Sharp Estimates Involving a Generalized Symmetric Sălăgean q-Differential Operator for Harmonic Functions via Quantum Calculus

Abstract

In this study, we apply q-symmetric calculus operator theory and investigate a generalized symmetric Sălăgean q-differential operator for harmonic functions in an open unit disk. We consider a newly defined operator and establish new subclasses of harmonic functions in complex order. We determine the sharp results, such as the sufficient necessary coefficient bounds, the extreme of closed convex hulls, and the distortion theorems for a new family of harmonic functions. Further, we illustrate how we connect the findings of previous studies and the results of this article.

1. Introduction

In an open unit disk $\mathcal{U}=\left\{\tau :\left|\tau \right|<1\right\}$, the set $\mathcal{A}$ represents the set of analytic functions h with the normalization $h\left(0\right)=h^{{}^{\prime }}\left(0\right)-1=0$, and has the following series expansion of the form:

$$h\left(\tau \right)=\tau +\sum _{n=2}^{\infty }{a}_n{\tau }^n,\tau \in \mathcal{U}.$$

(1)

To further clarify, let $\mathcal{S}$ denote the subclass of $\mathcal{A}$ that contains only univalent functions in $\mathcal{U}$.

Let u and v be harmonic functions, then the continuous functions $\xi =u+iv$ are complex valued harmonic functions in $\mathcal{U}$. If both h and g are analytic in $\mathcal{U}$, then we obtain $\xi \left(z\right)=h+\stackrel{\_}{g}$ (see [1]). Let us designate as the family $\mathcal{H}$ the orientation-preserving univalent harmonic functions $\xi =h+\stackrel{\_}{g}$ in $\mathcal{U}$, where h is defined as in (1) and its co-analytic part g has the following power series expansion:

$$g\left(\tau \right)=\sum _{n=1}^{\infty }{b}_n{\tau }^n,\left|b_1\right|<1.$$

(2)

A necessary and sufficient condition for $\xi$ to be locally univalent and sense-preserving in $\mathcal{U}$ is

$$\left|h^{{}^{\prime }}\left(z\right)\right|>\left|g^{{}^{\prime }}\left(z\right)\right|$$

and the harmonic functions $\xi \left(z\right)=h+\stackrel{\_}{g}$ will be normalized if $\xi \left(0\right)={\xi }^{{}^{\prime }}\left(0\right)-1=0.$ We point out that if the co-analytic component of $\xi$ is identically zero, i.e., $g\equiv 0$, then the family $\mathcal{H}$ of orientation-preserving, normalized harmonic univalent functions reduces to the well-known class $\mathcal{S}$ of normalized univalent functions.

Let $\overline{\mathcal{H}}$ be the subclass of $\mathcal{H}$ that contains the harmonic functions $\xi =h+\stackrel{\_}{g}$ whose h and g are of the form:

$$h\left(\tau \right)=\tau -\sum _{n=2}^{\infty }{a}_n{\tau }^n,g\left(\tau \right)=\sum _{n=1}^{\infty }{b}_n{\tau }^n{a}_n>0\mathrm{and}{b}_n>0.$$

After the groundbreaking work of Clunie and Sheil-Small [1] on harmonic mappings, several studies were published on different types of harmonic univalent functions with complex values. In particular, the features of specific types of harmonic univalent functions were explored in [2,3,4,5,6,7,8,9,10,11,12,13,14].

Jackson [15,16] pioneered the field of q-quantum calculus, which has a long history. It is crucial to the study of black holes and cosmic strings [15,16], conformal quantum mechanics [17,18], nuclear physics, high-energy physics, and more. We recommend [17,19] as an overview of this calculus. In particular, quantum mechanics has benefited from the applicability of q-symmetric quantum calculus [20]. As pointed out in [21], the regular q-symmetric integral has to be extended to the fundamental integral described in order to be compatible with q-deformed theory. Fractional q-symmetric integrals and fractional q-symmetric derivatives were introduced by Sun et al. in [22]. By studying a symmetric q-derivative operator, Kanas et al. [23] created an entirely novel class of analytic functions in $\mathcal{U}$ and investigated some of its potential uses in the conic domain. A new version of the generalized symmetric conic domains was recently developed by Khan et al. [24] using the ideas of symmetric q-calculus and conic regions. This domain was then used to characterize a new subclass of q-starlike functions in $\mathcal{U}$ and established numerous new results. In 2022, Khan et al. [25] introduced a symmetric q-difference operator for m-fold symmetric functions, and its study yielded interesting findings for m-fold symmetric bi-univalent functions. Khan et al. proposed many novel applications of multivalent q-starlike functions and developed the idea of a multivalent q-symmetric derivative operator in [26]. New classes of harmonic functions were developed by applying the q-symmetric difference operator to harmonic univalent functions (see [27,28] for more information).

Using certain q-operators on complex harmonic functions, Jahangiri [29] recently derived sharp coefficient bounds, distortion, and covering results. Porwal and Gupta [30] alternatively addressed using the q-calculus to analyze harmonic univalent functions. Here, we use the symmetric q-calculus to construct a q analogous to the symmetric Salagean differential operator that works for complex harmonic functions, as well as to introduce and study of new classes of harmonic univalent functions.

For $0<q<1,$ the q-symmetric difference operator of h is defined for the function $h\in \mathcal{S}$:

$${\tilde{\partial }}_{q}h\left(\tau \right)=\left\{\begin{array}{c}\frac{h\left(q\tau \right)-h\left(q^{-1}\tau \right)}{\left(q-q^{-1}\right)\tau }\mathrm{for}\tau \ne 0,\\ \\ h^{\prime }\left(0\right)\mathrm{for}\tau =0\end{array}\right.$$

(3)

and

$${\tilde{\partial }}_q^{2}h\left(\tau \right)={\tilde{\partial }}_q\left({\tilde{\partial }}_{q}h\left(\tau \right)\right).$$

It can be seen that we have

$${\tilde{\partial }}_{q}h\left(\tau \right)=1+\sum _{n=2}^{\infty }{\widetilde{\left[n\right]}}_q{a}_n{\tau }^{n-1},$$

where

$${\widetilde{\left[n\right]}}_q=\frac{q^n-q^{-n}}{q-q^{-1}}\mathrm{for}\mathrm{all}n\ge 1.$$

(4)

If $q\to 1^{-}$, then, ${\widetilde{\left[n\right]}}_q\to n$. Note that, the q-symmetric number can not reduce to a q-number.

For $h\in \mathcal{A},$$m\in {\mathbb{N}}_0=\{0,1,2,...\}$ and $\tau \in \mathcal{U},$ Zhang et al. [27] defined the symmetric q-Sălăgean differential operator with regard to applications of the symmetric q-difference operator defined in (3) as follows:

$$\begin{array}{ccc}\hfill {\tilde{D}}_q^{0}h\left(\tau \right)& =& h\left(\tau \right),\hfill \\ \hfill {\tilde{D}}_q^{1}h\left(\tau \right)& =& \tau {\tilde{D}}_{q}h\left(\tau \right),\hfill \\ & & ⋮\hfill \\ \hfill {\tilde{D}}_q^{m}h\left(\tau \right)& =& \tau {\tilde{\partial }}_q\left({\tilde{D}}_q^{m-1}h\left(\tau \right)\right)=\tau +\sum _{n=2}^{\infty }{\widetilde{\left[n\right]}}_q^m{a}_n{\tau }^n.\hfill \end{array}$$

(5)

We note that if $q\to 1^{-}$, then,

$${\tilde{D}}_q^{m}h\left(\tau \right)\to D^{m}h\left(\tau \right)=\tau +\sum _{n=2}^{\infty }{n}^m{a}_n{\tau }^n(m\in {\mathbb{N}}_0,\tau \in \mathcal{U}),$$

where $D^{m}h\left(\tau \right)$ is the familiar Sălăgean differential operator studied in [31].

Let ${\tilde{D}}_q^{m}h\left(\tau \right)$ be defined by (5) and ${\tilde{D}}_q^{m}g\left(\tau \right)$ can be defined for harmonic functions $\xi =h+\stackrel{\_}{g}\in \mathcal{H}$ as follows:

$$\begin{array}{ccc}\hfill {\tilde{D}}_q^{0}g\left(\tau \right)& =& g\left(\tau \right),\hfill \\ \hfill {\tilde{D}}_q^{1}g\left(\tau \right)& =& \tau {\tilde{D}}_{q}g\left(\tau \right),\hfill \\ & & ⋮\hfill \end{array}$$

$${\tilde{D}}_q^{m}g\left(\tau \right)=\tau {\tilde{\partial }}_q\left({\tilde{D}}_q^{m-1}g\left(\tau \right)\right)=\sum _{n=1}^{\infty }{\widetilde{\left[n\right]}}_q^m{b}_n{\tau }^n,$$

(6)

where h and g are, respectively, provided by (1) and (2).

Zhang et al. [27] considered a symmetric q-Sălăgean differential operator and defined a class $\widetilde{{\mathcal{H}}_q^m}\left(\alpha \right)$ of functions $\xi$ satisfying the following inequality:

$$\Re \left(\frac{{\tilde{D}}_q^{m+1}\xi \left(\tau \right)}{{\tilde{D}}_q^m\xi \left(\tau \right)}\right)\ge \alpha ,0\le \alpha <1,$$

where ${\tilde{D}}_q^{m}h\left(\tau \right)$ and ${\tilde{D}}_q^{m}g\left(\tau \right)$ are, respectively, defined by (5) and (6), and

$${\tilde{D}}_q^m\xi \left(\tau \right)={\tilde{D}}_q^{m}h\left(\tau \right)+{(-1)}^m\overline{{\tilde{D}}_q^{m}g\left(\tau \right)},m>-1.$$

We denote by $\tilde{{\stackrel{\_}{\mathcal{H}}}_q^m}\left(\alpha \right)$ the set of harmonic functions ${\xi }_m=h+\stackrel{\_}{g_m}$ for which

$$g_m\left(\tau \right)={(-1)}^m\sum _{n=1}^{\infty }{b}_n{\tau }^n,b_n>0\mathrm{and}h\left(\tau \right)=\tau -\sum _{n=2}^{\infty }{a}_n{\tau }^n,a_n>0.$$

(7)

Note that $\tilde{\stackrel{\_}{{\mathcal{H}}_q^m}}\left(\alpha \right)\subset \widetilde{{\mathcal{H}}_q^m}\left(\alpha \right).$

Jahangiri [29] used the concept of q-calculus, the generalized Sălăgean q-differential operator, and defined a class of harmonic functions. Getting inspiration from this idea as presented in [29], we first use the concepts of symmetric q-calculus and the symmetric q-difference operator. We then define the generalized symmetric Sălăgean q-differential operator.

Definition 1. 

The generalized symmetric Sălăgean q-differential operator ${\tilde{D}}_q^{m,\lambda }:\mathcal{S}\to \mathcal{S}$ defined as:

$${\tilde{D}}_q^{m,\lambda }h\left(\tau \right)=\tau +\sum _{n=2}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m{a}_n{\tau }^n$$

(8)

where

$$\widetilde{{\varphi }_q^{\lambda }}\left(n\right)=\left(1+\lambda \left({\widetilde{\left[n\right]}}_q-1\right)\right),\lambda \ge 0\mathrm{and}m\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$$

(9)

and h is given in (1), and ${\widetilde{\left[n\right]}}_q$ is given in (4). It can easily be noted that

$$\begin{array}{ccc}\hfill {\tilde{D}}_q^{0,\lambda }h\left(\tau \right)& =& h\left(\tau \right),\hfill \\ \hfill {\tilde{D}}_q^{1,\lambda }h\left(\tau \right)& =& \left(1-\lambda \right)h\left(\tau \right)+\lambda \tau {\tilde{\partial }}_{q}h\left(\tau \right),\hfill \\ & & ⋮\hfill \\ \hfill {\tilde{D}}_q^{m,\lambda }h\left(\tau \right)& =& {\tilde{D}}_q^{\lambda }\left({\tilde{D}}_q^{m-1,\lambda }h\left(\tau \right)\right).\hfill \end{array}$$

Remark 1. 

For $q\to 1-,$ then we have the generalized Sălăgean differential operator defined by Oboudi in [32].

Remark 2. 

For $\lambda =1,$ then we have the symmetric Sălăgean q-differential operator defined by Zhang et al. in [27].

Remark 3. 

Let ${\tilde{D}}_q^{m,\lambda }h\left(\tau \right)$ be defined by (8) and ${\tilde{D}}_q^{m,\lambda }g\left(\tau \right)$ can be defined for the harmonic functions $\xi =h+\stackrel{\_}{g}\in \mathcal{H}$ as follows:

$$\begin{array}{ccc}\hfill {\tilde{D}}_q^{0,\lambda }g\left(\tau \right)& =& g\left(\tau \right),\hfill \\ \hfill {\tilde{D}}_q^{1,\lambda }g\left(\tau \right)& =& \left(1-\lambda \right)g\left(\tau \right)+\lambda \tau {\tilde{\partial }}_{q}g\left(\tau \right),\hfill \\ & & ⋮\hfill \end{array}$$

$${\tilde{D}}_q^{m,\lambda }g\left(\tau \right)={\tilde{D}}_q^{\lambda }\left({\tilde{D}}_q^{m-1,\lambda }g\left(\tau \right)\right)=\sum _{n=1}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m{b}_n{\tau }^n,$$

where ${\widetilde{\left[n\right]}}_q$ is given by (4) and $\widetilde{{\varphi }_q^{\lambda }}\left(n\right)$ is given by (9).

We develop new families of harmonic functions by considering the generalized symmetric Sălăgean q-differential operator ${\tilde{D}}_q^{m,\lambda }$:

Definition 2. 

Let $c\ne 0$ and $c\in \mathbb{C},$ with $\left|c\right|\le 1$, $\gamma \in \mathbb{R}$, $0<q<1,$$\lambda \ge 0$ and $0\le \alpha <1;$ we let $\widetilde{\mathcal{H}{\mathcal{S}}_q^m}(c,\gamma ,\lambda ,\alpha )$ be the subclass of $\mathcal{H}$ of the harmonic functions $\xi =h+\stackrel{\_}{g}$ satisfying

$$\Re \left(1+\frac{1}{c}\left(\left(1+e^{i\gamma }\right)\frac{{\tilde{D}}_q^{m+1,\lambda }\xi \left(\tau \right)}{{\tilde{D}}_q^{m,\lambda }\xi \left(\tau \right)}-e^{i\gamma }-1\right)\right)>\alpha .$$

(10)

We also let $\widetilde{\overline{\mathcal{H}}{\mathcal{S}}_q^m}(c,\gamma ,\lambda ,\alpha )\equiv \widetilde{\mathcal{H}{\mathcal{S}}_q^m}(c,\gamma ,\lambda ,\alpha )\cap \mathcal{H}.$

We note that $\widetilde{\mathcal{H}{\mathcal{S}}_q^m}(1,\gamma ,\lambda ,\alpha )\equiv \widetilde{\mathcal{H}{\mathcal{S}}_q^m}(\gamma ,\lambda ,\alpha )$ is a generalized class of harmonic starlike functions, which satisfies

$$\Re \left(\left(1+e^{i\gamma }\right)\frac{{\tilde{D}}_q^{m+1,\lambda }\xi \left(\tau \right)}{{\tilde{D}}_q^{m,\lambda }\xi \left(\tau \right)}-e^{i\gamma }\right)>\alpha$$

and $\widetilde{\mathcal{H}{\mathcal{S}}_q^m}(c,0,\lambda ,\alpha )\equiv \widetilde{\mathcal{H}{\mathcal{R}}_q^m}(c,\lambda ,\alpha )$ is a harmonic type of starlike function of complex order c, which satisfies

$$\Re \left(1+\frac{2}{c}\left(\frac{{\tilde{D}}_q^{m+1,\lambda }\xi \left(\tau \right)}{{\tilde{D}}_q^{m,\lambda }\xi \left(\tau \right)}-1\right)\right)>\alpha .$$

2. Main Result

The remaining part of this paper is devoted to obtaining sufficient coefficient conditions, extreme points, distortion bounds, and growth theorems for harmonic functions $\xi =h+\stackrel{\_}{g}$ in $\widetilde{\mathcal{H}{\mathcal{S}}_q^m}(c,\gamma ,\lambda ,\alpha )$. Moreover, we show that the sufficient coefficient conditions for $\xi \in \widetilde{\mathcal{H}{\mathcal{S}}_q^m}(c,\gamma ,\lambda ,\alpha )$ are also necessary for $\xi \in \widetilde{\overline{\mathcal{H}}{\mathcal{S}}_q^m}(c,\gamma ,\lambda ,\alpha )$.

We begin with a sufficient coefficient condition for $\mathcal{H}{\mathcal{S}}_q^m(c,\gamma ,\lambda ,\alpha )$.

Theorem 1. 

Let $\xi =h+\overline{g}\in \mathcal{H}$, where $c\in \mathbb{C}$ and $c\ne 0$ with $\left|c\right|\le 1,$$\gamma \in \mathbb{R},$$\lambda \ge 0,$$0<q<1$ and $0\le \alpha <1$. If

$$\sum _{n=1}^{\infty }\left(\begin{array}{c}\frac{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-2+(1-\alpha )\left|c\right|\right]}{\left(1-\alpha \right)\left|c\right|}\left|a_n\right|\\ \\ +\frac{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)+2-(1-\alpha )\left|c\right|\right]}{\left(1-\alpha \right)\left|c\right|}\left|b_n\right|\end{array}\right)\le 2,$$

(11)

then,

(i) $\xi$ is harmonic univalent and orientation-preserving in $\mathcal{U}$.

(ii) $\xi \in \widetilde{\mathcal{H}{\mathcal{S}}_q^m}(c,\gamma ,\lambda ,\alpha )$.

Proof. 

First, we establish part (i). That is to say, we have to prove that

$$\left|{\tilde{D}}_q^{m+1,\lambda }h\left(\tau \right)\right|\ge \left|{\tilde{D}}_q^{m+1,\lambda }g\left(\tau \right)\right|.$$

Absolute value theory and the coefficient inequality (11) allow us to do this.

$$\begin{array}{ccc}\hfill \left|{\tilde{D}}_q^{m+1,\lambda }h\left(\tau \right)\right|& \ge & 1-\sum _{n=2}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^{m+1}\left|a_n\right|r^{n-1}>1-\sum _{n=2}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^{m+1}\left|a_n\right|\hfill \\ & \ge & 1-\sum _{n=2}^{\infty }\left[\frac{2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-2+(1-\alpha )\left|c\right|}{\left(1-\alpha \right)\left|c\right|}\right]{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|a_n\right|\hfill \\ & \ge & \sum _{n=1}^{\infty }\left[\frac{2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)+2-(1-\alpha )\left|c\right|}{\left(1-\alpha \right)\left|c\right|}\right]{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|b_n\right|\hfill \\ & \ge & \sum _{n=1}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^{m+1}\left|b_n\right|\ge \sum _{n=1}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^{m+1}\left|b_n\right|r^{n-1}\hfill \\ & \ge & \left|{\tilde{D}}_q^{m+1,\lambda }g\left(\tau \right)\right|.\hfill \end{array}$$

We use a technique pioneered by Jahangiri [5] to demonstrate that $\xi$ is univalent in $\mathcal{U}$. We will prove that $\xi \left({\tau }_1\right)\ne \xi \left({\tau }_2\right)$ when ${\tau }_{\{1}\ne {\tau }_2$. We take the points ${\tau }_1$ and ${\tau }_2$ in $\mathcal{U}$, and ${\tau }_1\ne {\tau }_2.$ Since the unit disc $\mathcal{U}$ is simply connected and convex, we have

$$\tau \left(t\right)=(1-t){\tau }_1+t{\tau }_2$$

in $\mathcal{U}$ for $0\le t\le 1$. We can write

$$\begin{array}{ccc}& & \Re \left(\frac{{\tilde{D}}_q^{m+1,\lambda }\xi \left({\tau }_2\right)-{\tilde{D}}_q^{m+1,\lambda }\xi \left({\tau }_1\right)}{\left({\tau }_2-{\tau }_1\right)}\right)\hfill \\ & >& \underset{0}{\overset{1}{\int }}\Re \left[{\tilde{D}}_q^{m+1,\lambda }h\left(\tau \left(t\right)\right)-\left|{\tilde{D}}_q^{m+1,\lambda }g\left(\tau \left(t\right)\right)\right|\right]dt.\hfill \end{array}$$

(12)

On the other hand,

$$\begin{array}{ccc}& & \Re ({\tilde{D}}_q^{m+1,\lambda }h\left(\tau \left(t\right)\right)-\left|{\tilde{D}}_q^{m+1,\lambda }g\left(\tau \left(t\right)\right)\right|\hfill \\ & \ge & \Re ({\tilde{D}}_q^{m+1,\lambda }h\left(\tau \left(t\right)\right)-\sum _{n=1}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^{m+1}\left|b_n\right|\hfill \\ & \ge & 1-\sum _{n=2}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^{m+1}\left|a_n\right|-\sum _{n=1}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^{m+1}\left|b_n\right|\hfill \\ & \ge & 1-\sum _{n=2}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[\frac{2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-2+(1-\alpha )\left|c\right|}{\left(1-\alpha \right)\left|c\right|}\right]\left|a_n\right|\hfill \\ & & -\sum _{n=1}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[\frac{2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-2+(1-\alpha )\left|c\right|}{\left(1-\alpha \right)\left|c\right|}\right]\left|b_n\right|\hfill \\ & \ge & 0.\hfill \end{array}$$

This, together with the inequality (12), suggests that $\xi$ is univalent. Further, we prove that $\xi \in \widetilde{{\mathcal{HS}}_q^m}(c,\gamma ,\lambda ,\alpha )$ if (11) is true. That is to say, we need to demonstrate that if (11) is true, then (10) must also be true. Using the fact that

$$\Re \left(w\right(\tau \left)\right)\ge \alpha \iff |1-\alpha +w|\ge |1+\alpha -w|,\mathrm{for}0\le \alpha <1,$$

it suffices to show that

$$\begin{array}{ccc}& & \left|\begin{array}{c}\left(2c-\alpha c-e^{i\gamma }-1\right)\left({\tilde{D}}_q^{m,\lambda }h\left(\tau \right)+{(-1)}^m\overline{{\tilde{D}}_q^{m,\lambda }g\left(\tau \right)}\right)\\ +(1+e^{i\gamma })\left({\tilde{D}}_q^{m+1,\lambda }h\left(\tau \right)-{(-1)}^m\overline{{\tilde{D}}_q^{m+1,\lambda }g\left(\tau \right)}\right)\end{array}\right|\hfill \\ & & -\left|\begin{array}{c}\left(1+\alpha c+e^{i\gamma }\right)\left({\tilde{D}}_q^{m,\lambda }h\left(\tau \right)+{(-1)}^m\overline{{\tilde{D}}_q^{m,\lambda }g\left(\tau \right)}\right)\\ -(1+e^{i\gamma })\left({\tilde{D}}_q^{m+1,\lambda }h\left(\tau \right)-{(-1)}^m\overline{{\tilde{D}}_q^{m+1,\lambda }g\left(\tau \right)}\right)\end{array}\right|\hfill \\ & \ge & 0.\hfill \end{array}$$

Substituting ${\tilde{D}}_q^{m,\lambda }h\left(\tau \right)$ and ${\tilde{D}}_q^{m,\lambda }g\left(\tau \right)$ into the latter yields

$$\left|\left(2c-\alpha c-(1+e^{i\gamma })\right)\left[\tau +\sum _{n=2}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m{a}_n{\tau }^n+{(-1)}^m\sum _{n=1}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\overline{b_n{\tau }^n}\right]\right.$$

$$+\left.(1+e^{i\gamma })\left[\tau +\sum _{n=2}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^{m+1}{a}_n{\tau }^n-{(-1)}^m\sum _{n=1}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^{m+1}\overline{b_n{\tau }^n}\right]\right|$$

$$-\left|\left(1+\alpha c+e^{i\gamma }\right)\left[\tau +\sum _{n=2}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m{a}_n{\tau }^n+{(-1)}^m\sum _{n=1}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\overline{b_n{\tau }^n}\right]\right.$$

$$\left.-(1+e^{i\gamma })\left[\tau +\sum _{n=2}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^{m+1}{a}_n{\tau }^n-{(-1)}^m\sum _{n=1}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^{m+1}\overline{b_n{\tau }^n}\right]\right|$$

$$\ge (2-\alpha )\left|c\right|\left|\tau \right|-\sum _{n=2}^{\infty }\left|(2-\alpha )c+(1+e^{i\gamma })(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-1)\right|{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|a_n\right|{\left|\tau \right|}^n$$

$$-\sum _{n=1}^{\infty }\left|(1+e^{i\gamma })(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)+1)-(2-\alpha )c\right|{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|b_n\right|{\left|\tau \right|}^n$$

$$-\alpha \left|c\right|\left|\tau \right|-\sum _{n=2}^{\infty }\left|(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-1)(1+e^{i\gamma })-\alpha c\right|{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|a_n\right|{\left|\tau \right|}^n$$

$$-\sum _{n=1}^{\infty }\left|(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-1)(1+e^{i\gamma })+\alpha c\right|{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|b_n\right|{\left|\tau \right|}^n$$

$$\ge 2(1-\alpha )\left|c\right|\left|\tau \right|\left(1-\sum _{n=2}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[\frac{2\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-2+(1-\alpha )\left|c\right|\right]}{2\left(1-\alpha \right)\left|c\right|}\left|a_n\right|\right]\right)$$

$$-2(1-\alpha )\left|c\right|\left|\tau \right|\sum _{n=1}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[\frac{2\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)+2-(1-\alpha )\left|c\right|\right]}{2\left(1-\alpha \right)\left|c\right|}\left|b_n\right|\right]\ge 0\mathrm{by}\left(11\right).$$

□

Remark 4. 

The function

$$\begin{array}{ccc}\hfill \xi \left(\tau \right)& =& \tau +\sum _{n=2}^{\infty }\left[\frac{\left(1-\alpha \right)\left|c\right|}{2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-2+(1-\alpha )\left|c\right|}\right]x_n{\tau }^n\hfill \\ & & \\ & & +\sum _{n=1}^{\infty }\left[\frac{\left(1-\alpha \right)\left|c\right|}{2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)+2-(1-\alpha )\left|c\right|}\right]\stackrel{\_}{y_n}{\tau }^n,\hfill \end{array}$$

where

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

shows that the coefficient bound given by (11) is sharp.

Example 1. 

The function $\xi =h+\overline{g}$ given by

$$\xi \left(\tau \right)=\tau +\sum _{n=2}^{\infty }{A}_n{\tau }^n+\sum _{n=1}^{\infty }{B}_n{\overline{\tau }}^n$$

where

$$A_n=\frac{\left(2+\delta \right)\left(1-\alpha \right)\left|c\right|{ϵ}_n}{\left(n+\delta \right)\left(n+1+\delta \right){\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-2+(1-\alpha )\left|c\right|\right]}$$

and

$$A_n=\frac{\left(2+\delta \right)\left(1-\alpha \right)\left|c\right|{ϵ}_n}{\left(n+\delta \right)\left(n+1+\delta \right){\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)+2-(1-\alpha )\left|c\right|\right]}$$

belonging to the class $\widetilde{\mathcal{H}{\mathcal{S}}_q^m}(c,\gamma ,\lambda ,\alpha ),$ for $\delta >-2,$${ϵ}_n\in \mathbb{C}$, $\left|{ϵ}_n\right|=1.$ Because we know that

$$\begin{array}{ccc}& & \sum _{n=2}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-2+(1-\alpha )\left|c\right|\right]\left|A_n\right|\hfill \\ & & +\sum _{n=1}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)+2-(1-\alpha )\left|c\right|\right]\left|B_n\right|\hfill \\ & \le & \left(2+\delta \right)\left(1-\alpha \right)\left|c\right|\sum _{n=2}^{\infty }\frac{1}{\left(n+\delta \right)\left(n+1+\delta \right)}\hfill \\ & & +\left(1+\delta \right)\left(1-\alpha \right)\left|c\right|\sum _{n=1}^{\infty }\frac{1}{\left(n+\delta \right)\left(n+1+\delta \right)}\hfill \\ & =& \left(2+\delta \right)\left(1-\alpha \right)\left|c\right|\left(\sum _{n=2}^{\infty }\frac{1}{n+\delta }-\frac{1}{n+1+\delta }\right)\hfill \\ & & +\left(1+\delta \right)\left(1-\alpha \right)\left|c\right|\left(\sum _{n=1}^{\infty }\frac{1}{n+\delta }-\frac{1}{n+1+\delta }\right)\hfill \\ & =& 2\left(1-\alpha \right)\left|c\right|.\hfill \end{array}$$

The following theorem demonstrates that condition (11) is also necessary for $\xi \in \widetilde{\overline{\mathcal{H}}{\mathcal{S}}_q^m}(c,\gamma ,\lambda ,\alpha ).$

Theorem 2. 

Let ${\xi }_m=h+\overline{g_m}$ be defined by (7), where $c\in \mathbb{C}$ and $c\ne 0$ with $\left|c\right|\le 1,$$\gamma \in \mathbb{R},$$\lambda \ge 0,$$0<q<1$ and $0\le \alpha <1$. Then, ${\xi }_m$ is harmonic univalent and orientation-preserving in $\mathcal{U}$ and $\xi \in \widetilde{\overline{\mathcal{H}}{\mathcal{S}}_q^m}(c,\gamma ,\lambda ,\alpha ),$ if and only if,

$$\sum _{n=2}^{\infty }\left(\begin{array}{c}\frac{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-2+(1-\alpha )\left|c\right|\right]}{\left(1-\alpha \right)\left|c\right|}{a}_n\\ \\ +\frac{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)+2-(1-\alpha )\left|c\right|\right]}{\left(1-\alpha \right)\left|c\right|}{b}_n\end{array}\right)\le 2.$$

(13)

Proof. 

Since $\widetilde{\overline{\mathcal{H}}{\mathcal{S}}_q^m}(c,\gamma ,\lambda ,\alpha )\subset \widetilde{{\mathcal{HS}}_q^m}(c,\gamma ,\lambda ,\alpha ),$ from Theorem 1, we may deduce that the if clause of Theorem 2 holds. To demonstrate only the if part, we shall first show that if (13) does not hold, then ${\xi }_m\notin \widetilde{\overline{\mathcal{H}}{\mathcal{S}}_q^m}(c,\gamma ,\lambda ,\alpha )$.

For ${\xi }_m\in \widetilde{\overline{\mathcal{H}}{\mathcal{S}}_q^m}(c,\gamma ,\lambda ,\alpha )$, we have

$$\Re \left(1+\frac{1}{c}\left((1+e^{i\gamma })\frac{{\tilde{D}}_q^{m+1,\lambda }h\left(\tau \right)-{(-1)}^m\overline{{\tilde{D}}_q^{m+1,\lambda }{g}_m\left(\tau \right)}}{{\tilde{D}}_q^{m,\lambda }h\left(\tau \right)+{(-1)}^m\overline{{\tilde{D}}_q^{m,\lambda }{g}_m\left(\tau \right)}}-(1+e^{i\gamma })\right)\right)\ge \alpha .$$

Or equivalently,

$$\Re \left(\frac{\left(1-\alpha \right)c\tau -{\displaystyle \sum _{n=2}^{\infty }}\left[\left(1-\alpha \right)c+\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-1\right)(1+e^{i\gamma })\right]{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|a_n\right|{\tau }^n}{c\left(\tau -{\displaystyle \sum _{n=2}^{\infty }}{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|a_n\right|{\tau }^n+{(-1)}^{2m}{\displaystyle \sum _{n=1}^{\infty }}{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|c_n\right|{\stackrel{\_}{\tau }}^n\right)}\right)$$

$$-\Re \left(\frac{{(-1)}^{2m}{\displaystyle \sum _{n=1}^{\infty }}\left[\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)+1\right)(1+e^{i\gamma })-(1-\alpha )c\right]{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|b_n\right|{\stackrel{\_}{\tau }}^n}{c\left(\tau -{\displaystyle \sum _{n=2}^{\infty }}{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|a_n\right|{\tau }^n+{(-1)}^{2m}{\displaystyle \sum _{n=1}^{\infty }}{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|b_n\right|{\stackrel{\_}{\tau }}^n\right)}\right)$$

$$=\Re \left(\frac{\left(1-\alpha \right){\left|c\right|}^2-{\displaystyle \sum _{n=2}^{\infty }}\left[\left(1-\alpha \right)c+\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-1\right)(1+e^{i\gamma })\right]\stackrel{\_}{c}{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|a_n\right|{\tau }^{n-1}}{{\left|c\right|}^2\left(1-{\displaystyle \sum _{n=2}^{\infty }}{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|a_n\right|{\tau }^{n-1}+\frac{\overline{\tau }}{\tau }{\displaystyle \sum _{n=1}^{\infty }}{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|b_n\right|{\stackrel{\_}{\tau }}^{n-1}\right)}\right)$$

$$-\Re \left(\frac{\frac{\stackrel{\_}{\tau }}{\tau }{\displaystyle \sum _{n=1}^{\infty }}\left[\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)+1\right)(1+e^{i\gamma })-(1-\alpha )c\right]\stackrel{\_}{c}{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|b_n\right|{\stackrel{\_}{\tau }}^{n-1}}{{\left|c\right|}^2\left(1-{\displaystyle \sum _{n=2}^{\infty }}{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|a_n\right|{\tau }^{n-1}+\frac{\stackrel{\_}{\tau }}{\tau }{\displaystyle \sum _{n=1}^{\infty }}{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|b_n\right|{\stackrel{\_}{\tau }}^{n-1}\right)}\right)\ge 0.$$

The above condition must hold for all values of γ, $\left|\tau \right|=r<1$ and $0<\left|c\right|<1$. For $\gamma =0$ and $\left|c\right|=c,$ let $\tau =r<1$ be on the positive real axis. The preceding condition is, thus,

$$\frac{\left(1-\alpha \right){\left|c\right|}^2-{\displaystyle \sum _{n=2}^{\infty }}\left[\left(2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-2\right)+\left(1-\alpha \right)c\right]\left|c\right|{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|a_n\right|r^{n-1}}{{\left|c\right|}^2\left(1-{\displaystyle \sum _{n=2}^{\infty }}{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|a_n\right|r^{n-1}+{\displaystyle \sum _{n=1}^{\infty }}{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|b_n\right|r^{n-1}\right)}$$

$$-\frac{{\displaystyle \sum _{n=1}^{\infty }}\left[\left(2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)+2\right)-(1-\alpha )c\right]\left|c\right|{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|b_n\right|r^{n-1}}{{\left|c\right|}^2\left(1-{\displaystyle \sum _{n=2}^{\infty }}{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|a_n\right|r^{n-1}+{\displaystyle \sum _{n=1}^{\infty }}{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left|b_n\right|r^{n-1}\right)}\ge 0.$$

(14)

Now, we see that if the condition (13) does not hold, the numerator in the above necessary inequality (14) is negative. This means that the quotient of the preceding inequalities is negative for certain values of ${\tau }_0=r_0\in (0,1).$ This contradicts the necessary requirement (10) for ${\xi }_m\in \widetilde{\overline{\mathcal{H}}{\mathcal{S}}_q^m}(c,\gamma ,\lambda ,\alpha )$. This concludes the proof. □

Corollary 1. 

Let ${\xi }_m=h+{\stackrel{\_}{g}}_m$ of the form (7). Then, ${\xi }_m\in \widetilde{\overline{\mathcal{H}}{\mathcal{S}}_q^m}(\gamma ,\lambda ,\alpha ),$ if and only if,

$$\sum _{n=1}^{\infty }\left(\frac{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-1-\alpha )\right]}{1-\alpha }{a}_n+\frac{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)+1+\alpha )\right]}{1-\alpha }{b}_n\right)\le 2.$$

The following theorem establishes the extreme points of the closed convex hull of $\widetilde{\overline{\mathcal{H}}{\mathcal{S}}_q^m}(c,\gamma ,\lambda ,\alpha ),$ represented by $clco\widetilde{\overline{\mathcal{H}}{\mathcal{S}}_q^m}(c,\gamma ,\lambda ,\alpha ).$

Theorem 3. 

${\xi }_m\in clco\widetilde{\overline{\mathcal{H}}{\mathcal{S}}_q^m}(c,\gamma ,\lambda ,\alpha ),$ if and only if,

$${\xi }_m\left(\tau \right)=\sum _{n=1}^{\infty }\left(X_n{h}_n+Y_n{g}_{m_n}\right)$$

(15)

where

$$h_1\left(\tau \right)=\tau ,h_n\left(\tau \right)=\tau -\frac{\left(1-\alpha \right)\left|c\right|}{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-2+\left(1-\alpha \right)\left|c\right|\right]}{\tau }^n,n=2,3,\cdots$$

$$g_{m_n}\left(\tau \right)=\tau +{(-1)}^m\frac{\left(1-\alpha \right)\left|c\right|}{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)+2-\left(1-\alpha \right)\left|c\right|\right]}{\stackrel{\_}{\tau }}^n,n=1,2,\cdots$$

and

$$\sum _{n=1}^{\infty }\left(X_n+Y_n\right)=1,X_n\ge 0,Y_n\ge 0.$$

In particular, the two extreme values of $clco\widetilde{\overline{\mathcal{H}}{\mathcal{S}}_q^m}(c,\gamma ,\lambda ,\alpha )$ are $h_n$ and $g_{m_n}.$

Proof. 

From (15), we have

$$\begin{array}{ccc}\hfill {\xi }_m\left(\tau \right)& =& \sum _{n=1}^{\infty }\left(X_n{h}_n+Y_n{g}_{m_n}\right)\hfill \\ & =& \sum _{n=1}^{\infty }\left(X_n+Y_n\right)\tau -\sum _{n=2}^{\infty }\frac{\left(1-\alpha \right)\left|c\right|}{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-2+\left(1-\alpha \right)\left|c\right|\right]}{X}_n{\tau }^n\hfill \\ & & +{(-1)}^m\sum _{n=1}^{\infty }\frac{\left(1-\alpha \right)\left|c\right|}{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)+2-\left(1-\alpha \right)\left|c\right|\right]}{Y}_n{\stackrel{\_}{\tau }}^n.\hfill \end{array}$$

Therefore,

$$\sum _{n=2}^{\infty }\frac{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-2+\left(1-\alpha \right)\left|c\right|\right]}{\left(1-\alpha \right)\left|c\right|}\left(\frac{\left(1-\alpha \right)\left|c\right|}{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-2+\left(1-\alpha \right)\left|c\right|\right]}\right)X_n$$

$$+\sum _{n=1}^{\infty }\frac{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)+2-\left(1-\alpha \right)\left|c\right|\right]}{\left(1-\alpha \right)\left|c\right|}\left(\frac{\left(1-\alpha \right)\left|c\right|}{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)+2-\left(1-\alpha \right)\left|c\right|\right]}\right)Y_n$$

$$=\sum _{n=2}^{\infty }{X}_n+\sum _{n=1}^{\infty }{Y}_n=1-X_1\le 1.$$

Thus, ${\xi }_m\in clco\widetilde{\overline{\mathcal{H}}{\mathcal{S}}_q^m}(c,\gamma ,\lambda ,\alpha )$. Conversely, suppose that ${\xi }_m\in clco\overline{\mathcal{H}}{\mathcal{S}}_q^m(c,\gamma ,\lambda ,\alpha )$. Set

$$X_n=\frac{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[2{\varphi }_q^{\lambda }\left(n\right)-2+\left(1-\alpha \right)\left|c\right|\right]}{\left(1-\alpha \right)\left|c\right|}\left|a_n\right|,n=2,3,\cdots ,$$

and

$$Y_n=\frac{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)+2-\left(1-\alpha \right)\left|c\right|\right]}{\left(1-\alpha \right)\left|c\right|}\left|b_n\right|,n=1,2,3,\cdots ,$$

where

$$\sum _{n=1}^{\infty }\left(X_n+Y_n\right)=1.$$

Then,

$$\begin{array}{ccc}\hfill {\xi }_m\left(\tau \right)& =& \tau -\sum _{n=2}^{\infty }{a}_n{\tau }^n+{(-1)}^m\sum _{n=1}^{\infty }{b}_n{\stackrel{\_}{\tau }}^n\hfill \\ & =& \tau -\sum _{n=2}^{\infty }\frac{\left(1-\alpha \right)\left|c\right|}{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-2+\left(1-\alpha \right)\left|c\right|\right]}{X}_n{\tau }^n\hfill \\ & & +{(-1)}^m\sum _{n=2}^{\infty }\frac{\left(1-\alpha \right)\left|c\right|}{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)+2-\left(1-\alpha \right)\left|c\right|\right]}{Y}_n{\stackrel{\_}{\tau }}^n\hfill \\ & =& \tau -\sum _{n=2}^{\infty }\left[X_n\left(h_n\left(\tau \right)\right)-\tau \right]+\sum _{n=1}^{\infty }\left[Y_n\left(g_{m_n}\left(\tau \right)\right)-\tau \right]\hfill \\ & =& \sum _{n=1}^{\infty }\left(X_n{h}_n+Y_n{g}_{m_n}\right).\hfill \end{array}$$

We may now derive that $0\le X_n\le 1,(n\ge 2)$ and $0\le Y_n\le 1,(n\ge 1)$ from Theorem 2. Thus,

$$X_1=1-\sum _{n=2}^{\infty }{X}_n-\sum _{n=1}^{\infty }{Y}_n\ge 0.$$

Therefore,

$$\sum _{n=1}^{\infty }(X_n{h}_n+Y_n{g}_{m_n})={\xi }_m\left(\tau \right)$$

as required in the theorem. □

Finally, we establish the distortion theorem $\xi \in \widetilde{\overline{\mathcal{H}}{\mathcal{S}}_q^m}(c,\gamma ,\lambda ,\alpha ).$

Theorem 4. 

Let ${\xi }_m\in \widetilde{\overline{\mathcal{H}}{\mathcal{S}}_q^m}(c,\gamma ,\lambda ,\alpha )$ where $\left|\tau \right|=r<1.$ Then,

$$\left|{\xi }_m\left(\tau \right)\right|\le (1+b_1)r+\left(\begin{array}{c}\frac{\left(1-\alpha \right)\left|c\right|}{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(2\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(2\right)-2+\left(1-\alpha \right)\left|c\right|\right]}\\ \\ -\frac{4-\left(1-\alpha \right)\left|c\right|}{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(2\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(2\right)-2+\left(1-\alpha \right)\left|c\right|\right]}\left|b_1\right|\end{array}\right)r^2$$

and

$$\left|{\xi }_m\left(\tau \right)\right|\ge (1-b_1)r-\left(\begin{array}{c}\frac{\left(1-\alpha \right)\left|c\right|}{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(2\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(2\right)-2+\left(1-\alpha \right)\left|c\right|\right]}\\ \\ -\frac{4-\left(1-\alpha \right)\left|c\right|}{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(2\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(2\right)-2+\left(1-\alpha \right)\left|c\right|\right]}\left|b_1\right|\end{array}\right)r^2.$$

Proof. 

The right-hand inequality will be shown. The left-hand inequality proof is assumed to be identical and is removed. Let ${\xi }_m\in \widetilde{\overline{\mathcal{H}}{\mathcal{S}}_q^m}(c,\gamma ,\lambda ,\alpha )$. By calculating the absolute value of ${\xi }_m$, we obtain

$$\begin{array}{ccc}\hfill \left|{\xi }_m\left(\tau \right)\right|& \le & \left(1+\left|b_1\right|\right)r+\sum _{n=2}^{\infty }\left[\left|a_n\right|+\left|b_n\right|\right]){\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m{r}^n\hfill \\ & \le & \left(1+\left|b_1\right|\right)r+r^2\sum _{n=2}^{\infty }\left(\left|a_n\right|+\left|b_n\right|\right){\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\hfill \end{array}$$

$$\begin{array}{ccc}& =& \left(1+\left|b_1\right|\right)r+\frac{\left(1-\alpha \right)\left|b\right|r^2}{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-2+\left(1-\alpha \right)\left|c\right|\right]}\hfill \\ & & \times \sum _{n=2}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left(\frac{2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-2+\left(1-\alpha \right)\left|c\right|}{\left(1-\alpha \right)\left|c\right|}\left|a_n\right|+\frac{2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-2+\left(1-\alpha \right)\left|c\right|}{\left(1-\alpha \right)\left|c\right|}\left|b_n\right|\right)\hfill \end{array}$$

$$\begin{array}{ccc}& \le & \left(1+\left|b_1\right|\right)r+\frac{\left(1-\alpha \right)\left|c\right|r^2}{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(2\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(2\right)-2+\left(1-\alpha \right)\left|c\right|\right]}\hfill \\ & & \times \sum _{n=2}^{\infty }{\left(\widetilde{{\varphi }_q^{\lambda }}\left(n\right)\right)}^m\left(\begin{array}{c}\frac{2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)-2+\left(1-\alpha \right)\left|c\right|}{\left(1-\alpha \right)\left|c\right|}\left|a_n\right|\\ \\ +\frac{2\widetilde{{\varphi }_q^{\lambda }}\left(n\right)+2-\left(1-\alpha \right)\left|c\right|}{\left(1-\alpha \right)\left|c\right|}\left|b_n\right|\end{array}\right)\hfill \end{array}$$

$$\begin{array}{ccc}& \le & \left(1+\left|b_1\right|\right)r+\frac{\left(1-\alpha \right)\left|c\right|}{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(2\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(2\right)-2+\left(1-\alpha \right)\left|c\right|\right]}\hfill \\ & & \times \left(1-\frac{\left[4-(1-\alpha )\left|c\right|\right]}{(1-\alpha )\left|c\right|}\left|b_1\right|\right)r^2\hfill \\ & \le & \left(1+\left|b_1\right|\right)r+\left(\begin{array}{c}\frac{\left(1-\alpha \right)\left|c\right|}{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(2\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(2\right)-2+\left(1-\alpha \right)\left|c\right|\right]}\\ \\ -\frac{4-(1-\alpha )\left|c\right|}{{\left({\varphi }_q^{\lambda }\left(2\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(2\right)-2+\left(1-\alpha \right)\left|c\right|\right]}\left|b_1\right|\end{array}\right)r^2.\hfill \end{array}$$

The result is sharp for

$$\xi \left(\tau \right)=\tau +\left|b_1\right|\stackrel{\_}{\tau }+\left(\begin{array}{c}\frac{\left(1-\alpha \right)\left|c\right|}{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(2\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(2\right)-2+\left(1-\alpha \right)\left|c\right|\right]}\\ \\ -\frac{4-(1-\alpha )\left|c\right|}{{\left(\widetilde{{\varphi }_q^{\lambda }}\left(2\right)\right)}^m\left[2\widetilde{{\varphi }_q^{\lambda }}\left(2\right)-2+\left(1-\alpha \right)\left|c\right|\right]}\left|b_1\right|\end{array}\right){\stackrel{\_}{\tau }}^2,$$

where

$$\left|b_1\right|\le \frac{(1-\alpha )\left|c\right|}{4-(1-\alpha )\left|c\right|}.$$

□

3. Conclusions

Studies of q-calculus and symmetric q-calculus in connection with geometric function theory and, in particular, harmonic univalent functions, are relatively new, and only a small number of papers have been written on the subject so far. We used the symmetric quantum, or the symmetric (or q-), calculus operator theory for the study of a new family of harmonic functions. First, we developed a generalized symmetric Sălăgean q-differential operator for harmonic functions in an open unit disk. By using the newly developed q-analogue of the differential operator, we investigated and explored new sub-classes of harmonic functions in the open unit disk. Some useful results, such as sufficient necessary coefficient bounds, closed convex hulls, and distortion theorems were established in this article for newly defined classes of harmonic functions. In addition, we proved that these results are sharp, and, for the special values of the parameters, we determined certain consequences of our main results. More information related to this work can be found at [33,34,35,36,37].