Starlike Functions with Respect to (ℓ, κ)-Symmetric Points Associated with the Vertical Domain

Abstract

The study of various subclasses of univalent functions involving the solutions to various differential equations is not totally new, but studies of analytic functions with respect to $(\mathsf{\ell },\kappa )$-symmetric points are rarely conducted. Here, using a differential operator which was defined using the Hadamard product of Mittag–Leffler function and general analytic function, we introduce a new class of starlike functions with respect to $(\mathsf{\ell },\kappa )$-symmetric points associated with the vertical domain. To define the function class, we use a Carathéodory function which was recently introduced to study the impact of various conic regions on the vertical domain. We obtain several results concerned with integral representations and coefficient inequalities for functions belonging to this class. The results obtained by us here not only unify the recent studies associated with the vertical domain but also provide essential improvements of the corresponding results.

1. Introduction and Definition

We will begin with a short introduction of a very vast theory that led to the definition of a generalized M-series (see [1], Equation (1)). For ${\tau }_i\in \mathbb{C}(i=1,\dots ,r)\mathrm{and}{\sigma }_j\in \mathbb{C}\setminus Z_0^{-}=\{0,-1,\dots \}(j=1,\dots ,s)$, the Fox–Wright function ${}_r\mathsf{\Lambda }_s$ is defined by [2] (Equation (1.6)), [3] (p. 21) and [4] (p. 19).

$${}_r\mathsf{\Lambda }_s\left[\begin{array}{cccc}({\tau }_1,A_1)& \dots & ({\tau }_r,A_r)& \\ ({\sigma }_1,B_1)& \dots & ({\sigma }_s,B_s)\end{array};\xi \right]=\sum _{n=0}^{\infty }{\displaystyle \frac{{\prod }_{i=1}^r\mathsf{\Gamma }({\tau }_j+A_{j}n)}{{\prod }_{j=1}^s\mathsf{\Gamma }({\sigma }_j+B_{j}n)}}{\displaystyle \frac{{\xi }^n}{n!}}.$$

(1)

where $Re\left(A_i\right)>0,(i=1,\dots ,r)\mathrm{and}Re\left(B_j\right)>0\in \mathbb{C}(j=1,\dots ,s)$. The defined series (1) converges in the entire complex plane if $\Delta :=Re\left({\sum }_{j=1}^s{B}_j-{\sum }_{i=1}^r{A}_i\right)>-1$. Further if $\Delta :=-1$, then the series in (1) converges for $\left|z\right|<L$, and $\left|z\right|=L$ under the condition $Re\left({\sum }_{j=1}^s{\sigma }_j-{\sum }_{i=1}^r{\tau }_i+{\displaystyle \frac{r+s}{2}}\right)>{\displaystyle \frac{1}{2}}$, where $L=\left({\prod }_{j=1}^r{A}_j^{-A_j}\right)\left({\prod }_{j=1}^s{B}_j^{B_j}\right)$ (see [5], Definition 2).

Lin and Srivastava ([6], Equation (8)) studied a generalization of the Hurwitz–Lerch zeta function ${\varphi }_{\tau ,\sigma }^{k,ϵ}(\xi ,m,\vartheta )$ given by

$${\varphi }_{\tau ,\sigma }^{k,ϵ}(\xi ,m,\vartheta )=\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(\tau \right)}_{kn}}{{\left(\sigma \right)}_{ϵn}}}{\displaystyle \frac{{\xi }^n}{{(n+\vartheta )}^m}},$$

where $\tau \in \mathbb{C};\sigma ,\tau \in \mathbb{C}\setminus Z_0^{-};k,ϵ\in {\mathbb{R}}^{+};k<ϵ$ when $m,\xi \in \mathbb{C};k=ϵ$ and $m\in \mathbb{C}$ when $\left|\xi \right|<1$; $k=ϵ$ and $Re(m-\tau +\sigma )>1$ when $\left|\xi \right|=1$. Various extensions of the existing special functions which naturally arise from the solutions of the ordinary differential equations have proven to be very useful tools in analysis.

Using the Hadamard product or convolution, Karthikeyan et al. [7] introduced the operator $D_{\delta }^m({\tau }_1,{\sigma }_1;\eta ,\vartheta )\phi :\mathcal{A}\to \mathcal{A}$ which is defined by

$$D_{\delta }^m({\tau }_1,{\sigma }_1;\eta ,\vartheta )\phi =\xi +\sum _{n=2}^{\infty }{\left[1+(n-1)\delta \right]}^m{\displaystyle \frac{{\left({\tau }_1\right)}_{n-1}\dots {\left({\tau }_r\right)}_{n-1}}{{\left({\sigma }_1\right)}_{n-1}\dots {\left({\sigma }_s\right)}_{n-1}}}{\displaystyle \frac{\mathsf{\Gamma }\left(\vartheta \right)a_n{\xi }^n}{(n-1)!\mathsf{\Gamma }\left(\eta \right(n-1)+\vartheta )}}$$

(2)

where ${\tau }_j\in \mathbb{C}(j=1,\dots ,r);{\sigma }_j\in \mathbb{C}\setminus Z_0^{-}=\{0,-1,\dots \}(j=1,\dots ,s);m\in N_0;\delta \ge 0;\eta ,\vartheta \in \mathbb{C}$ and $Re\left(\eta \right)>0$. Letting $r=2$, $s=1$, ${\tau }_1={\sigma }_1$ and ${\tau }_2=1$ in (2), we get the operator

$$D_{\delta }^m(\eta ,\vartheta )\phi \left(\xi \right)=\xi +\sum _{n=2}^{\infty }{\left[1+(n-1)\delta \right]}^m{\displaystyle \frac{\mathsf{\Gamma }\left(\vartheta \right)a_n{\xi }^n}{\mathsf{\Gamma }\left(\eta \right(n-1)+\vartheta )}}.$$

The operator $D_{\delta }^m(\eta ,\vartheta )\phi$ was introduced by Elhaddad et al. ([8], Equation (1.6)) and was further studied by Mashwan et al. ([9], Equation (16)). For the choice of $m=0$ in (2), the operator $D_{\delta }^m({\tau }_1,{\sigma }_1;\eta ,\vartheta )\phi$ reduces to the well-known Dziok–Srivastava operator [10].

Fixing generalization and unification as our primary objectives, we will study a subclass of analytic functions which will be defined involving the operator $D_{\delta }^m({\tau }_1,{\sigma }_1;\eta ,\vartheta )\phi$. Most of the geometrically defined subclasses of univalent functions have analytic characterizations which comprise derivatives of one order or more. Hence, any study involving a versatile differential operator $D_{\delta }^m({\tau }_1,{\sigma }_1;\eta ,\vartheta )\phi$ not only generalizes the well-known subclasses of analytic functions but also generates new families.

We denote by $\mathcal{A}$ the class of analytic functions in $\mathsf{\Theta }$ with a Taylor series expansion of the form

$$\phi \left(\xi \right)=\xi +\sum _{n=2}^{\infty }{a}_n{\xi }^n,\xi \in \mathsf{\Theta }.$$

(3)

Let $\mathcal{S}$ be the class of functions $\phi \in \mathcal{A}$ which are one–one in $\mathsf{\Theta }$. The following are the well-known subclasses associated with $\mathcal{S}$:

• Carathéodory’s function (see [11]): $\mathcal{P}=\left\{p\in \mathcal{H}(1,1);Re\left(p\left(\xi \right)\right)>0;\forall \xi \in \mathsf{\Theta }\right\}.$
• Starlike function class [12,13]: ${\mathcal{S}}^{*}=\left\{\phi \in \mathcal{A};Re\left({\displaystyle \frac{\xi {\phi }^{\prime }\left(\xi \right)}{\phi \left(\xi \right)}}\right)>0;\forall \xi \in \mathsf{\Theta }\right\}.$
• Convex function class (see [12,13]): $\mathcal{C}=\left\{\phi \in \mathcal{A};Re\left(1+{\displaystyle \frac{\xi {\phi }^{″}\left(\xi \right)}{{\phi }^{\prime }\left(\xi \right)}}\right)>0;\forall \xi \in \mathsf{\Theta }\right\}.$
• Close-to-convex function class [14]: $\mathcal{K}=\left\{\phi \in \mathcal{A};Re\left({\displaystyle \frac{\xi {\phi }^{\prime }\left(\xi \right)}{\chi \left(\zeta \right)}}\right)>0;\chi \left(\xi \right)\in {\mathcal{S}}^{*}\right\}.$
• Quasi-convex function class [14]: ${\mathcal{C}}^{*}=\left\{\phi \in \mathcal{A};Re\left({\displaystyle \frac{{\left(\xi {\phi }^{\prime }\left(\xi \right)\right)}^{\prime }}{{\chi }^{\prime }\left(\xi \right)}}\right)>0;\chi \left(\zeta \right)\in \mathcal{C}\right\}.$

All the subclasses characterized above were geometrically defined; here, we omit those details since they can be found in any standard text on univalent function theory. Refer to [12,13] for a detailed study.

For $\phi \in \mathcal{A}$, Sakaguchi ([15], Theorem 3) established that functions satisfying

$$Re\left\{{\displaystyle \frac{\xi {\phi }^{\prime }\left(\xi \right)}{\frac{1}{\kappa }{\sum }_{\nu =0}^{\kappa -1}\frac{\phi \left({\omega }^{\nu }\xi \right)}{{\omega }^{\nu \mathsf{\ell }}}}}\right\}>0,(\xi \in \mathsf{\Theta }),$$

are univalent and close to convex in $\mathsf{\Theta }$, provided that $\kappa$ is a positive integer and $\omega =exp\left({\displaystyle \frac{2\pi i}{\kappa }}\right)$. Extending this result, Wang et al. [16] (see also [17]) studied the respective classes of starlike and convex functions with respect to $\kappa$-symmetric points, which are defined by subjecting $\phi \in \mathcal{A}$ to satisfy the conditions

$${\displaystyle \frac{\xi {\phi }^{\prime }\left(\xi \right)}{{\phi }_{\kappa }\left(\xi \right)}}\prec \mathsf{\Psi }\left(\xi \right)\mathrm{and}{\displaystyle \frac{{\left(\xi {\phi }^{\prime }\left(\xi \right)\right)}^{\prime }}{{\phi }_{\kappa }^{\prime }\left(\xi \right)}}\prec \mathsf{\Psi }\left(\xi \right),(\kappa \in \mathcal{N}),$$

(4)

where ≺ denotes subordination, ${\phi }_{\kappa }\left(\xi \right)={\displaystyle \frac{1}{\kappa }}{\sum }_{\nu =0}^{\kappa -1}{\displaystyle \frac{\phi \left({\omega }^{\nu }\xi \right)}{{\omega }^{\nu }}},(\phi \in \mathcal{A})$ and $\mathsf{\Psi }\left(\xi \right)\in \mathcal{P}$ is symmetric with respect to the real axis which has a series expansion of the form

$$\mathsf{\Psi }\left(\xi \right)=1+{\psi }_1\xi +{\psi }_2{\xi }^2+{\psi }_3{\xi }^3+\cdots ,({\psi }_1\ne 0;\xi \in \mathsf{\Theta }).$$

(5)

Here ${\mathcal{S}}_s^{\kappa }\left(\mathsf{\Psi }\right)$ and ${\mathcal{C}}_s^{\kappa }\left(\mathsf{\Psi }\right)$ denote the classes of starlike and convex functions with respect to $\kappa$-symmetric points. Karthikeyan [18] further extended the ${\mathcal{S}}_s^{\kappa }\left(\mathsf{\Psi }\right)$ and ${\mathcal{C}}_s^{\kappa }\left(\mathsf{\Psi }\right)$ by replacing ${\phi }_{\kappa }\left(\xi \right)$ with ${\phi }_{\mathsf{\ell },\kappa }\left(\xi \right)$ in (4), where ${\phi }_{\mathsf{\ell },\kappa }\left(\xi \right)$ is defined as follows:

$${\phi }_{\mathsf{\ell },\kappa }\left(\xi \right)={\displaystyle \frac{1}{\kappa }}\sum _{\nu =0}^{\kappa -1}{\displaystyle \frac{\phi \left({\omega }^{\nu }\xi \right)}{{\omega }^{\nu \mathsf{\ell }}}},\left(\phi \in \mathcal{A}\right),$$

(6)

$\omega =exp\left({\displaystyle \frac{2\pi i}{\kappa }}\right)$ and $\kappa \ge 1$ is a fixed integer, $\mathsf{\ell }=0,1,2,\dots ,\kappa -1$. The decomposition (6) was introduced by Liczberski and Połubiński in [19], and its use to study various subclasses of univalent function theory was initiated by Karthikeyan [18]. For $\phi \left(\xi \right)=\xi +{\sum }_{n=2}^{\infty }{a}_n{\xi }^n$, we can get

$$\begin{array}{c}\hfill {\phi }_{\mathsf{\ell },\kappa }\left(\xi \right)=\sum _{n=1}^{\infty }{a}_n{{\rm Y}}_{n,\mathsf{\ell }}^{\kappa }{\xi }^n,(a_1=1),{{\rm Y}}_{n,\mathsf{\ell }}^{\kappa }={\displaystyle \frac{1}{\kappa }}\sum _{\nu =0}^{\kappa -1}{\left[exp\left({\displaystyle \frac{2\pi i}{\kappa }}\right)\right]}^{(n-\mathsf{\ell })\nu }.\end{array}$$

(7)

If $\nu$ is an integer, then the following identities follow directly from (6):

$$\begin{array}{cc}\hfill {\phi }_{\mathsf{\ell },\kappa }\left({\omega }^{\nu }\xi \right)& ={\omega }^{\nu \mathsf{\ell }}{\phi }_{\mathsf{\ell },\kappa }\left(\xi \right),\hfill \\ \hfill {\phi }_{\mathsf{\ell },\kappa }^{\prime }\left({\omega }^{\nu }\xi \right)& ={\omega }^{\nu \mathsf{\ell }-\nu }{\phi }_{\mathsf{\ell },\kappa }^{\prime }\left(\xi \right)={\displaystyle \frac{1}{\kappa }}\sum _{\nu =0}^{\kappa -1}{\displaystyle \frac{{\phi }^{\prime }\left({\omega }^{\nu }\xi \right)}{{\omega }^{\nu \mathsf{\ell }-\nu }}},\hfill \\ \hfill {\phi }_{\mathsf{\ell },\kappa }^{″}\left({\omega }^{\nu }\xi \right)& ={\omega }^{\nu \mathsf{\ell }-2\nu }{\phi }_{\mathsf{\ell },\kappa }^{″}\left(\xi \right)={\displaystyle \frac{1}{\kappa }}\sum _{\nu =0}^{\kappa -1}{\displaystyle \frac{{\phi }^{″}\left({\omega }^{\nu }\xi \right)}{{\omega }^{\nu \mathsf{\ell }-2\nu }}}.\hfill \end{array}$$

(8)

For various studies related to analytic functions with respect to $(\mathsf{\ell },\kappa )$-symmetric points, refer to [20,21].

Further, the classes ${\mathcal{S}}_s^{\kappa }\left(\mathsf{\Psi }\right)$ and ${\mathcal{C}}_s^{\kappa }\left(\mathsf{\Psi }\right)$ were restudied by various prominent authors [22,23] after restricting the superordinate function $\mathsf{\Psi }\left(\xi \right)$ in (4). In particular, studies where the function was chosen to map the unit disc onto the conic region were interesting. In this direction, Aracı et al. [24] recently studied the impact of various conic regions when they become sandwiched in a vertical domain. They defined the following function to study the effect of the vertical domain on the conic region:

$$\mathsf{\Lambda }[x,y;\mathsf{\Psi }\left(\xi \right)]=1+i{\displaystyle \frac{y-x}{\pi }}log\left[{\displaystyle \frac{\mathsf{\Psi }\left(\xi \right)(1-{\lambda }^2)+(1+{\lambda }^2)}{2}}\right],\left(\xi \in \mathsf{\Theta }=\left\{\xi :\left|\xi \right|<1\right\}\right)$$

(9)

where $\mathsf{\Psi }\left(\xi \right)\in \mathcal{P}$ (Carathéodory’s function class; see [11]), and $\lambda$ is defined by

$$\lambda =exp\left(i{\displaystyle \frac{1-x}{y-x}}\pi \right)=cos\theta +isin\theta ,\theta ={\displaystyle \frac{1-x}{y-x}}\pi \in (0,\pi ),$$

(10)

and these powers are considered at the main branch, that is $log1=0$. In [24], the authors established that $\mathsf{\Lambda }[x,y;\mathsf{\Psi }(\xi \left)\right]\in \mathcal{P}$ if $\mathsf{\Psi }\left(\xi \right)\in \mathcal{P}$. If we let $\mathsf{\Psi }\left(\xi \right)={\displaystyle \frac{1+\xi }{1-\xi }}$ (which is an extremal function in the class $\mathcal{P}$) in (9), we will see $\mathsf{\Lambda }[x,y;\mathsf{\Psi }(\xi \left)\right]$ reducing to $p_{x,y}\left(\xi \right)$, which maps $\mathsf{\Theta }$ onto a vertical domain and is of the form

$$p_{x,y}\left(\xi \right)=1+i{\displaystyle \frac{y-x}{\pi }}log\left({\displaystyle \frac{1-{\lambda }^2\xi }{1-\xi }}\right).$$

(11)

Note that the function $p_{x,y}\left(\xi \right)$ is a conformal mapping of $\overline{\mathsf{\Theta }}\setminus \left\{1,{\lambda }^{-2}\right\}$ with

$$p_{x,y}\left(\mathsf{\Theta }\right)=\left\{w\in \mathbb{C}:x<Re\left(w\right)<y\right\}={\mathsf{\Omega }}_{x,y}.$$

Another prominent study involving the vertical domain was presented by Karger et al. [25], who defined an analytic function ${\mathcal{H}}_x$ and the vertical strip ${\mathsf{\Omega }}_x$ which is as follows:

$${\mathcal{H}}_x={\displaystyle \frac{1}{2isinx}}log\left({\displaystyle \frac{1+e^{ix}\xi }{1+e^{-ix}\xi }}\right)$$

and

$${\mathsf{\Omega }}_x=\left\{w\in \mathbb{C}:{\displaystyle \frac{x-\pi }{2sinx}}<Re\left(w\right)<{\displaystyle \frac{x}{2sinx}}\right\},$$

where $\pi /2\le x<\pi$. The function ${\mathcal{H}}_x$ is convex and univalent in $\mathsf{\Theta }$. In addition, ${\mathcal{H}}_x$ maps the unit disc onto ${\mathsf{\Omega }}_x$ or onto the convex hull of three points (one of which may be at infinity) on the boundary of ${\mathsf{\Omega }}_x$.

In [26], the authors presented various interesting estimates which have indeed spurred renewed interest in studying the impact of various classes of functions when embedded inside a vertical domain. In Figure 1 and Figure 2, we illustrate the impact of $\mathsf{\Lambda }[x,y;\mathsf{\Psi }(\xi \left)\right]$ on ${\displaystyle \frac{2\sqrt{1+\xi }}{1+e^{-\xi }}}$, which is defined using the principal branch of the square root function, with a branch cut along the interval $(-\infty ,-1]$ (the balloon-shaped region was recently studied by Ahmed et al. [27]) and ${\displaystyle \frac{2+tanz}{2}}$, respectively. Figure 1a is the mapping of $\mathsf{\Theta }$ under the transformation ${\displaystyle \frac{2\sqrt{1+\xi }}{1+e^{-\xi }}}$, whereas Figure 1b exhibits the impact of $\mathsf{\Lambda }\left[x,y;\mathsf{\Psi }\left(\xi \right)\right]$ on $\mathsf{\Psi }\left(\xi \right)={\displaystyle \frac{2\sqrt{1+\xi }}{1+e^{-\xi }}}$ for a choice of the parameters $x={\displaystyle \frac{3}{4}}$ and $y=2$. Similarly, Figure 2a is the mapping of $\mathsf{\Theta }$ under the transformation ${\displaystyle \frac{2+tanz}{2}}$, whereas Figure 2b exhibits the impact of $\mathsf{\Lambda }\left[x,y;\mathsf{\Psi }\left(\xi \right)\right]$ on $\mathsf{\Psi }\left(\xi \right)={\displaystyle \frac{2+tanz}{2}}$ for a choice of the parameters $x={\displaystyle \frac{3}{4}}$ and $y=2$. Refer to [25,28] for information on the geometric and analytic properties of analytic functions associated with the vertical domain.

Throughout this paper, we assume that

$$m,r,s\in N_0,\omega =exp\left({\displaystyle \frac{2\pi i}{\kappa }}\right)$$

and

$${\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )={\displaystyle \frac{1}{\kappa }}\sum _{\nu =0}^{\kappa -1}{\omega }^{-\nu \mathsf{\ell }}{D}_{\delta }^m({\tau }_1,{\sigma }_1;\eta ,\vartheta )\phi \left({\omega }^{\nu }\xi \right).$$

(12)

Clearly, for $\mathsf{\ell }=0$ and $\kappa =1$, we have

$${\phi }_{0,1}^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )=D_{\delta }^m({\tau }_1,{\sigma }_1;\eta ,\vartheta )\phi \left(\xi \right).$$

Also for simplification, we will denote

$${\mathrm{T}}_n={\left[1+(n-1)\delta \right]}^m{\displaystyle \frac{{\left({\tau }_1\right)}_{n-1}\dots {\left({\tau }_r\right)}_{n-1}}{{\left({\sigma }_1\right)}_{n-1}\dots {\left({\sigma }_s\right)}_{n-1}}}{\displaystyle \frac{\mathsf{\Gamma }\left(\vartheta \right)}{(n-1)!\mathsf{\Gamma }\left(\eta \right(n-1)+\vartheta )}}.$$

(13)

Studies involving the operator $D_{\delta }^m({\tau }_1,{\sigma }_1;\eta ,\vartheta )\phi \left(\xi \right)$ will not only generalize the classes stated in this section but also generalize the well-known classes like the Pascu class [29] and Mocanu class [30]. Motivated by Breaz et al. [31], now, we will define presumably new subclasses of starlike functions associated with the vertical domain using the function ${\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )$ defined in (12).

Definition 1.

A function $\phi \left(\xi \right)\in \mathcal{A}$ belongs to the class ${\mathcal{RS}}_s^{\delta ,m}(x,y;{\tau }_1,{\sigma }_1;\eta ,\vartheta ;\mathsf{\Psi }\left(\xi \right))$ if and only if it satisfies

$${\displaystyle \frac{\xi {\left[D_{\delta }^m({\tau }_1,{\sigma }_1;\eta ,\vartheta )\phi \left(\xi \right)\right]}^{\prime }}{{\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )}}\prec \mathsf{\Lambda }[x,y;\mathsf{\Psi }\left(\xi \right)],$$

(14)

where the parameter range is as given in the operator $D_{\delta }^m({\tau }_1,{\sigma }_1;\eta ,\vartheta )\phi$ and $\mathsf{\Lambda }[x,y;\mathsf{\Psi }(\xi \left)\right]$ is defined by (9).

Remark 1.

Some special cases of the ${\mathcal{RS}}_s^{\delta ,m}(x,y;{\tau }_1,{\sigma }_1;\eta ,\vartheta ;\mathsf{\Psi }\left(\xi \right))$ are as follows:

1. 

Letting $r=2$, $s=1$, ${\tau }_1={\sigma }_1$, ${\tau }_2=1$, $m=\eta =\mathsf{\ell }=0$, $\delta =\kappa =1$ and $\mathsf{\Psi }\left(\xi \right)=(1+\xi )/(1-\xi )$ in Definition 1, the ${\mathcal{RS}}_s^{\delta ,m}(x,y;{\tau }_1,{\sigma }_1;\eta ,\vartheta ;\mathsf{\Psi }\left(\xi \right))$ will reduce to the $\mathcal{S}(x,y)$ which satisfies the condition

$$x<Re\left({\displaystyle \frac{\xi {\phi }^{\prime }\left(\xi \right)}{\phi \left(\xi \right)}}\right)<y,\left(0\le x<1<y;\xi \in \mathbb{B}\right).$$

The class $\mathcal{S}(x,y)$ was studied by Sim and Kwon ([32], Definition 1).

2. 

Letting $r=2$, $s=1$, ${\tau }_1={\sigma }_1$, ${\tau }_2=1$, $m=\eta =0$, $\delta =1$, $\mathsf{\ell }=1$, $\kappa =2$ and $\mathsf{\Psi }\left(\xi \right)=(1+\xi )/(1-\xi )$ in Definition 1, the ${\mathcal{RS}}_s^{\delta ,m}(x,y;{\tau }_1,{\sigma }_1;\eta ,\vartheta ;\mathsf{\Psi }\left(\xi \right))$ will reduce to the ${\mathcal{S}}_s(x,y)$ which satisfies the condition

$$x<Re\left({\displaystyle \frac{2\xi {\phi }^{\prime }\left(\xi \right)}{\phi \left(\xi \right)-\phi (-\xi )}}\right)<y,\left(0\le x<1<y;\xi \in \mathbb{B}\right).$$

3. 

Letting $\eta =\mathsf{\ell }=0$ and $m=\kappa =1$ in Definition 1, the ${\mathcal{RS}}_s^{\delta ,m}(x,y;{\tau }_1,{\sigma }_1;\eta ,\vartheta ;\mathsf{\Psi }\left(\xi \right))$ will reduce to the ${\mathcal{M}}_s^{\delta }(x,y;{\tau }_1,{\sigma }_1;\eta ,\vartheta ;\mathsf{\Psi }\left(\xi \right))$ which satisfies the condition

$${\displaystyle \frac{(1-\delta )\xi {\left[D^1({\tau }_1,{\sigma }_1;\eta ,\vartheta )\phi \left(\xi \right)\right]}^{\prime }+\delta \xi {\left(\xi {\left[D^1({\tau }_1,{\sigma }_1;\eta ,\vartheta )\phi \left(\xi \right)\right]}^{\prime }\right)}^{\prime }}{(1-\delta )D^1({\tau }_1,{\sigma }_1;\eta ,\vartheta )\phi \left(\xi \right)+\delta \xi {\left[D^1({\tau }_1,{\sigma }_1;\eta ,\vartheta )\phi \left(\xi \right)\right]}^{\prime }}}\prec \mathsf{\Lambda }[x,y;\mathsf{\Psi }\left(\xi \right)].$$

The class ${\mathcal{M}}_s^{\delta }(x,y;{\tau }_1,{\sigma }_1;\eta ,\vartheta ;\mathsf{\Psi }\left(\xi \right))$ is closely related to the subclass studied by Breaz et al. [31].

For completeness, we define the following.

Definition 2.

A function $\phi \left(\xi \right)\in \mathcal{A}$ belongs to the class ${\mathcal{RC}}_s^{\delta ,m}(x,y;{\tau }_1,{\sigma }_1;\eta ,\vartheta ;\mathsf{\Psi }\left(\xi \right))$ if and only it satisfies

$${\displaystyle \frac{{\left(\xi {\left[D_{\delta }^m({\tau }_1,{\sigma }_1;\eta ,\vartheta )\phi \left(\xi \right)\right]}^{\prime }\right)}^{\prime }}{{\left({\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )\right)}^{\prime }}}\prec \mathsf{\Lambda }[x,y;\mathsf{\Psi }\left(\xi \right)],$$

(15)

where the parameter range is as given in the operator $D_{\delta }^m({\tau }_1,{\sigma }_1;\eta ,\vartheta )\phi$ and $\mathsf{\Lambda }[x,y;\mathsf{\Psi }(\xi \left)\right]$ is defined by (9).

Remark 2.

Note that classes $\phi \left(\xi \right)\in {\mathcal{RC}}_s^{\delta ,m}(x,y;{\tau }_1,{\sigma }_1;\eta ,\vartheta ;\mathsf{\Psi }\left(\xi \right))$ if and only if $\xi {\phi }^{\prime }\left(\xi \right)\in {\mathcal{RS}}_s^{\delta ,m}(x,y;{\tau }_1,{\sigma }_1;\eta ,\vartheta ;\mathsf{\Psi }\left(\xi \right))$.

2. Integral Representations

We begin with the following.

By replacing $\xi$ with ${\omega }^{\nu }\xi$ in (14), then (3) will be of the form

$${\displaystyle \frac{{\omega }^{\nu }\xi {\left[D_{\delta }^m({\tau }_1,{\sigma }_1;\eta ,\vartheta )\phi \left({\omega }^{\nu }\xi \right)\right]}^{\prime }}{{\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;{\omega }^{\nu }\xi )}}=\mathsf{\Lambda }[x,y;\mathsf{\Psi }\left\{k\left({\omega }^{\nu }\xi \right)\right\}],(\nu =0,1,2,\dots ,\kappa -1).$$

(16)

where $k\left(\xi \right)$ is the Schwartz function. Using (8) in (16), we get

$${\displaystyle \frac{{\omega }^{\nu }\xi {\left[D_{\delta }^m({\tau }_1,{\sigma }_1;\eta ,\vartheta )\phi \left({\omega }^{\nu }\xi \right)\right]}^{\prime }}{{\omega }^{\nu \mathsf{\ell }}{\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )}}=\mathsf{\Lambda }[x,y;\mathsf{\Psi }\left(k\left({\omega }^{\nu }\xi \right)\right)],(\xi \in \mathsf{\Theta }).$$

(17)

Let $\nu =0,1,2,\dots ,\kappa -1$ in (17), respectively, and summing them, we get

$${\displaystyle \frac{\xi {\sum }_{\nu =0}^{\kappa -1}{\omega }^{\nu -\nu \mathsf{\ell }}{\left[D_{\delta }^m({\tau }_1,{\sigma }_1;\eta ,\vartheta )\phi \left(\xi \right)\right]}^{\prime }}{{\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )}}=\sum _{\nu =0}^{\kappa -1}\mathsf{\Lambda }[x,y;\mathsf{\Psi }\left(k\left({\omega }^{\nu }\xi \right)\right)],(\xi \in \mathsf{\Theta }).$$

Or equivalently,

$${\displaystyle \frac{{\left[{\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )\right]}^{\prime }}{{\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )}}-{\displaystyle \frac{1}{\xi }}={\displaystyle \frac{1}{\xi }}\left\{{\displaystyle \frac{1}{\kappa }}\sum _{\nu =0}^{\kappa -1}\mathsf{\Lambda }[x,y;\mathsf{\Psi }\left(k\left({\omega }^{\nu }\xi \right)\right)]-1\right\},(\xi \in \mathsf{\Theta }).$$

On integrating the above expression, we have

$$log\left({\displaystyle \frac{{\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )}{\xi }}\right)={\displaystyle \frac{1}{\kappa }}\sum _{\nu =0}^{\kappa -1}{\int }_0^{{\omega }^{\nu }\xi }{\displaystyle \frac{\left\{\mathsf{\Lambda }[x,y;\mathsf{\Psi }(k\left(t\right)\left)\right]-1\right\}}{t}}dt,(\xi \in \mathsf{\Theta }).$$

On further simplifying the expression, we get

$${\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )=\xi exp\left\{{\displaystyle \frac{1}{\kappa }}\sum _{\nu =0}^{\kappa -1}{\int }_0^{{\omega }^{\nu }\xi }{\displaystyle \frac{\left\{\mathsf{\Lambda }[x,y;\mathsf{\Psi }(k\left(t\right)\left)\right]-1\right\}}{t}}dt\right\},(\xi \in \mathsf{\Theta }).$$

From the above discussion, we can conclude that

Theorem 1.

Let $\phi \left(\xi \right)\in {\mathcal{RS}}_s^{\delta ,m}(x,y;{\tau }_1,{\sigma }_1;\eta ,\vartheta ;\mathsf{\Psi }\left(\xi \right))$ with ${\displaystyle \frac{{\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )}{\xi }}\ne 0$. Then

$${\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )=\xi exp\left\{{\displaystyle \frac{1}{\kappa }}\sum _{\nu =0}^{\kappa -1}{\int }_0^{{\omega }^{\nu }\xi }{\displaystyle \frac{\left\{\mathsf{\Lambda }[x,y;\mathsf{\Psi }(k\left(t\right)\left)\right]-1\right\}}{t}}dt\right\},(\xi \in \mathsf{\Theta }),$$

where ${\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )$ is defined by equality (12), $k\left(\xi \right)$ is analytic in Θ and $k\left(0\right)=0$, and $\mid k\left(\xi \right)\mid <1$.

Since ${\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )\in {\mathcal{RC}}_s^{\delta ,m}(x,y;{\tau }_1,{\sigma }_1;\eta ,\vartheta ;\mathsf{\Psi }\left(\xi \right))$ if and only if

$\xi {\left[{\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )\right]}^{\prime }\in {\mathcal{RS}}_s^{\delta ,m}(x,y;{\tau }_1,{\sigma }_1;\eta ,\vartheta ;\mathsf{\Psi }\left(\xi \right))$, we have the following result analogous to Theorem 1.

Theorem 2.

Let $\phi \left(\xi \right)\in {\mathcal{RC}}_s^{\delta ,m}(x,y;{\tau }_1,{\sigma }_1;\eta ,\vartheta ;\mathsf{\Psi }\left(\xi \right))$ with ${\displaystyle \frac{{\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )}{\xi }}\ne 0$. Then

$${\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )={\int }_0^{\xi }exp\left\{{\displaystyle \frac{1}{\kappa }}\sum _{\nu =0}^{\kappa -1}{\int }_0^{{\omega }^{\nu }\zeta }{\displaystyle \frac{\left\{\mathsf{\Lambda }[x,y;\mathsf{\Psi }(k\left(t\right)\left)\right]-1\right\}}{t}}dt\right\}d\zeta ,(\xi \in \mathsf{\Theta })$$

where ${\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )$ is defined by equality (12), $k\left(\xi \right)$ is analytic in Θ and $k\left(0\right)=0$, and $\mid k\left(\xi \right)\mid <1$.

3. Coefficient Estimates

We need the following result due to Araci et al. [24].

Lemma 1

([24]). Let the function $\mathsf{\Lambda }[x,y;\mathsf{\Psi }\left(\xi \right)]=1+i{\displaystyle \frac{y-x}{\pi }}log\left[{\displaystyle \frac{\mathsf{\Psi }\left(\xi \right)(1-{\lambda }^2)+(1+{\lambda }^2)}{2}}\right]$ be convex univalent in Θ where the function Ψ is defined as in (5). If $\mathsf{\ell }\left(\xi \right)=1+\sum _{k=1}^{\infty }{\mathsf{\ell }}_k{\xi }^k$ is analytic in Θ and holds the following condition

$$\mathsf{\ell }\left(\xi \right)\prec 1+i{\displaystyle \frac{y-x}{\pi }}log\left[{\displaystyle \frac{\mathsf{\Psi }\left(\xi \right)(1-{\lambda }^2)+(1+{\lambda }^2)}{2}}\right],$$

(18)

then

$$\left|{\mathsf{\ell }}_k\right|\le {\displaystyle \frac{{\psi }_1(y-x)}{\pi }}sin\left({\displaystyle \frac{\pi (1-x)}{(y-x)}}\right),(k\ge 1;{\psi }_1>0).$$

(19)

Theorem 3.

Let $\mathsf{\Lambda }[x,y;\mathsf{\Psi }(\xi \left)\right]$ be convex univalent in Θ. If $\phi \in {\mathcal{RS}}_s^{\delta ,m}(x,y;{\tau }_1,{\sigma }_1;\eta ,\vartheta ;\mathsf{\Psi }\left(\xi \right))$, then for $n\ge 2$

$$\begin{array}{c}\hfill |a_n|\le {\displaystyle \frac{1}{\left|{\mathrm{T}}_n\right|}}\prod _{t=1}^{n-1}\left[{\displaystyle \frac{\pi \left|\left(t-{{\rm Y}}_{t,\mathsf{\ell }}^{\kappa }\right)\right|+\left|{\psi }_1{{\rm Y}}_{t,\mathsf{\ell }}^{\kappa }\right|(y-x)sin\left(\frac{\pi (1-x)}{(y-x)}\right)}{\pi \left|\left(t+1-{{\rm Y}}_{t+1,\mathsf{\ell }}^{\kappa }\right)\right|}}\right],\end{array}$$

(20)

where ${\mathrm{T}}_n$ and ${{\rm Y}}_{n,\mathsf{\ell }}^{\kappa }$ are given by (13) and (7), respectively.

Proof. 

By the definition of ${\mathcal{RS}}_s^{\delta ,m}(x,y;{\tau }_1,{\sigma }_1;\eta ,\vartheta ;\mathsf{\Psi }\left(\xi \right))$, we have

$${\displaystyle \frac{\xi {\left[D_{\delta }^m({\tau }_1,{\sigma }_1;\eta ,\vartheta )\phi \left(\xi \right)\right]}^{\prime }}{{\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )}}=p\left(\xi \right)\prec 1+i{\displaystyle \frac{y-x}{\pi }}log\left[{\displaystyle \frac{\mathsf{\Psi }\left(\xi \right)(1-{\lambda }^2)+(1+{\lambda }^2)}{2}}\right],$$

(21)

where $p\left(\xi \right)={\sum }_{n=0}^{\infty }{p}_n{\xi }^n\in \mathcal{P}$. From (12), we have

$${\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )=\xi +\sum _{n=2}^{\infty }{\mathrm{T}}_n{a}_n{{\rm Y}}_{n,\mathsf{\ell }}^{\kappa }{\xi }^n{{\rm Y}}_{n,\mathsf{\ell }}^{\kappa }={\displaystyle \frac{1}{\kappa }}\sum _{\nu =0}^{\kappa -1}{\left[exp\left({\displaystyle \frac{2\pi i}{\kappa }}\right)\right]}^{(n-\mathsf{\ell })\nu },$$

where ${\mathrm{T}}_n$ is given by (13). Equation (21) can be equivalently rewritten as

$$\xi +\sum _{n=2}^{\infty }\left(n-{{\rm Y}}_{n,\mathsf{\ell }}^{\kappa }\right){\mathrm{T}}_n{a}_n{\xi }^n=\left[\xi +\sum _{n=2}^{\infty }{\mathrm{T}}_n{a}_n{{\rm Y}}_{n,\mathsf{\ell }}^{\kappa }{\xi }^n\right]\left[\sum _{n=1}^{\infty }{p}_n{\xi }^n\right].$$

On equating the coefficient of ${\xi }^n$, we get

$$\begin{array}{c}\hfill \left(n-{{\rm Y}}_{n,\mathsf{\ell }}^{\kappa }\right){\mathrm{T}}_n{a}_n=p_{n-1}+p_{n-2}{\mathrm{T}}_2{{\rm Y}}_{2,\mathsf{\ell }}^{\kappa }{a}_2+\cdots +p_1{\mathrm{T}}_{n-1}{{\rm Y}}_{n-1,\mathsf{\ell }}^{\kappa }{a}_{n-1}.\end{array}$$

On computation, we have

$$\left|\left(n-{{\rm Y}}_{n,\mathsf{\ell }}^{\kappa }\right){\mathrm{T}}_n{a}_n\right|\le \sum _{j=1}^{n-1}\left|a_{n-j}{\mathrm{T}}_{n-j}{{\rm Y}}_{n-j,\mathsf{\ell }}^{\kappa }\right|\left|p_j\right|.$$

(22)

From (21), it implies that $p\left(\xi \right)\prec \mathsf{\Lambda }[x,y;\mathsf{\Psi }(\xi \left)\right]$. By using Lemma 1, we have $|p_j|\le {\displaystyle \frac{{\psi }_1(y-x)}{\pi }}sin\left({\displaystyle \frac{\pi (1-x)}{(y-x)}}\right),(\forall j\ge 1)$. Now using the upper bound of $|p_j|$ in the inequality (22), we have (for $a_1=1$, ${\mathrm{T}}_1=1$)

$$\left|\left(n-{{\rm Y}}_{n,\mathsf{\ell }}^{\kappa }\right){\mathrm{T}}_n{a}_n\right|\le {\displaystyle \frac{{\psi }_1(y-x)}{\pi }}sin\left({\displaystyle \frac{\pi (1-x)}{(y-x)}}\right)\sum _{j=1}^{n-1}\left|{\mathrm{T}}_j{a}_j{{\rm Y}}_{j,\mathsf{\ell }}^{\kappa }\right|.$$

(23)

Hence, we can rewrite (23) as

$$\left|a_n\right|\le {\displaystyle \frac{{\psi }_1(y-x)}{\pi \left|\left(n-{{\rm Y}}_{n,\mathsf{\ell }}^{\kappa }\right){\mathrm{T}}_n\right|}}sin\left({\displaystyle \frac{\pi (1-x)}{(y-x)}}\right)\sum _{j=1}^{n-1}\left|{\mathrm{T}}_j{a}_j{{\rm Y}}_{j,\mathsf{\ell }}^{\kappa }\right|.$$

(24)

Let $n=2$ in (24), then

$$\left|a_2\right|\le {\displaystyle \frac{{\psi }_1(y-x)}{\pi \left|\left(2-{{\rm Y}}_{2,\mathsf{\ell }}^{\kappa }\right){\mathrm{T}}_2\right|}}sin\left({\displaystyle \frac{\pi (1-x)}{(y-x)}}\right).$$

(25)

Letting $n=2$ in (20), we get

$$\begin{array}{cc}\hfill |a_2|& \le {\displaystyle \frac{1}{\left|{\mathrm{T}}_2\right|}}\prod _{t=1}^{2-1}\left[{\displaystyle \frac{\pi \left|\left(t-{{\rm Y}}_{t,\mathsf{\ell }}^{\kappa }\right)\right|+\left|{\psi }_1{{\rm Y}}_{t,\mathsf{\ell }}^{\kappa }\right|(y-x)sin\left(\frac{\pi (1-x)}{(y-x)}\right)}{\pi \left|\left(t+1-{{\rm Y}}_{t+1,\mathsf{\ell }}^{\kappa }\right)\right|}}\right]\hfill \\ & ={\displaystyle \frac{1}{\left|{\mathrm{T}}_2\right|}}\left[{\displaystyle \frac{0+|{\psi }_1{{\rm Y}}_{1,\mathsf{\ell }}^{\kappa }|(y-x)sin\left(\frac{\pi (1-x)}{(y-x)}\right)}{\pi \left|\left(2-{{\rm Y}}_{2,\mathsf{\ell }}^{\kappa }\right)\right|}}\right]={\displaystyle \frac{{\psi }_1(y-x)}{\pi \left|\left(2-{{\rm Y}}_{2,\mathsf{\ell }}^{\kappa }\right){\mathrm{T}}_2\right|}}sin\left({\displaystyle \frac{\pi (1-x)}{(y-x)}}\right).\hfill \end{array}$$

(26)

From (25) and (26), we conclude that (20) is correct for $n=2$. Now let $n=3$ in (24), and we have

$$\begin{array}{cc}\hfill \left|a_3\right|& \le {\displaystyle \frac{{\psi }_1(y-x)}{\pi \left|\left(3-{{\rm Y}}_{3,\mathsf{\ell }}^{\kappa }\right){\mathrm{T}}_3\right|}}sin\left({\displaystyle \frac{\pi (1-x)}{(y-x)}}\right)\sum _{j=1}^{3-1}\left|{\mathrm{T}}_j{a}_j{{\rm Y}}_{j,\mathsf{\ell }}^{\kappa }\right|\hfill \\ & ={\displaystyle \frac{{\psi }_1(y-x)}{\pi \left|\left(3-{{\rm Y}}_{3,\mathsf{\ell }}^{\kappa }\right){\mathrm{T}}_3\right|}}sin\left({\displaystyle \frac{\pi (1-x)}{(y-x)}}\right)\left[1+\left|{\mathrm{T}}_2{a}_2{{\rm Y}}_{2,\mathsf{\ell }}^{\kappa }\right|\right]\hfill \\ & \le {\displaystyle \frac{{\psi }_1(y-x)}{\pi \left|\left(3-{{\rm Y}}_{3,\mathsf{\ell }}^{\kappa }\right){\mathrm{T}}_3\right|}}sin\left({\displaystyle \frac{\pi (1-x)}{(y-x)}}\right)\left[1+{\displaystyle \frac{\left|{{\rm Y}}_{2,\mathsf{\ell }}^{\kappa }\right|{\psi }_1(y-x)sin\left(\frac{\pi (1-x)}{(y-x)}\right)}{\pi \left|\left(2-{{\rm Y}}_{2,\mathsf{\ell }}^{\kappa }\right)\right|}}\right].\hfill \end{array}$$

(27)

If we let $n=3,$ in (20), we have

$$\begin{array}{cc}\hfill |a_3|& \le {\displaystyle \frac{1}{\left|{\mathrm{T}}_3\right|}}\prod _{t=1}^{3-1}\left[{\displaystyle \frac{\pi \left|\left(t-{{\rm Y}}_{t,\mathsf{\ell }}^{\kappa }\right)\right|+\left|{\psi }_1{{\rm Y}}_{t,\mathsf{\ell }}^{\kappa }\right|(y-x)sin\left(\frac{\pi (1-x)}{(y-x)}\right)}{\pi \left|\left(t+1-{{\rm Y}}_{t+1,\mathsf{\ell }}^{\kappa }\right)\right|}}\right]\hfill \\ \hfill & \le {\displaystyle \frac{1}{\left|{\mathrm{T}}_3\right|}}\left[{\displaystyle \frac{|{\psi }_1|(y-x)sin\left(\frac{\pi (1-x)}{(y-x)}\right)}{\pi \left|\left(2-{{\rm Y}}_{2,\mathsf{\ell }}^{\kappa }\right)\right|}}\times {\displaystyle \frac{\pi \left|\left(2-{{\rm Y}}_{2,\mathsf{\ell }}^{\kappa }\right)\right|+\left|{\psi }_1{{\rm Y}}_{2,\mathsf{\ell }}^{\kappa }\right|(y-x)sin\left(\frac{\pi (1-x)}{(y-x)}\right)}{\pi \left|\left(3-{{\rm Y}}_{3,\mathsf{\ell }}^{\kappa }\right)\right|}}\right]\hfill \\ & \le {\displaystyle \frac{{\psi }_1(y-x)}{\pi \left|\left(3-{{\rm Y}}_{3,\mathsf{\ell }}^{\kappa }\right){\mathrm{T}}_3\right|}}sin\left({\displaystyle \frac{\pi (1-x)}{(y-x)}}\right)\left[1+{\displaystyle \frac{\left|{{\rm Y}}_{2,\mathsf{\ell }}^{\kappa }\right|{\psi }_1(y-x)sin\left(\frac{\pi (1-x)}{(y-x)}\right)}{\pi \left|\left(2-{{\rm Y}}_{2,\mathsf{\ell }}^{\kappa }\right)\right|}}\right].\hfill \end{array}$$

Hence the hypothesis is correct for $n=3$. Assume that (20) is valid for $n=2,3,\dots r$. So from (20), we have

$$\begin{array}{c}\hfill |a_r|\le {\displaystyle \frac{1}{\left|{\mathrm{T}}_r\right|}}\prod _{t=1}^{r-1}\left[{\displaystyle \frac{\pi \left|\left(t-{{\rm Y}}_{t,\mathsf{\ell }}^{\kappa }\right)\right|+\left|{\psi }_1{{\rm Y}}_{t,\mathsf{\ell }}^{\kappa }\right|(y-x)sin\left(\frac{\pi (1-x)}{(y-x)}\right)}{\pi \left|\left(t+1-{{\rm Y}}_{t+1,\mathsf{\ell }}^{\kappa }\right)\right|}}\right].\end{array}$$

By the induction hypothesis, we have

$$\begin{array}{c}|a_r|\le {\displaystyle \frac{1}{\left|{\mathrm{T}}_r\right|}}\prod _{t=1}^{r-1}\left[{\displaystyle \frac{\pi \left|\left(t-{{\rm Y}}_{t,\mathsf{\ell }}^{\kappa }\right)\right|+\left|{\psi }_1{{\rm Y}}_{t,\mathsf{\ell }}^{\kappa }\right|(y-x)sin\left(\frac{\pi (1-x)}{(y-x)}\right)}{\pi \left|\left(t+1-{{\rm Y}}_{t+1,\mathsf{\ell }}^{\kappa }\right)\right|}}\right]\\ ={\displaystyle \frac{{\psi }_1(y-x)}{\pi \left|\left(r-{{\rm Y}}_{r,\mathsf{\ell }}^{\kappa }\right){\mathrm{T}}_r\right|}}sin\left({\displaystyle \frac{\pi (1-x)}{(y-x)}}\right)\sum _{j=1}^{r-1}\left|{\mathrm{T}}_j{a}_j{{\rm Y}}_{j,\mathsf{\ell }}^{\kappa }\right|.\end{array}$$

(28)

Now letting $n=r+1$ in (24), we have

$$\begin{array}{c}|a_{r+1}|\le {\displaystyle \frac{{\psi }_1(y-x)}{\pi \left|\left(r+1-{{\rm Y}}_{r+1,\mathsf{\ell }}^{\kappa }\right){\mathrm{T}}_{r+1}\right|}}sin\left({\displaystyle \frac{\pi (1-x)}{(y-x)}}\right)\sum _{j=1}^r\left|{\mathrm{T}}_j{a}_j{{\rm Y}}_{j,\mathsf{\ell }}^{\kappa }\right|\\ ={\displaystyle \frac{{\psi }_1(y-x)}{\pi \left|\left(r+1-{{\rm Y}}_{r+1,\mathsf{\ell }}^{\kappa }\right){\mathrm{T}}_{r+1}\right|}}sin\left({\displaystyle \frac{\pi (1-x)}{(y-x)}}\right)\left[\sum _{j=1}^{r-1}\left|{\mathrm{T}}_j{a}_j{{\rm Y}}_{j,\mathsf{\ell }}^{\kappa }\right|+\left|{\mathrm{T}}_r{a}_r{{\rm Y}}_{r,\mathsf{\ell }}^{\kappa }\right|\right].\end{array}$$

(29)

Using (28) in (29), we can obtain

$$\begin{array}{c}|a_{r+1}|\le {\displaystyle \frac{{\psi }_1(y-x)}{\pi \left|\left(r+1-{{\rm Y}}_{r+1,\mathsf{\ell }}^{\kappa }\right){\mathrm{T}}_{r+1}\right|}}sin\left({\displaystyle \frac{\pi (1-x)}{(y-x)}}\right)\left[\sum _{j=1}^{r-1}\left|{\mathrm{T}}_j{a}_j{{\rm Y}}_{j,\mathsf{\ell }}^{\kappa }\right|\right.\\ \left.+{\displaystyle \frac{\left|{{\rm Y}}_{r,\mathsf{\ell }}^{\kappa }\right|{\psi }_1(y-x)}{\pi \left|\left(r-{{\rm Y}}_{r,\mathsf{\ell }}^{\kappa }\right)\right|}}sin\left({\displaystyle \frac{\pi (1-x)}{(y-x)}}\right)\sum _{j=1}^{r-1}\left|{\mathrm{T}}_j{a}_j{{\rm Y}}_{j,\mathsf{\ell }}^{\kappa }\right|\right]\\ ={\displaystyle \frac{{\psi }_1(y-x)sin\left(\frac{\pi (1-x)}{(y-x)}\right)}{\pi \left|\left(r+1-{{\rm Y}}_{r+1,\mathsf{\ell }}^{\kappa }\right){\mathrm{T}}_{r+1}\right|}}\sum _{j=1}^{r-1}\left|{\mathrm{T}}_j{a}_j{{\rm Y}}_{j,\mathsf{\ell }}^{\kappa }\right|\left[{\displaystyle \frac{\pi \left|\left(r-{{\rm Y}}_{r,\mathsf{\ell }}^{\kappa }\right)\right|+\left|{{\rm Y}}_{r,\mathsf{\ell }}^{\kappa }\right|{\psi }_1(y-x)sin\left(\frac{\pi (1-x)}{(y-x)}\right)}{\pi \left|\left(r-{{\rm Y}}_{r,\mathsf{\ell }}^{\kappa }\right)\right|}}\right]\end{array}$$

$$\begin{array}{c}={\displaystyle \frac{{\psi }_1(y-x)sin\left(\frac{\pi (1-x)}{(y-x)}\right){\sum }_{j=1}^{r-1}\left|{\mathrm{T}}_j{a}_j{{\rm Y}}_{j,\mathsf{\ell }}^{\kappa }\right|}{\pi \left|\left(r-{{\rm Y}}_{r,\mathsf{\ell }}^{\kappa }\right){\mathrm{T}}_{r+1}\right|}}\left[{\displaystyle \frac{\pi \left|\left(r-{{\rm Y}}_{r,\mathsf{\ell }}^{\kappa }\right)\right|+\left|{{\rm Y}}_{r,\mathsf{\ell }}^{\kappa }\right|{\psi }_1(y-x)sin\left(\frac{\pi (1-x)}{(y-x)}\right)}{\pi \left|\left(r+1-{{\rm Y}}_{r+1,\mathsf{\ell }}^{\kappa }\right)\right|}}\right]\\ ={\displaystyle \frac{1}{\left|{\mathrm{T}}_{r+1}\right|}}\prod _{t=1}^{r-1}\left[{\displaystyle \frac{\pi \left|\left(t-{{\rm Y}}_{t,\mathsf{\ell }}^{\kappa }\right)\right|+\left|{\psi }_1{{\rm Y}}_{t,\mathsf{\ell }}^{\kappa }\right|(y-x)sin\left(\frac{\pi (1-x)}{(y-x)}\right)}{\pi \left|\left(t+1-{{\rm Y}}_{t+1,\mathsf{\ell }}^{\kappa }\right)\right|}}\right]\times \\ \left[{\displaystyle \frac{\pi \left|\left(r-{{\rm Y}}_{r,\mathsf{\ell }}^{\kappa }\right)\right|+\left|{{\rm Y}}_{r,\mathsf{\ell }}^{\kappa }\right|{\psi }_1(y-x)sin\left(\frac{\pi (1-x)}{(y-x)}\right)}{\pi \left|\left(r+1-{{\rm Y}}_{r+1,\mathsf{\ell }}^{\kappa }\right)\right|}}\right]\\ ={\displaystyle \frac{1}{\left|{\mathrm{T}}_{r+1}\right|}}\prod _{t=1}^r\left[{\displaystyle \frac{\pi \left|\left(t-{{\rm Y}}_{t,\mathsf{\ell }}^{\kappa }\right)\right|+\left|{\psi }_1{{\rm Y}}_{t,\mathsf{\ell }}^{\kappa }\right|(y-x)sin\left(\frac{\pi (1-x)}{(y-x)}\right)}{\pi \left|\left(t+1-{{\rm Y}}_{t+1,\mathsf{\ell }}^{\kappa }\right)\right|}}\right],\end{array}$$

implies that inequality (20) is true for $n=r+1$. Hence the proof of the theorem is complete. □

Corollary 1

([33] Corollary 3.2, [26] Theorem 1). If $\phi \in {\mathcal{S}}^{*}(x,y)$, then

$$|a_n|\le \prod _{t=2}^n\left[{\displaystyle \frac{t-2+|M_1|}{t-1}}\right],\left(n=2,3,\dots \right).$$

where $\left|M_1\right|={\displaystyle \frac{2(y-x)}{\pi }}sin\left({\displaystyle \frac{\pi (1-x)}{(y-x)}}\right)$.

Proof. 

Letting $r=2$, $s=1$, ${\tau }_1={\sigma }_1$, ${\tau }_2=1$, $m=\eta =\mathsf{\ell }=0$, $\delta =\kappa =1$ and $\mathsf{\Psi }\left(\xi \right)=(1+\xi )/(1-\xi )$ in Theorem 3, then (21) will be of the form

$${\displaystyle \frac{\xi {\phi }^{\prime }\left(\xi \right)}{\phi \left(\xi \right)}}\prec 1+\sum _{n=1}^{\infty }{\displaystyle \frac{(y-x)}{n\pi }}i\left(1-e^{2ni{\displaystyle \frac{\pi (1-x)}{(y-x)}}}\right)z^n.$$

Retracing the steps mentioned in Theorem 3, we get the result obtained by [26]. □

For completeness, we state the following result.

Theorem 4.

Let $\mathsf{\Lambda }[x,y;\mathsf{\Psi }(\xi \left)\right]$ be convex univalent in Θ. If $\phi \in {\mathcal{RC}}_s^{\delta ,m}(x,y;{\tau }_1,{\sigma }_1;\eta ,\vartheta ;\mathsf{\Psi }\left(\xi \right))$, then for $n\ge 2$

$$\begin{array}{c}\hfill |a_n|\le {\displaystyle \frac{1}{n\left|{\mathrm{T}}_n\right|}}\prod _{t=1}^{n-1}\left[{\displaystyle \frac{\pi \left|\left(t-{{\rm Y}}_{t,\mathsf{\ell }}^{\kappa }\right)\right|+\left|{\psi }_1{{\rm Y}}_{t,\mathsf{\ell }}^{\kappa }\right|(y-x)sin\left(\frac{\pi (1-x)}{(y-x)}\right)}{\pi \left|\left(t+1-{{\rm Y}}_{t+1,\mathsf{\ell }}^{\kappa }\right)\right|}}\right].\end{array}$$

(30)

Proof. 

By the definition of ${\mathcal{RC}}_s^{\delta ,m}(x,y;{\tau }_1,{\sigma }_1;\eta ,\vartheta ;\mathsf{\Psi }\left(\xi \right))$, we have

$${\displaystyle \frac{{\left(\xi {\left[D_{\delta }^m({\tau }_1,{\sigma }_1;\eta ,\vartheta )\phi \left(\xi \right)\right]}^{\prime }\right)}^{\prime }}{{\left({\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )\right)}^{\prime }}}=p\left(\xi \right)\prec 1+i{\displaystyle \frac{y-x}{\pi }}log\left[{\displaystyle \frac{\mathsf{\Psi }\left(\xi \right)(1-{\lambda }^2)+(1+{\lambda }^2)}{2}}\right],$$

(31)

where $p\left(\xi \right)={\sum }_{n=0}^{\infty }{p}_n{\xi }^n\in \mathcal{P}$. Equation (21) can be equivalently rewritten as

$$1+\sum _{n=2}^{\infty }n\left(n-{{\rm Y}}_{n,\mathsf{\ell }}^{\kappa }\right){\mathrm{T}}_n{a}_n{\xi }^{n-1}=\left[1+\sum _{n=2}^{\infty }n{\mathrm{T}}_n{a}_n{{\rm Y}}_{n,\mathsf{\ell }}^{\kappa }{\xi }^{n-1}\right]\left[\sum _{n=1}^{\infty }{p}_n{\xi }^n\right].$$

On equating the coefficient of ${\xi }^{n-1}$, we get

$$\begin{array}{c}\hfill n\left(n-{{\rm Y}}_{n,\mathsf{\ell }}^{\kappa }\right){\mathrm{T}}_n{a}_n=p_{n-1}+2p_{n-2}{\mathrm{T}}_2{{\rm Y}}_{2,\mathsf{\ell }}^{\kappa }{a}_2+\cdots +(n-1)p_1{\mathrm{T}}_{n-1}{{\rm Y}}_{n-1,\mathsf{\ell }}^{\kappa }{a}_{n-1}.\end{array}$$

On computation, we have

$$\left|a_n\right|\le {\displaystyle \frac{{\psi }_1(y-x)}{n\pi \left|\left(n-{{\rm Y}}_{n,\mathsf{\ell }}^{\kappa }\right){\mathrm{T}}_n\right|}}sin\left({\displaystyle \frac{\pi (1-x)}{(y-x)}}\right)\sum _{j=1}^{n-1}\left|j{\mathrm{T}}_j{a}_j{{\rm Y}}_{j,\mathsf{\ell }}^{\kappa }\right|.$$

Following the steps in Theorem 1, we can obtain the assertion 30. □

Remark 3.

The inequalities (20) and (30) will be sharp if

$${\displaystyle \frac{\xi {\left[D_{\delta }^m({\tau }_1,{\sigma }_1;\eta ,\vartheta )\phi \left(\xi \right)\right]}^{\prime }}{{\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )}}=\mathsf{\Lambda }\left[x,y;{\displaystyle \frac{1+\xi }{1-\xi }}\right]$$

and

$${\displaystyle \frac{{\left(\xi {\left[D_{\delta }^m({\tau }_1,{\sigma }_1;\eta ,\vartheta )\phi \left(\xi \right)\right]}^{\prime }\right)}^{\prime }}{{\left({\phi }_{\mathsf{\ell },\kappa }^{\delta ,m}({\tau }_1,{\sigma }_1;\eta ,\vartheta ;\xi )\right)}^{\prime }}}=\mathsf{\Lambda }\left[x,y;{\displaystyle \frac{1+\xi }{1-\xi }}\right]$$

respectively.

4. Discussion

As in the case of the results obtained by various authors [25,26,28,32], our results will not unify the results of the well-known classes of starlike and convex functions. But they help us obtain the results of various other new classes like ${\mathcal{S}}_s(x,y)$, ${\mathcal{M}}_s^{\delta }(x,y;{\tau }_1,{\sigma }_1;\eta ,\vartheta ;\mathsf{\Psi }\left(\xi \right))$ and so on (see Remark 1). The class recently studied by Breaz et al. ([31], Definition 1.1) may look more versatile, but it is not so. Here the superordinate function $\mathsf{\Lambda }[x,y;\mathsf{\Psi }(\xi \left)\right]$ in the class ${\mathcal{RS}}_s^{\delta ,m}(x,y;{\tau }_1,{\sigma }_1;\eta ,\vartheta ;\mathsf{\Psi }\left(\xi \right))$ is Carathéodory’s provided the class $\mathsf{\Psi }\left(\xi \right)\in \mathcal{P}$, whereas the superordinate function used by Breaz et al. [31] to define their function class need not belong to the class $\mathcal{P}$. Apart from this marked difference, the operator used here to define the function class ${\mathcal{RS}}_s^{\delta ,m}(x,y;{\tau }_1,{\sigma }_1;\eta ,\vartheta ;\mathsf{\Psi }\left(\xi \right))$ is more versatile since it helps us to unify several well-known classes.

5. Conclusions

This area of study has received limited attention in the literature; in particular, coefficient inequalities for the classes involving functions with respect to symmetric points have been attempted by very few authors. It is still open to various researchers to explore the studies analogous to Bazilevič, pseudo-starlike, gamma-starlike, Pascu classes, and so on with respect to symmetric points. The Fekete–Szego inequality, logarithmic coefficients and inverse coefficients which are computationally cumbersome should be very interesting as a scope for future research.