued, continuous and convex function on G. Then

$$max\left\{\mathcal{J}\left(g\right):g\in G\right\}=max\left\{\mathcal{J}\left(g\right):g\in EG\right\}.$$

Lemma 2.

A class $G\mathcal{H}$ is compact if and only if G is closed and locally uniformly bounded.

Lemma 3

([43]). Let G be a non-empty compact subclass of the class $\mathcal{H}$, then $EG$ is non-empty and $\overline{co}EG=\overline{co}G$.

3. Main Results

We derive necessary and sufficient conditions in this section, and then we evaluate some inequality regarding the coefficients of the functions along with some examples for justifications.

Theorem 1.

Let $g\in {\mathcal{H}}_0$ be a harmonic function with the form (3), then $g\in$${\mathcal{S}}_{\mathcal{H}}^{\alpha ,\beta }(m,\delta ,q,M,L)$ if and only if

$${\mathcal{S}}_{\mathcal{H}}^{\alpha ,\beta }(m,\delta ,q,M,L)={\left\{{\mathcal{D}}^m(\alpha ,\beta ,\delta ,q){\phi }_{\varsigma }\left(\xi \right),\left|\varsigma \right|=1\right\}}^{\ast },$$

where

$${\phi }_{\varsigma }\left(\xi \right)=\frac{\xi \left\{1+L\varsigma -\left(1+M\varsigma \right)\left(1-\xi \right)\right\}}{\left(1-\xi \right)\left(1-q\xi \right)}-\frac{\overline{\xi }\left\{1+L\varsigma -\left(1+M\varsigma \right)\left(1-\overline{\xi }\right)\right\}}{\left(1-\overline{\xi }\right)\left(1-q\overline{\xi }\right)},\xi \in E.$$

Proof. 

Let $g\in {\mathcal{H}}_0,$ then $g\in$${\mathcal{S}}_{\mathcal{H}}^{\alpha ,\beta }(m,\delta ,q,M,L)$ if and only if the following holds

$$\frac{{\mathcal{D}}_{\mathcal{H}}\left({\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right)\right)}{{\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right)}\ne \frac{1+M\varsigma }{1+L\varsigma },\varsigma \in C,\left|\varsigma \right|=1.$$

Now as

$${\mathcal{D}}_{\mathcal{H}}\left({\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)k\left(\xi \right)\right)={\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)k\left(\xi \right)\ast \frac{\xi }{\left(1-\xi \right)\left(1-q\xi \right)}$$

and

$${\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)k\left(\xi \right)={\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)k\left(\xi \right)\ast \frac{\xi }{1-\xi }.$$

Thus

$$\begin{array}{ccc}& & \left(1+L\varsigma \right){\mathcal{D}}_{\mathcal{H}}\left({\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right)\right)-\left(1+M\varsigma \right){\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right),\hfill \\ & & \\ & =& \left(1+L\varsigma \right){\mathcal{D}}_{\mathcal{H}}\left({\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)k\left(\xi \right)\right)-\left(1+M\varsigma \right){\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)k\left(\xi \right)\hfill \\ & & \\ & & -\left[\left(1+L\varsigma \right){\mathcal{D}}_{\mathcal{H}}\left(\overline{{\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)l\left(\xi \right)}\right)+\left(1+M\varsigma \right)\overline{{\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)l\left(\xi \right)}\right],\hfill \\ & & \\ & =& {\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)k\left(\xi \right)\ast \left(\frac{\left(1+L\varsigma \right)\xi }{\left(1-\xi \right)\left(1-q\xi \right)}-\frac{\left(1+M\varsigma \right)\xi }{1-\xi }\right)\hfill \\ & & \\ & & -\overline{{\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)l\left(\xi \right)}\ast \left(\frac{\left(1+L\varsigma \right)\overline{\xi }}{\left(1-\overline{\xi }\right)\left(1-q\overline{\xi }\right)}-\frac{\left(1+M\varsigma \right)\overline{\xi }}{1-\overline{\xi }}\right),\hfill \\ & & \\ & =& g\left(\xi \right)\ast {\mathcal{D}}^m(\alpha ,\beta ,\delta ,q){\phi }_{\varsigma }\left(\xi \right)\ne 0.\hfill \end{array}$$

So $g\in$${\mathcal{S}}_H^{\alpha ,\beta }(m,\delta ,q,M,L)$ if and only if $g\left(\xi \right)\ast {\mathcal{D}}^m(\alpha ,\beta ,\delta ,q){\phi }_{\varsigma }\left(\xi \right)\ne 0$ for $\xi \in E_0,\left|\varsigma \right|=1,$ that is

$${\mathcal{S}}_H^{\alpha ,\beta }(m,\delta ,q,M,L)={\left\{{\mathcal{D}}^m(\alpha ,\beta ,\delta ,q){\phi }_{\varsigma }\left(\xi \right),\left|\varsigma \right|=1\right\}}^{\ast }.$$

□

In particular, if $q\to 1^{-}$, then the above Theorem 1 gives the following result.

Corollary 1

([9]). Let $g\in {\mathcal{H}}_0$ be given by (3), then $g\in$${\mathcal{S}}_{\mathcal{H}}^{\alpha ,\beta }(m,\delta ,M,L)$ if and only if

$${\mathcal{S}}_{\mathcal{H}}^{\alpha ,\beta }(m,\delta ,M,L)={\left\{{\mathcal{D}}^m(\alpha ,\beta ,\delta ){\phi }_{\varsigma }\left(\xi \right),\left|\varsigma \right|=1\right\}}^{\ast },$$

where

$${\phi }_{\varsigma }\left(\xi \right)=\frac{\xi \left\{1+L\varsigma -\left(1+M\varsigma \right)\left(1-\xi \right)\right\}}{{\left(1-\xi \right)}^2}-\frac{\overline{\xi }\left\{1+L\varsigma -\left(1+M\varsigma \right)\left(1-\overline{\xi }\right)\right\}}{{\left(1-\overline{\xi }\right)}^2},\xi \in E.$$

3.1. Coefficient Bounds for the Class ${\mathcal{S}}_{\mathcal{H}}^{\alpha ,\beta }(m,\delta ,q,M,L)$ of Harmonic Functions

In the following theorem we present a sufficient coefficient bound for the class ${\mathcal{S}}_{\mathcal{H}}^{\alpha ,\beta }(m,\delta ,q,M,L).$

Theorem 2.

Let $g\in {\mathcal{H}}_0$ of the form (3) and satisfies the condition

$$\sum _{n=2}^{\infty }\left({\mathrm{{\rm Y}}}_n\left|a_n\right|+{\mathrm{\Psi }}_n\left|b_n\right|\right)\le L-M,$$

(16)

with

$$\begin{array}{ccc}\hfill {\mathrm{{\rm Y}}}_n& =& \left|{\mathrm{{\rm Y}}}_{\delta }^q\left(n,\alpha ,\beta \right)\right|\left\{\left(1+L\right){\left[n\right]}_q-\left(1+M\right)\right\},\hfill \\ & & \hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill {\mathrm{\Psi }}_n& =& \left|{\mathrm{{\rm Y}}}_{\delta }^q\left(n,\alpha ,\beta \right)\right|\left\{\left(1+L\right){\left[n\right]}_q+\left(1+M\right)\right\}.\hfill \end{array}$$

(18)

Then g is harmonic univalent in E and $g\in$${\mathcal{S}}_H^{\alpha ,\beta }(m,\delta ,q,M,L)$ if the inequality (16) holds. The equality is held for

$$g\left(\xi \right)=\xi +\sum _{n=2}^{\infty }\frac{L-M}{{\mathrm{{\rm Y}}}_n}{x}_n{\xi }^n+\overline{\sum _{n=1}^{\infty }\frac{L-M}{{\mathrm{\Psi }}_n}{y}_n{\xi }^n},$$

where

$$\sum _{n=2}^{\infty }\left|x_n\right|+\sum _{n=1}^{\infty }\left|y_n\right|=1$$

and ${\mathrm{{\rm Y}}}_{\delta }^q\left(n,\alpha ,\beta \right)$ is given by (11).

Proof. 

First of all, we have to show that $g=k+\overline{l}$ is locally univalent and sense preserving in E. It is enough show that

$$\left|w\right|=\left|l^{{}^{\prime }}/k^{{}^{\prime }}\right|<1\mathrm{in}E.$$

For $\xi =r{e}^{i\theta }\in E$, we have

$$\begin{array}{ccc}\hfill \left|{\partial }_{q}k\left(\xi \right)\right|& \ge & 1-\sum _{n=2}^{\infty }{\left[n\right]}_q\left|a_n\right|r^{n-1}>1-\sum _{n=2}^{\infty }{\left[n\right]}_q\left|a_n\right|\ge 1-\sum \sum _{n=2}^{\infty }\frac{{\mathrm{{\rm Y}}}_n}{L-M}\left|a_n\right|\hfill \\ & \ge & \sum _{n=1}^{\infty }\frac{{\mathrm{\Psi }}_n}{M-L}\left|b_n\right|\ge \sum _{n=1}^{\infty }{\left[n\right]}_q\left|b_n\right|\ge \sum _{n=1}^{\infty }{\left[n\right]}_q\left|b_n\right|r^{n-1}\ge \left|{\partial }_{q}l\left(\xi \right)\right|.\hfill \end{array}$$

If $q\to 1^{-},$ then

$$\left|{\partial }_{q}k\left(\xi \right)\right|\ge \left|{\partial }_{q}l\left(\xi \right)\right|⟹\left|k^{{}^{\prime }}\left(\xi \right)\right|>\left|l^{{}^{\prime }}\left(\xi \right)\right|\mathrm{in}E.$$

Therefore g is locally univalent and sense-preserving in $E.$

To show that $g=k+\overline{l}$ is univalent in E we can use an argument that is due to author [44].

Let ${\xi }_1$,${\xi }_2\in E$ so that ${\xi }_1\ne {\xi }_2.$ Since E is simply connected and convex, we have

$$\xi \left(t\right)=\left(1-t\right){\xi }_1+t{\xi }_2\in E$$

for $0\le t\le 1.$ Then for ${\xi }_1-{\xi }_2\ne 0,$ we can write

$$Re\left(\frac{g\left({\xi }_2\right)-g\left({\xi }_1\right)}{{\xi }_2-{\xi }_1}\right){>}_0^1\left(Re{\partial }_{q}k\left(\xi \left(t\right)\right)-\left|{\partial }_{q}l\left(\xi \left(t\right)\right)\right|\right)dt.$$

On the other hand, we observe that

$$Re\left({\partial }_{q}k\left(\xi \right)\right)-\left|{\partial }_{q}l\left(\xi \right)\right|\ge Re{\partial }_{q}k\left(\xi \right)-\sum _{n=1}^{\infty }{\left[n\right]}_q\left|b_n\right|\ge 1-\sum _{n=2}^{\infty }{\left[n\right]}_q\left|a_n\right|-\sum _{n=1}^{\infty }{\left[n\right]}_q\left|b_n\right|$$

$$\ge 1-\sum _{n=2}^{\infty }\frac{{\mathrm{{\rm Y}}}_n}{L-M}\left|a_n\right|-\sum _{n=1}^{\infty }\frac{{\mathrm{\Psi }}_n}{M-L}\left|b_n\right|\ge 0.$$

Therefore, $g=k+\overline{l}$ is univalent in E and $g\in {\mathcal{H}}_0.$ Therefore $g\in$${\mathcal{S}}_{\mathcal{H}}^{\alpha ,\beta }(m,\delta ,q,M,L)$ if and only if there exists a complex-valued function $w,$ $w\left(0\right)=0,$ $\left|w\left(\xi \right)\right|<1$ ($\xi \in E),$ such that

$$\frac{{\mathcal{D}}_{\mathcal{H}}\left({\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right)\right)}{{\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right)}=\frac{1+Mw\left(\xi \right)}{1+Lw\left(\xi \right)},$$

or equivalently

$$\left|\frac{{\mathcal{D}}_{\mathcal{H}}\left({\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right)\right)-{\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right)}{L{\mathcal{D}}_{\mathcal{H}}\left({\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right)\right)-M{\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right)}\right|<1.$$

Thus, it is sufficient to prove that

$$\begin{array}{ccc}& & \left|{\mathcal{D}}_{\mathcal{H}}\left({\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right)\right)-{\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right)\right|\hfill \\ & & \\ & & -\left|L{\mathcal{D}}_{\mathcal{H}}\left({\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right)\right)-M{\mathcal{D}}^m(\alpha ,\beta ,\delta ,q)g\left(\xi \right)\right|\hfill \\ & =& \left|\begin{array}{c}\sum _{n=2}^{\infty }{\mathrm{{\rm Y}}}_{\delta }^q\left(n,\alpha ,\beta \right)\left({\left[n\right]}_q-1\right)a_n{\xi }^n\\ -\sum _{n=2}^{\infty }{\mathrm{{\rm Y}}}_{\delta }^q\left(n,\alpha ,\beta \right)\left({\left[n\right]}_q+1\right)\overline{b_n{\xi }^n}\end{array}\right|\hfill \\ & & \\ & & -\left|\begin{array}{c}\left(L-M\right)\xi +\sum _{n=2}^{\infty }{\mathrm{{\rm Y}}}_{\delta }^q\left(n,\alpha ,\beta \right)\left(L{\left[n\right]}_q-M\right)a_n{\xi }^n\\ -\sum _{n=2}^{\infty }{\mathrm{{\rm Y}}}_{\delta }^q\left(n,\alpha ,\beta \right)\left(L{\left[n\right]}_q+M\right)\overline{b_n{\xi }^n}\end{array}\right|\hfill \\ & & \\ & \le & \sum _{n=2}^{\infty }\left|{\mathrm{{\rm Y}}}_{\delta }^q\left(n,\alpha ,\beta \right)\right|\left({\left[n\right]}_q-1\right)\left|a_n\right|r^n\hfill \\ & & +\sum _{n=2}^{\infty }\left|{\mathrm{{\rm Y}}}_{\delta }^q\left(n,\alpha ,\beta \right)\right|\left({\left[n\right]}_q+1\right)\left|b_n\right|r^n\hfill \\ & & \\ & & -\left(L-M\right)r+\sum _{n=2}^{\infty }\left|{\mathrm{{\rm Y}}}_{\delta }^q\left(n,\alpha ,\beta \right)\right|\left(L{\left[n\right]}_q-M\right)\left|a_n\right|r^n\hfill \\ & & +\sum _{n=2}^{\infty }\left|{\mathrm{{\rm Y}}}_{\delta }^q\left(n,\alpha ,\beta \right)\right|\left(L{\left[n\right]}_q+M\right)\left|b_n\right|r^n\hfill \\ & & \\ & \le & r\left(\sum _{n=2}^{\infty }\left({\mathrm{{\rm Y}}}_n\left|a_n\right|+{\mathrm{\Psi }}_n\left|b_n\right|\right)r^n-\left(L-M\right)\right)<0.\hfill \end{array}$$

Hence $g\in$${\mathcal{S}}_{\mathcal{H}}^{\alpha ,\beta }(m,\delta ,q,M,L).$ □

In particular, if $q\to 1^{-}$, the previous Theorem 2 gives the following result.

Corollary 2

([9]). Let $g\in {\mathcal{H}}_0$ of the form (3), then g satisfies the following condition,

$$\sum _{n=2}^{\infty }\left({\mathrm{{\rm Y}}}_n\left|a_n\right|+{\mathrm{\Psi }}_n\left|b_n\right|\right)\le L-M,$$

with

$$\begin{array}{ccc}\hfill {\mathrm{{\rm Y}}}_n& =& \left|{\mathrm{{\rm Y}}}_{\delta }\left(n,\alpha ,\beta \right)\right|\left\{\left(1+L\right)n-\left(1+M\right)\right\},\hfill \\ \hfill {\mathrm{\Psi }}_n& =& \left|{\mathrm{{\rm Y}}}_{\delta }\left(n,\alpha ,\beta \right)\right|\left\{\left(1+L\right)n+\left(1+M\right)\right\}.\hfill \end{array}$$

Example 1.

Given the function

$$g\left(\xi \right)=\xi +\sum _{n=2}^{\infty }{p}_n\left(\frac{L-M}{{\mathrm{{\rm Y}}}_n}\right){\xi }^n+\sum _{n=2}^{\infty }{t}_n\left(\frac{L-M}{{\mathrm{\Psi }}_n}\right){\overline{\xi }}^n(\xi \in E),$$

such that $\sum _{n=2}^{\infty }\left(\left|p_n\right|+\left|t_n\right|\right)=1,$ we have

$$\begin{array}{ccc}\hfill \sum _{n=2}^{\infty }\left({\mathrm{{\rm Y}}}_n\left|a_n\right|+{\mathrm{\Psi }}_n\left|b_n\right|\right)& =& \sum _{n=2}^{\infty }\left(\left|p_n\right|\left(L-M\right)+\left|t_n\right|\left(L-M\right)\right)\hfill \\ & =& \left(L-M\right)\sum _{n=2}^{\infty }\left(\left|p_n\right|+\left|t_n\right|\right)=\left(L-M\right).\hfill \end{array}$$

It follows that $g\in$${\mathcal{S}}_{\mathcal{H}}^{\alpha ,\beta }(m,\delta ,q,M,L)$.

Theorem 3.

Let $g\in \tau$ of the form (14). Then $g\in$${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L)$ if and only if the condition (16) holds true.

Proof. 

By Theorem 2, we need to show that $g\in$${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L)$ satisfies the coefficient inequality (3).

If $g\in$${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L),$ then it is of the form (14) and satisfies (13) or equivalently

$$\left|\frac{-\sum _{n=2}^{\infty }{Q}_1\left(n,\alpha ,\beta ,q\right)a_n{\xi }^n-\sum _{n=2}^{\infty }{Q}_2\left(n,\alpha ,\beta ,q\right)\overline{b_n{\xi }^n}}{\left(L-M\right)\xi -\sum _{n=2}^{\infty }{Q}_3\left(n,\alpha ,\beta ,q\right)a_n{\xi }^n-\sum _{n=2}^{\infty }{Q}_4\left(n,\alpha ,\beta ,q\right)\overline{b_n{\xi }^n}}\right|<1,$$

where

$$\begin{array}{ccc}\hfill Q_1\left(n,\alpha ,\beta ,q\right)& =& {\mathrm{{\rm Y}}}_{\delta }^q\left(n,\alpha ,\beta \right)\left({\left[n\right]}_q-1\right),\hfill \\ \hfill Q_2\left(n,\alpha ,\beta ,q\right)& =& {\mathrm{{\rm Y}}}_{\delta }^q\left(n,\alpha ,\beta \right)\left({\left[n\right]}_q+1\right),\hfill \\ \hfill Q_3\left(n,\alpha ,\beta ,q\right)& =& {\mathrm{{\rm Y}}}_{\delta }^q\left(n,\alpha ,\beta \right)\left(L{\left[n\right]}_q-M\right),\hfill \\ \hfill Q_4\left(n,\alpha ,\beta ,q\right)& =& {\mathrm{{\rm Y}}}_{\delta }^q\left(n,\alpha ,\beta \right)\left(L{\left[n\right]}_q+M\right).\hfill \end{array}$$

Therefore by putting $\xi =r$, $r_1\in [0,1)$, we get

$$\frac{\sum _{n=2}^{\infty }\left|{\mathrm{{\rm Y}}}_{\delta }^q\left(n,\alpha ,\beta \right)\right|\left\{\left({\left[n\right]}_q-1\right)\left|a_n\right|+\left({\left[n\right]}_q+1\right)\left|b_n\right|\right\}{r}^{n-1}}{\left(L-M\right)+\sum _{n=2}^{\infty }\left|{\mathrm{{\rm Y}}}_{\delta }^q\left(n,\alpha ,\beta \right)\right|\left\{\left(L{\left[n\right]}_q-M\right)\left|a_n\right|+\left(L{\left[n\right]}_q+M\right)\left|b_n\right|\right\}}<1.$$

(19)

This shows that the denominator of the left hand side cannot vanishes for $r\in (0,1)$. Furthermore, it is positive for $r=0$, and in consequence for $r\in (0,1)$. Thus, by (19), we have

$$\sum _{n=2}^{\infty }\left({\mathrm{{\rm Y}}}_n\left|a_n\right|+{\mathrm{\Psi }}_n\left|b_n\right|\right)r^{n-1}\le L-M,r\in (0,1).$$

(20)

The sequence of partial sums $\left\{{\mathcal{S}}_n\right\}$ associated with the series

$$\sum _{n=2}^{\infty }\left({\mathrm{{\rm Y}}}_n\left|a_n\right|+{\mathrm{\Psi }}_n\left|b_n\right|\right)$$

is a non-decreasing sequence. Moreover, using (20), it is bounded by $L-M.$ Hence, the sequence $\left\{{\mathcal{S}}_n\right\}$ is convergent and

$$\sum _{n=2}^{\infty }\left({\mathrm{{\rm Y}}}_n\left|a_n\right|+{\mathrm{\Psi }}_n\left|b_n\right|\right)r^{n-1}=\underset{n\to \infty }{lim}{\mathcal{S}}_n\le L-M,$$

which leads to the assertion 16). □

In particular, if $q\to 1^{-},$ then Theorem 3 provides the result ([9]) in the following example.

Example 2.

For the function

$$g\left(\xi \right)=\xi -\sum _{n=2}^{\infty }{c}_n\left(\frac{L-M}{{\mathrm{{\rm Y}}}_n}\right){\xi }^n+\sum _{n=2}^{\infty }{d}_n\left(\frac{L-M}{{\mathrm{\Psi }}_n}\right){\overline{\xi }}^n(\xi \in E),$$

such that $\sum _{n=2}^{\infty }\left(\left|c_n\right|+\left|d_n\right|\right)=1,$ we have

$$\begin{array}{ccc}\hfill \sum _{n=2}^{\infty }\left({\mathrm{{\rm Y}}}_n\left|a_n\right|+{\mathrm{\Psi }}_n\left|b_n\right|\right)& =& \sum _{n=2}^{\infty }\left(\left|c_n\right|\left(L-M\right)+\left|c_n\right|\left(L-M\right)\right)\hfill \\ & =& \left(L-M\right)\sum _{n=2}^{\infty }\left(\left|c_n\right|+\left|d_n\right|\right)\hfill \\ & =& \left(L-M\right).\hfill \end{array}$$

Thus $g\in$${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L)$.

Theorem 4.

The class ${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L)$ is a convex and compact subset of $\mathcal{H}.$

Proof. 

Let $g_i\in$${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L)$ be a functions of the form

$$g_i\left(\xi \right)=\xi -\sum _{n=2}^{\infty }\left(\left|a_{i,n}\right|{\xi }^n-\left|b_{i,n}\right|{\overline{\xi }}^n\right)\left(\xi \in E,i\in \mathbb{N}\right),$$

(21)

and $0\le \gamma \le 1.$ Since

$$\begin{array}{ccc}& & \gamma g_1\left(\xi \right)+\left(1-\gamma \right)g_2\left(\xi \right)\hfill \\ & & \\ & =& \xi -\sum _{n=2}^{\infty }\left\{\left(\gamma \left|a_{1,n}\right|+\left(1-\gamma \right)\left|a_{2,n}\right|\right){\xi }^n+\left(\gamma \left|b_{1,n}\right|+\left(1-\gamma \right)\left|b_{2,n}\right|\right){\overline{\xi }}^n\right\}\hfill \end{array}$$

and by Theorem 3, we have

$$\begin{array}{ccc}& & \sum _{n=2}^{\infty }\left\{{\alpha }_n\left(\gamma \left|a_{1,n}\right|+\left(1-\gamma \right)\left|a_{2,n}\right|\right){\xi }^n+{\beta }_n\left(\gamma \left|b_{1,n}\right|+\left(1-\gamma \right)\left|b_{2,n}\right|\right){\overline{\xi }}^n\right\}\hfill \\ & =& \gamma \sum _{n=2}^{\infty }\left\{{\alpha }_n\left|a_{1,n}\right|+{\beta }_n\left|b_{1,n}\right|\right\}+\left(1-\gamma \right)\sum _{n=2}^{\infty }\left\{{\alpha }_n\left|a_{2,n}\right|+{\beta }_n\left|b_{2,n}\right|\right\}\hfill \\ & \le & \gamma \left(L-M\right)+\left(1-\gamma \right)\left(L-M\right)=\left(L-M\right),\hfill \end{array}$$

the function $\mathrm{\Psi }=\gamma g_1+\left(1-\gamma \right)g_2\in$${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L).$

Hence, the class is convex. Furthermore, for $g\in$${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L),$$\left|\xi \right|\le r,$$r\in \left(0,1\right),$ we have

$$\left|g\left(\xi \right)\right|\le r+\sum _{n=2}^{\infty }\left(\left|a_n\right|+\left|b_n\right|\right)r^n\le r+\sum _{n=2}^{\infty }\left({\alpha }_n\left|a_n\right|+{\beta }_n\left|b_n\right|\right)\le r+L-M.$$

(22)

Thus, we conclude that the class ${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L)$ is locally uniformly bounded.

By Lemma 2, we need to show that it is closed. For this, if $g_l\to g,$ then $g\in$${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L).$ Let $g_i$ and g be given by (21) and (14), respectively.

Using Theorem 3, we have

$$\sum _{n=2}^{\infty }\left\{{\alpha }_n\left|a_{i,n}\right|+{\beta }_n\left|b_{i,n}\right|\right\}\le L-M,\left(i\in N\right).$$

(23)

Since $g_i\to g$, we conclude that $\left|a_{i,n}\right|\to \left|a_n\right|$ and $\left|b_{i,n}\right|\to \left|b_n\right|$ as $i\to \infty$ ($n\in N).$

Therefore, the sequence of partial sums $\left\{{\mathcal{S}}_n\right\}$ associated with the series

$$\sum _{n=2}^{\infty }\left\{{\alpha }_n\left|a_{i,n}\right|+{\beta }_n\left|b_{i,n}\right|\right\}$$

is non-decreasing sequence and by (23) it is bounded by $L-M$. Therefore, the sequence $\left\{{\mathcal{S}}_n\right\}$ is convergent and

$$\sum _{n=2}^{\infty }\left\{{\alpha }_n\left|a_{i,n}\right|+{\beta }_n\left|b_{i,n}\right|\right\}=\underset{n\to \infty }{lim}\left\{{\mathcal{S}}_n\right\}\le L-M.$$

This gives condition (16) and in $g\in$${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L).$ Which completes the proof. □

If we consider $q\to 1^{-}$ in Theorem 4, we can obtain the result proved in [9].

Theorem 5.

The following identity is true,

$$E{\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L)=\left\{k_n^{\ast }:n\in N\right\}\cup \left\{l_n^{\ast }:n\in \{2,3,\dots \right\}\},$$

where

$$k_1^{\ast }\left(\xi \right)=\xi ,k_n^{\ast }\left(\xi \right)=\xi -\frac{L-M}{{\mathrm{{\rm Y}}}_n}{\xi }^n,l_n^{\ast }\left(\xi \right)=\xi +\frac{L-M}{{\mathrm{\Psi }}_n}{\overline{\xi }}^n,n\in (2,3,\dots ,\xi \in E).$$

(24)

Proof. 

Let $0<\gamma <1$ and

$$l_n^{\ast }\left(\xi \right)=\gamma g_1+\left(1-\gamma \right)g_2,$$

where $g_1,g_2\in$${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L)$ are functions of the form (21). Then, by (16),

$$\left|b_{1,n}\right|=\left|b_{2,n}\right|=\frac{L-M}{{\mathrm{\Psi }}_n}$$

and, in consequence

$$a_{1,i}=a_{2,i}=0\mathrm{for}i=\left\{2,3,\dots \right\}\mathrm{and}{b}_{1,i}=b_{2,i}=0\mathrm{for}i\in \left\{2,3,\dots \right\}\backslash \left\{n\right\}.$$

It follows that

$$l_n^{\ast }=g_1=g_2,$$

and consequently

$$l_n^{\ast }\in E{\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L).$$

Similarly, we verify that the functions $k_n^{\ast }$ of the form (24) are extreme points of the class ${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L).$

Now we suppose that $g\in E$${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L)$ and g is not of the form (24). Then there exists $i\in \left\{2,3,\dots \right\}$

$$0<\left|a_i\right|<\frac{L-M}{{\mathrm{{\rm Y}}}_i}\mathrm{or}0<\left|b_i\right|<\frac{L-M}{{\mathrm{\Psi }}_i}.$$

If $0<\left|a_i\right|<\frac{L-M}{{\mathrm{{\rm Y}}}_i},$ then putting

$$\gamma =\frac{\left|a_i\right|{\mathrm{{\rm Y}}}_i}{L-M},\Phi =\frac{1}{1-\gamma }\left(g-\gamma k_i^{\ast }\right),$$

we have that $0<\gamma <1,$ $k_i^{\ast },\Phi \in$ ${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L),$ $k_i^{\ast }\ne \Phi$ and

$$g=\gamma k_i^{\ast }+\left(1-\gamma \right)\Phi .$$

Thus, $g\notin E$ ${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L).$

Similarly, if $0<\left|b_i\right|<\frac{L-M}{{\mathrm{\Psi }}_i},$ then putting

$$\gamma =\frac{\left|b_i\right|{\mathrm{\Psi }}_i}{L-M},\mathrm{\Psi }=\frac{1}{1-\gamma }\left(g-\gamma l_i^{\ast }\right),$$

we have that

$$0<\gamma <1,k_i^{\ast },\Phi \in {\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L),k_i^{\ast }\ne \Phi$$

and

$$g=\gamma l_i^{\ast }+\left(1-\gamma \right)\mathrm{\Psi }.$$

Thus, $g\notin E$ ${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L)$. Hence this completes the proof. □

If we consider $q\to 1^{-}$ in Theorem 5, we also obtain the result in [9].

Theorem 6.

The radii of starlikeness of order ${\alpha }_0$ for the class ${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L)$ is given by

$$R_a^{\ast }\left({\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L)\right)=\underset{n\ge 2}{inf}{\left(\frac{1-{\alpha }_0}{L-M}min\left(\frac{{\mathrm{{\rm Y}}}_n}{n-{\alpha }_0},\frac{{\mathrm{\Psi }}_n}{n+{\alpha }_0}\right)\right)}^{\frac{1}{n-1}},$$

(25)

where ${\mathrm{{\rm Y}}}_n$ and ${\mathrm{\Psi }}_n$ are defined in (17) and (18), respectively.

Proof. 

Let $g\in$ ${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L)$ of the form (14). Then, for $\left|\xi \right|=r<1,$ we have

$$\begin{array}{ccc}& & \left|\frac{{\mathcal{D}}_{\mathcal{H}}g\left(\xi \right)-\left(1+{\alpha }_0\right)g\left(\xi \right)}{{\mathcal{D}}_{\mathcal{H}}g\left(\xi \right)+\left(1+{\alpha }_0\right)g\left(\xi \right)}\right|\hfill \\ & & \\ & =& \left|\frac{-{\alpha }_0\xi +{\displaystyle \sum _{n=2}^{\infty }}\left\{\left({\left[n\right]}_q-1-{\alpha }_0\right)\left|a_n\right|{\xi }^n-\left({\left[n\right]}_q+1+{\alpha }_0\right)\left|b_n\right|\overline{{\xi }^n}\right\}}{\left(2-{\alpha }_0\right)\xi -{\displaystyle \sum _{n=2}^{\infty }}\left\{\left({\left[n\right]}_q+1-{\alpha }_0\right)\left|a_n\right|{\xi }^n-\left({\left[n\right]}_q-1+{\alpha }_0\right)\left|b_n\right|\overline{{\xi }^n}\right\}}\right|\hfill \\ & & \\ & \le & \frac{{\alpha }_0+{\displaystyle \sum _{n=2}^{\infty }}\left\{\left({\left[n\right]}_q-1-{\alpha }_0\right)\left|a_n\right|-\left({\left[n\right]}_q+1+{\alpha }_0\right)\left|b_n\right|\right\}{r}^{n-1}}{\left(2-{\alpha }_0\right)-{\displaystyle \sum _{n=2}^{\infty }}\left\{\left({\left[n\right]}_q+1-{\alpha }_0\right)\left|a_n\right|-\left({\left[n\right]}_q-1+{\alpha }_0\right)\left|b_n\right|\right\}{r}^{n-1}}.\hfill \end{array}$$

Thus the condition (15) is true if and only if

$$\sum _{n=2}^{\infty }\left(\frac{n-{\alpha }_0}{1-{\alpha }_0}\left|a_n\right|+\frac{n+{\alpha }_0}{1-{\alpha }_0}\left|b_n\right|\right)r^{n-1}\le 1.$$

(26)

Applying for Theorem 2,

$$\sum _{n=2}^{\infty }\left(\frac{{\mathrm{{\rm Y}}}_n}{L-M}\left|a_n\right|+\frac{{\mathrm{\Psi }}_n}{L-M}\left|b_n\right|\right)\le 1,$$

(27)

where ${\mathrm{{\rm Y}}}_n$ and ${\mathrm{\Psi }}_n$ are defined in (17) and (18), respectively. So the condition (26) is true if

$$\frac{n-{\alpha }_0}{1-{\alpha }_0}{r}^{n-1}\le \frac{{\mathrm{{\rm Y}}}_n}{L-M},\frac{n+{\alpha }_0}{1-{\alpha }_0}{r}^{n-1}\le \frac{{\mathrm{\Psi }}_n}{L-M}(n=\left\{2,3,..\right\}).$$

That is,

$$r\le {\left(\frac{1-{\alpha }_0}{L-M}min\left(\frac{{\mathrm{{\rm Y}}}_n}{n-{\alpha }_0},\frac{{\mathrm{\Psi }}_n}{n-{\alpha }_0}\right)\right)}^{\frac{1}{n-1}}(n\in \left\{2,3,\dots \right\}.$$

This shows that the function g is starlike of order ${\alpha }_0$ in the disc $E\left(r^{\ast }\right),$ where

$$r^{\ast }=inf\left(\frac{1-{\alpha }_0}{L-M}min{\left(\frac{{\mathrm{{\rm Y}}}_n}{n-{\alpha }_0},\frac{{\mathrm{\Psi }}_n}{n-{\alpha }_0}\right)}^{\frac{1}{n-1}}\right).$$

The functions $k_n^{\ast }\left(\xi \right)$ and $l_n^{\ast }\left(\xi \right)$ defined by (24) show that the equality (27) and the radius $r^{\ast }$ cannot be larger, and hence we have (25). □

In particular, if we consider $q\to 1^{-}$ in Theorem 6, then we can get the result in [9].

Theorem 7.

The radii of convexity of order ${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L)$ is given by

$$R_a^c\left({\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L)\right)=\underset{n\ge 2}{inf}{\left(\frac{1-{\alpha }_0}{L-M}min\left(\frac{{\mathrm{{\rm Y}}}_n}{n-{\alpha }_0},\frac{{\mathrm{\Psi }}_n}{n+{\alpha }_0}\right)\right)}^{\frac{1}{n-1}},$$

where ${\mathrm{{\rm Y}}}_n$ and ${\mathrm{\Psi }}_n$ are defined in (17) and (18), respectively.

Proof. 

The proof of Theorem (7) is similar as Theorem 6, so we omit the proof here. □

3.2. Applications

Here we discuss some applications of our work in the form of some results and examples. It is clear that if the class

$$G=\left\{g_n\in \mathcal{H}:n\in \mathbb{N}\right\}$$

is locally uniformly bounded, then

$$\overline{co}G=\left\{\sum _{n=1}^{\infty }{\gamma }_n{g}_n:\sum _{n=2}^{\infty }{\gamma }_n=1,{\gamma }_n\ge 0,n\in \mathbb{N}\right\}.$$

(28)

Corollary 3.

It is true that

$${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L)=\left\{\sum _{n=2}^{\infty }\left({\gamma }_n{k}_n^{\ast }+{\delta }_n{l}_n^{\ast }\right):\sum _{n=2}^{\infty }\left({\gamma }_n+{\delta }_n\right)=1,{\delta }_1=0,{\delta }_n,{\gamma }_n\ge 0n\in \mathbb{N}\right\},$$

(29)

where $k_n^{\ast }$ and $l_n^{\ast }$ are defined by Equation (24).

Proof. 

By Theorem 4 and Lemma 3,

$${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L)=\overline{co}{\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L)=\overline{co}E{\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L).$$

Thus, by Theorem 5 and by (28), we get (29). □

We notice that for each $n\in \mathbb{N}$ and $\xi \in E$ the following real-valued functional $\mathcal{J}$ is continuous and convex on $\mathcal{H}.$

$$\mathcal{J}\left(G\right)=\left|a_n\right|,\mathcal{J}\left(G\right)=\left|b_n\right|,\mathcal{J}\left(G\right)=\left|g\left(\xi \right)\right|,\mathcal{J}\left(G\right)=\left|{\mathcal{D}}_{\mathcal{H}}g\left(\xi \right)\right|(g\in \mathcal{H}),$$

and

$$\mathcal{J}\left(G\right)=\left(\frac{1}{2\pi }{\int }_0^{2\pi }{\left|g(r{e}^{i\theta }\right|}^{\gamma }{d}_q\theta \right),(g\in \mathcal{H},\gamma \ge 1,0<r<1).$$

Therefore, by using Lemma 1 and Theorem 5, we obtain the following two corollaries.

Corollary 4.

Let $g\in {\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L)$ be a function of the form (14). Then we have the following sharp inequalities,

$$\left|a_n\right|\le \frac{L-M}{{\mathrm{{\rm Y}}}_n},\left|b_n\right|\le \frac{L-M}{{\mathrm{\Psi }}_n},(n=2,3,4\dots ),$$

(30)

where ${\mathrm{{\rm Y}}}_n$ and ${\mathrm{\Psi }}_n$ are defined by (17) and (18), respectively; and the function $k_n^{\ast }$ and $l_n^{\ast }$ of the form (24) are extremal functions.

Proof. 

Since $k_n^{\ast }$ and $l_n^{\ast }$ are the extremal functions, we have

$$\left|a_n\right|=\frac{L-M}{{\mathrm{{\rm Y}}}_n},\mathrm{and}\left|b_n\right|=\frac{L-M}{{\mathrm{\Psi }}_n}.$$

Thus, by Lemma 1, we get (30). □

Example 3.

Since $\frac{L-M+2}{{\mathrm{{\rm Y}}}_2}>\frac{L-M}{{\mathrm{{\rm Y}}}_2},$ by Corollary 4, the polynomial

$$K\left(\xi \right)=\xi -\frac{L-M+2}{{\mathrm{{\rm Y}}}_2}{\xi }^2,\left(\xi \in E\right)$$

is clearly $K\left(\xi \right)\notin$ ${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L).$

Corollary 5.

Let $g\in$ ${\mathcal{S}}_{\mathcal{H},\tau }^{\alpha ,\beta }(m,\delta ,q,M,L),$ $\left|\xi \right|=r<1.$ Then

$$\begin{array}{ccc}& & r-\left|\frac{{\mathrm{\Gamma }}_q(\alpha +\beta )}{{\left(1+q\delta \right)}^m{\mathrm{\Gamma }}_q\left(\beta \right)}\right|\frac{L-M}{1+2L-M}{r}^2\hfill \\ & \le & \left|g\left(\xi \right)\right|\le r+\left|\frac{{\mathrm{\Gamma }}_q(\alpha +\beta )}{{\left(1+q\delta \right)}^m{\mathrm{\Gamma }}_q\left(\beta \right)}\right|\frac{L-M}{1+2L-M}{r}^2,\hfill \\ & & \\ & & r-\left|\frac{{\mathrm{\Gamma }}_q(\alpha +\beta )}{{\left(1+q\delta \right)}^m{\mathrm{\Gamma }}_q\left(\beta \right)}\right|\frac{2\left(L-M\right)}{1+2L-M}{r}^2\hfill \\ & \le & \left|D_{\mathcal{H}}g\left(\xi \right)\right|\le r+\left|\frac{{\mathrm{\Gamma }}_q(\alpha +\beta )}{{\left(1+q\delta \right)}^m{\mathrm{\Gamma }}_q\left(\beta \right)}\right|\frac{2\left(L-M\right)}{1+2L-M}{r}^2.\hfill \end{array}$$

4. Conclusions

In our current study, we are inspired by the usage of the basic (or q-) calculus, and the fractional (or q-) calculus in geometric function theory as described in the survey-cum-expository review article by Srivastava [19]. We successfully defined the q-analogous value of the differential operator for harmonic functions. We then studied a new subclass of harmonic functions associated with Janowski functions and quantum (or q-) Mittag–Leffler functions. We also examined several useful properties of this new class of harmonic functions, such as necessary and sufficient conditions, criteria for analyticity, compactness and convexity, extreme points, radii of starlikeness, and radii of convexity, distortion bounds, and integral mean inequality. Moreover, we discussed some applications of this study in the form of some results and examples. The importance of the results established in this paper is clear from the fact that these findings would generalize and extend various recently known results from many earlier works. In view of motivation and encouragement for further research on the subject of our investigation, we cite several recent articles (for example, [16,25,28,45,46]) on a wide range of developments in geometric function theory of Complex Analysis associated with quantum (q-) calculus and quantum (q-) symmetric calculus operator theory.

However, Srivastava has already highlighted in [19] that the $(p,q)$ variations of the proposed q-results will lead to insignificant research because the extra parameter p is redundant. This observation by Srivastava [19] will indeed also apply to any attempt to produce the relatively straightforward $(p,q)$-variations of the results which we have presented in this paper.