Starlike Functions in the Space of Meromorphic Harmonic Functions

Abstract

The Geometric Theory of Analytic Functions was initially developed for the space of functions that are analytic in the unit disk. The convexity and starlikeness of functions are the first geometric ideas considered in this theory. We can notice a symmetry between the subjects considered in the space of analytic functions and those in the space of harmonic functions. In the presented paper, we consider the starlikeness of functions in the space of meromorphic harmonic functions.

1. Introduction

The Geometric Theory of Analytic Functions was initially developed for the space $\mathcal{A}$ of analytic functions in the disk $\mathbb{U}:=\mathbb{U}\left(1\right),$ where $\mathbb{U}$$\left(r\right):=\{$$\varsigma$$\in \mathbb{C}:\left|\varsigma \right|<r\}$. The convexity and starlikeness of functions are the first geometric ideas considered in this theory. We say that a function $f\in \mathcal{A}$ of the form

$$f\left(\varsigma \right)=\varsigma +\sum _{n=2}^{\infty }{a}_n{\varsigma }^n\left(\varsigma \in \mathbb{E}\right)$$

is starlike in $\mathbb{U}$ if it is univalent in the unit disc $\mathbb{U}$ and $f\left(\mathbb{U}\right)$ is a starlike domain with respect to the origin. By Jack’s Lemma, we obtain that the starlikeness of a function f in the unit disk is preserved inside this disk, i.e., f maps each disk $\mathbb{U}$$\left(r\right)\subset \mathbb{U}$ onto a starlike domain with respect to the origin. It is equivalent to the following analytic condition:

$${\displaystyle \frac{\partial }{\partial t}}\left(argf\left(r{e}^{it}\right)\right)\ge 0\left(0\le t\le 2\pi \right).$$

(1)

Thus, we can obtain the following analytic definition of starlikeness. A function $f\in \mathcal{A}$, $f\left(0\right)=f^{\prime }\left(0\right)-1=0,$ is starlike in $\mathbb{U}$ if and only if

$$\mathrm{Re}{\displaystyle \frac{\varsigma f^{\prime }\left(\varsigma \right)}{f\left(\varsigma \right)}}>0(\varsigma \in \mathbb{U}).$$

Similar topics are considered in the space of analytic functions in $\mathbb{E}:=\{$$\varsigma \in \mathbb{C}$$:\left|\varsigma \right|>1\}$, i.e., in the space of meromorphic functions.

We can also notice a symmetry between the subjects considered in the space of analytic functions and those in the space of harmonic functions. Clunie and Sheil-Small [1] transferred the ideas of convexity and starlikeness to the space of harmonic functions. Harmonic functions are famous for their use in the study of minimal surfaces and also play important roles in a variety of problems in applied mathematics (e.g., see Choquet [2], Dorff [3], or Lewy [4]). We say that $f:D\to \mathbb{C}$ is a harmonic function in a domain $D\subset \mathbb{C}$ if it has continuous second-order partial derivatives and satisfies the Laplace equation

$$\mathsf{\Delta }f:={\displaystyle \frac{{\partial }^{2}f}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}f}{\partial y^2}}=0.$$

The function f defined in the domain $D=\mathbb{E}\left(r\right):=\{\varsigma \in \mathbb{C}:\left|\varsigma \right|>r\}$ is called a meromorphic harmonic function. Let ${\mathcal{M}}_{\mathcal{H}}$ denote the class of all such functions, with the normalization $f(\infty )=\infty .$ The object of this paper is to generalize and investigate starlike functions in the space of meromorphic harmonic functions.

A function $f\in {\mathcal{M}}_{\mathcal{H}}$ that preserves the orientation of a Jordan curve is called sense-preserving or orientation-preserving. It is equivalent to saying that the Jacobian is strictly positive at every point of the domain $\mathbb{E}$.

Let $f\in {\mathcal{M}}_{\mathcal{H}}$ be a harmonic, sense-preserving, and univalent function in $\mathbb{E}$ Then, Hengartner and Schober [5] showed that there exist $B\in \mathbb{C}$ and functions $h,g$ of the form

$$h\left(\varsigma \right)=\alpha \varsigma +\sum _{n=1}{a}_n{\varsigma }^{-n},g\left(\varsigma \right)=\beta \varsigma +\sum _{n=1}^{\infty }{b}_n{\varsigma }^{-n}\left(0\le \left|\alpha \right|<\left|\beta \right|,\varsigma \in \mathbb{E}\right),$$

such that

$$f\left(\varsigma \right)=\phi \left(\varsigma \right)+\overline{\psi \left(\varsigma \right)}+Blog\left|\varsigma \right|\left(\varsigma \in \mathbb{E}\right),$$

where and $\overline{f_{\overline{\varsigma }}}/f_{\varsigma }$ is analytic and bounded by 1 in $\mathbb{E}$.

Finally, we denote by $\mathcal{M}$ the class of functions $f\in {\mathcal{M}}_{\mathcal{H}}$ of the form

$$f\left(\varsigma \right)=h+\overline{g},h\left(\varsigma \right)=\varsigma +\sum _{n=1}^{\infty }{a}_n{\varsigma }^{-n},g\left(\varsigma \right)=\sum _{n=1}^{\infty }{b}_n{\varsigma }^{-n}\left(\varsigma \in \mathbb{E}\right)$$

(2)

which are sense-preserving and univalent in $\mathbb{E}.$

It should be noted that meromorphic harmonic functions have been extensively studied by researchers (see [6,7,8,9,10,11]).

A function $f\in \mathcal{M}$ is said to be meromorphic harmonic starlike in $\mathbb{E}\left(r\right)$ if f maps $\partial \mathbb{E}\left(r\right)$ onto a curve that is starlike with respect to the origin, i.e., it satisfies the condition (1). If we define the following meromorphic harmonic derivative

$$D_{\mathcal{M}}f\left(\varsigma \right):=\varsigma h^{\prime }\left(\varsigma \right)-\overline{\varsigma g^{\prime }\left(\varsigma \right)}\left(\varsigma \in \mathbb{E}\right),$$

then we can write the condition (1) as follows:

$$\mathrm{Re}{\displaystyle \frac{D_{\mathcal{M}}f\left(\varsigma \right)}{f\left(\varsigma \right)}}\ge 0\left(\left|\varsigma \right|=r\right).$$

Researchers have extensively studied the class of starlike and convex functions and their generalizations in the space of analytic functions and the space of harmonic functions. They use various techniques and tools, e.g., subordination, convolution, and linear operators. Also, special functions play an important role in this aspect. The main one is the generalized hypergeometric function ${\hspace{0.17em}}_q{F}_s$. We transfer these ideas to the space of meromorphic harmonic functions.

The well-known definition of subordination in the space $\mathcal{A}$ is closely related to Jack’s Lemma, which does not apply in the space of meromorphic harmonic functions. Therefore, we adapt the definition of weak subordination introduced by Mauir [12].

We say that a function $f:\mathbb{E}\to \mathbb{C}$ is weakly subordinate to a function $F:\mathbb{E}\to \mathbb{C}$, and we write $f\left(\varsigma \right)⪯F\left(\varsigma \right)$ if $f\left(\mathbb{E}\right)\subset F\left(\mathbb{E}\right).$

Let

$$f_j\left(\varsigma \right)=\sum _{k=-1}^{\infty }\left(a_{j,k}{\varsigma }^{-k}+\overline{b_{j,k}{\varsigma }^{-k}}\right)\left(\varsigma \in \mathbb{E},j=1,2\right)$$

(3)

By $f_1\ast f_2$, we denote the convolution of $f_1$ and $f_2$ defined by

$$\left(f_1\ast f_2\right)\left(\varsigma \right)=\sum _{n=-1}^{\infty }\left(a_{1,n}{a}_{2,n}{\varsigma }^{-n}+\overline{b_{1,n}{b}_{2,n}{\varsigma }^{-n}}\right)\left(\varsigma \in \mathbb{E}\right).$$

Let ${\alpha }_1,\dots ,{\alpha }_s\in \mathbb{C}$, ${\beta }_1,\dots ,{\beta }_s\in \mathbb{C}\setminus \left\{0\right\}$ be the complex parameters. Using the generalized hypergeometric function ${\hspace{0.17em}}_q{F}_s$, we define the function

$$F_s\left(\varsigma \right)={\varsigma }^{-1}{{\hspace{0.17em}}_{s+1}{F}_s({\alpha }_1,\dots ,{\alpha }_s,1;{\beta }_1,\dots ,{\beta }_s;\varsigma }^{-1}):=\sum _{n=1}^{\infty }{\mathrm{\Gamma }}_n{\varsigma }^{-n}\left(\varsigma \in \mathbb{U}\right),$$

(4)

where

$${\mathrm{\Gamma }}_n={\displaystyle \frac{{\left({\alpha }_1\right)}_{n-1}\cdots {\left({\alpha }_{s+1}\right)}_{n-1}}{{\left({\beta }_1\right)}_{n-1}\cdots {\left({\beta }_s\right)}_{n-1}}},$$

(5)

and ${\left(\lambda \right)}_n$ is the Pochhammer symbol defined, in terms of the Gamma function $\mathrm{\Gamma }$, as

$${\left(\lambda \right)}_n:={\displaystyle \frac{\mathrm{\Gamma }(\lambda +n)}{\mathrm{\Gamma }\left(\lambda \right)}}=\left\{\begin{array}{cc}\hfill 1& (n=0)\\ \hfill \lambda (\lambda +1)···(\lambda +n-1)& (n\in \mathbb{N}).\end{array}\right.$$

Corresponding to the generalized hypergeometric function Dziok and Srivastava [13] introduced the linear operator $H_s$ defined in the space $\mathcal{A}$ of analytic functions by the convolution $H_{s}f:=F_s\ast f.$The linear operator $H_s$ includes other linear operators considered in earlier works, for example, the Hohlov operator, the Carlson–Shaffer operator, Ruscheweyh derivatives, the generalized Bernardi–Libera–Livingston integral operator, and the Srivastava and Owa fractional derivative operator (for details, see [13]). Harmonic generalizations of the operator $H_s$ were considered by Al-Kharsani and Al-Khal [14] (see also [15,16]). We define a similar operator in the space $\mathcal{M}$ of meromorphic harmonic functions.

Let $\left|\tau \right|=1$ and

$${\mathrm{\Phi }}_{\mathcal{M}}^{s,\tau }\left(\varsigma \right):=\varsigma +F_s\left(\varsigma \right)+\overline{\tau F_s\left(\varsigma \right)}\left(\varsigma \in \mathbb{E}\right).$$

(6)

Due to Dziok and Srivastava [13], we define the operator$H_{\mathcal{M}}^{s,\tau }=H_{\mathcal{M}}^{s,\tau }({\alpha }_1,\dots ,{\alpha }_s;{\beta }_1,\dots ,{\beta }_s):\mathcal{M}\to \mathcal{M}$ by the convolution formula

$$H_{\mathcal{M}}^{s,\tau }f:={\mathrm{\Phi }}_{\mathcal{M}}^{s,\tau }\ast f.$$

In particular, we have

$$H_{\mathcal{M}}^{1,1}(1;1)f=f,H_{\mathcal{M}}^{1,-1}(2;1)f=D_{\mathcal{M}}f,$$

and

$$H_{\mathcal{M}}^{s+1,-\tau }({\alpha }_1,\dots ,{\alpha }_s,2;{\beta }_1,\dots ,{\beta }_s,1)f=D_{\mathcal{M}}{H}_{\mathcal{M}}^{s,\tau }({\alpha }_1,\dots ,{\alpha }_s;{\beta }_1,\dots ,{\beta }_s)f.$$

The operator

$${\mathcal{D}}_{\mathcal{M}}^{n}f:=H_{\mathcal{M}}^{1,{\left(-1\right)}^n}(1+n;1)f\left(n\in {\mathbb{N}}_0\right),$$

is associated with the Ruscheweyh derivatives (see [17]), and the operator

$$J_{\mathcal{M}}^0\left(f\right)=f,J_{\mathcal{M}}^{n}f:=H_{\mathcal{M}}^{n,{\left(-1\right)}^n}(2,\dots ,2;1,\dots ,1)f\left(n\in \mathbb{N}\right),$$

is related to the harmonic Sălăgean operator [18] (see also [19,20]). For a function $f\in \mathcal{M}$, we have

$$\begin{array}{cc}\hfill {\mathcal{D}}_{\mathcal{M}}^{n}f\left(\varsigma \right)& =\varsigma +{\displaystyle \frac{\varsigma {\left({\varsigma }^{-n}h\left(\varsigma \right)\right)}^{\left(n\right)}}{n!}}+{\left(-1\right)}^n\overline{{\displaystyle \frac{\varsigma {\left({\varsigma }^{-n}g\left(\varsigma \right)\right)}^{\left(n\right)}}{n!}}}\left(n\in {\mathbb{N}}_0,\varsigma \in \mathbb{E}\right),\hfill \\ \hfill J_{\mathcal{M}}^0\left(f\right)& =f,J_{\mathcal{M}}^n\left(f\right)=D_{\mathcal{M}}\left(J_{\mathcal{M}}^{n-1}\left(f\right)\right)\left(n\in \mathbb{N}\right).\hfill \end{array}$$

Using the above tools, we define generalizations of convex and starlike functions in the space $\mathcal{M}.$ Next, we consider classical extremal problems for the defined classes of functions with correlated coefficients. Some applications of the main results are also discussed.

2. Definitions of Main Subclasses of $\mathcal{M}$

In this section, we define the main subclasses of the space $\mathcal{M}$ and formulate alternative definitions for these classes of functions. Let us assume

$$\left|{\mathrm{\Gamma }}_n\right|\ge \left|{\mathrm{\Gamma }}_1\right|=1,N>max\{0,M\}.$$

(7)

We denote by${\mathcal{W}}_{\mathcal{M}}={\mathcal{W}}_{\mathcal{M}}^{s,\tau }(M,N;{\alpha }_1,\dots ,{\alpha }_s;{\beta }_1,\dots ,{\beta }_s)$the class of functions $f\in \mathcal{M}$ for which

$${\displaystyle \frac{H_{\mathcal{M}}^{s+1,-\tau }f\left(\varsigma \right)}{H_{\mathcal{M}}^{s\tau }f\left(\varsigma \right)}}⪯{\displaystyle \frac{M+\varsigma }{N+\varsigma }}\left({\alpha }_{s+1}=2,{\beta }_{s+1}=1\right),$$

(8)

or, equivalently,

$${\displaystyle \frac{D_{\mathcal{M}}\left(H_{\mathcal{M}}^{s\tau }f\right)\left(\varsigma \right)}{H_{\mathcal{M}}^{s\tau }f\left(\varsigma \right)}}⪯{\displaystyle \frac{M+\varsigma }{N+\varsigma }}.$$

(9)

Also, by ${\mathcal{R}}_{\mathcal{M}}={\mathcal{R}}_{\mathcal{M}}^{s,\tau }(M,N;{\alpha }_1,\dots ,{\alpha }_s;{\beta }_1,\dots ,{\beta }_s)$, we denote the class of functions $f\in \mathcal{M}$ such that

$${\varsigma }^{-1}{H}_{\mathcal{M}}^{s,\tau }f\left(\varsigma \right)⪯{\displaystyle \frac{M+\varsigma }{N+\varsigma }}.$$

In particular, the classes

$$\begin{array}{cc}\hfill {\mathcal{W}}_{\mathcal{R}}^n(M,N)& :={\mathcal{W}}_{\mathcal{M}}^{1,{\left(-1\right)}^n}(M,N;1,1+n;1),\hfill \\ \hfill {\mathcal{R}}_{\mathcal{R}}^n(M,N)& :={\mathcal{R}}_{\mathcal{M}}^{1,{\left(-1\right)}^n}(M,N;1,1+n;1)\hfill \end{array}$$

are associated with the Ruscheweyh derivatives ${\mathcal{D}}_{\mathcal{M}}^{n}f$, and the classes

$$\begin{array}{cc}\hfill {\mathcal{W}}_{\mathcal{S}}^n(M,N)& :={\mathcal{W}}_{\mathcal{M}}^{n,{\left(-1\right)}^n}(M,N;2,\dots ,2;1,\dots ,1),\hfill \\ \hfill {\mathcal{R}}_{\mathcal{S}}^n(M,N)& :={\mathcal{R}}_{\mathcal{M}}^{n,{\left(-1\right)}^n}(M,N;2,\dots ,2;1,\dots ,1)\hfill \end{array}$$

are related to the harmonic Sălăgean operator $J_{\mathcal{M}}^{n}f$. The classes

$$\begin{array}{cc}\hfill {\mathcal{M}}^{\ast }(M,N)& :={\mathcal{W}}_{\mathcal{R}}^0(M,N)={\mathcal{W}}_{\mathcal{S}}^0(M,N),\hfill \\ \hfill {\mathcal{M}}^c(M,N)& :={\mathcal{W}}_{\mathcal{R}}^1(M,N)={\mathcal{W}}_{\mathcal{S}}^1(M,N),\hfill \\ \hfill {\mathcal{R}}_{\mathcal{M}}(M,N)& :={\mathcal{R}}_{\mathcal{M}}^1(M,N)={\mathcal{W}}_{\mathcal{M}}^1(M,N)\hfill \end{array}$$

are related to the Janowski functions [21] (see also [10,22]). Putting $M=1-2\alpha ,$$n=1$, we obtain the classes

$${\mathcal{M}}^{\ast }\left(\alpha \right):={\mathcal{M}}^{\ast }\left(1-2\alpha ,1\right),{\mathcal{M}}^c\left(\alpha \right):={\mathcal{M}}^c\left(1-2\alpha ,1\right)$$

of meromorphic harmonic starlike and convex functions of order $\alpha$, respectively. Finally, ${\mathcal{M}}^{\ast }:={\mathcal{W}}_{\mathcal{D}}^0(-1,1)$is the class of meromorphic harmonic starlike functions, and ${\mathcal{M}}^c:={\mathcal{M}}^c(-1,1)$ is the class of meromorphic harmonic convex functions.

Motivated by Ruscheweyh [23], we define the dual set of the class $\mathcal{B}\subset {\mathcal{M}}_{\mathcal{H}}$ by

$${\mathcal{B}}^{\ast }=\left\{f\in \mathcal{M}|\left(f\ast \phi \right)\left(\varsigma \right)\ne 0,\phi \in \mathcal{B},\varsigma \in \mathbb{E}\right\}.$$

Now, we show that the main classes of functions can be defined as dual sets.

Theorem 1. 

$${\mathcal{W}}_{\mathcal{M}}={\left\{{\varphi }_{\zeta }:\left|\zeta \right|=1\right\}}^{\ast },$$

where

$$\begin{array}{cc}\hfill {\varphi }_{\zeta }\left(\varsigma \right):=& \varsigma +\sum _{n=1}^{\infty }{\displaystyle \frac{nN-M+\left(n-1\right)\zeta }{N-M}}{\mathrm{\Gamma }}_n{\varsigma }^{-n}\hfill \\ \hfill & -\sum _{n=1}^{\infty }{\displaystyle \frac{nN+M+\left(n+1\right)\zeta }{N-M}}\overline{\tau {\mathrm{\Gamma }}_n}\overline{{\varsigma }^{-n}}\left(\varsigma \in \mathbb{E}\right).\hfill \end{array}$$

(10)

Proof. 

Let

$$\psi \left(\varsigma \right):={\displaystyle \frac{D_{\mathcal{M}}\left(H_{\mathcal{M}}^{s\tau }f\right)\left(\varsigma \right)}{H_{\mathcal{M}}^{s\tau }f\left(\varsigma \right)}},J\left(\varsigma \right):={\displaystyle \frac{M+\varsigma }{N+\varsigma }}\left(\varsigma \in \mathbb{E}\right).$$

Then, $\psi ⪯J$, i.e., $\psi \left(\mathbb{E}\right)\subset J\left(\mathbb{E}\right)$ if and only if

$$\psi \left(\mathbb{E}\right)\cap \partial J\left(\mathbb{E}\right)=\varnothing ,$$

or, equivalently,

$${\displaystyle \frac{D_{\mathcal{M}}\left(H_{\mathcal{M}}^{s\tau }f\right)\left(\varsigma \right)}{H_{\mathcal{M}}^{s\tau }f\left(\varsigma \right)}}\ne {\displaystyle \frac{M+\zeta }{N+\zeta }}\left(\varsigma \in \mathbb{E},\left|\zeta \right|=1\right).$$

(11)

Thus, $f\in {\mathcal{W}}_{\mathcal{M}}$ if and only if

$$\left(N+\zeta \right)D_{\mathcal{M}}\left(H_{\mathcal{M}}^{s\tau }f\right)\left(\varsigma \right)-\left(M+\zeta \right)H_{\mathcal{M}}^{s\tau }f\left(\varsigma \right)\ne 0\left(\varsigma \in \mathbb{E},\left|\zeta \right|=1\right).$$

Since

$$\begin{array}{cc}\hfill D_{\mathcal{M}}\left(H_{\mathcal{M}}^{s\tau }f\right)\left(\varsigma \right)& =\left(H_{\mathcal{M}}^{s\tau }f\right)\left(\varsigma \right)\ast \left(\varsigma +\sum _{n=1}^{\infty }n\left({\varsigma }^{-n}-\overline{{\varsigma }^{-n}}\right)\right),\hfill \\ \hfill \left(H_{\mathcal{M}}^{s\tau }f\right)\left(\varsigma \right)& =\left(H_{\mathcal{M}}^{s\tau }f\right)\left(\varsigma \right)\ast \left(\varsigma +\sum _{n=1}^{\infty }\left({\varsigma }^{-n}+\overline{{\varsigma }^{-n}}\right)\right),\hfill \end{array}$$

by (11), we have

$$\begin{array}{cc}\hfill & \left(N+\zeta \right)D_{\mathcal{M}}\left(H_{\mathcal{M}}^{s\tau }f\right)\left(\varsigma \right)-\left(M+\zeta \right)H_{\mathcal{M}}^{s\tau }f\left(\varsigma \right)\hfill \\ \hfill & =\left(H_{\mathcal{M}}^{s\tau }f\right)\left(\varsigma \right)\ast \left(\left(N+\zeta \right)\varsigma +\left(N+\zeta \right)\sum _{n=1}^{\infty }n\left({\varsigma }^{-n}-\overline{{\varsigma }^{-n}}\right)\right)\hfill \\ \hfill & -\left(H_{\mathcal{M}}^{s\tau }f\right)\left(\varsigma \right)\ast \left(\left(M+\zeta \right)\varsigma +\left(M+\zeta \right)\sum _{n=1}^{\infty }\left({\varsigma }^{-n}+\overline{{\varsigma }^{-n}}\right)\right)\hfill \\ \hfill & =\left(H_{\mathcal{M}}^{s\tau }f\right)\left(\varsigma \right)\ast \left(\left(N-M\right)\varsigma +\sum _{n=1}^{\infty }\left(nN-M+\left(n-1\right)\zeta \right){\varsigma }^{-n}\right.\hfill \\ \hfill & \left.-\sum _{n=1}^{\infty }\left(nN+M+\left(n+1\right)\zeta \right)\overline{{\varsigma }^{-n}}\right)\hfill \\ \hfill & =\left(N-M\right)·f\left(\varsigma \right)\ast H_{\mathcal{M}}^{s\tau }\left(\varsigma +\sum _{n=1}^{\infty }{\displaystyle \frac{nN-M+\left(n-1\right)\zeta }{N-M}}{\varsigma }^{-n}\right.\hfill \\ \hfill & -\left.\sum _{n=1}^{\infty }{\displaystyle \frac{nN+M+\left(n+1\right)\zeta }{N-M}}\overline{{\varsigma }^{-n}}\right)\hfill \\ \hfill & =\left(N-M\right)·f\left(\varsigma \right)\ast {\varphi }_{\zeta }\left(\varsigma \right)\ne 0\left(\varsigma \in \mathbb{E},\left|\zeta \right|=1\right).\hfill \end{array}$$

Therefore, $f\in {\mathcal{W}}_{\mathcal{M}}$ if and only if $f\left(\varsigma \right)\ast {\varphi }_{\zeta }\left(\varsigma \right)\ne 0$, i.e., $f\in {\left\{{\varphi }_{\zeta }:\left|\zeta \right|=1\right\}}^{\ast }.$ □

Theorem 2. 

$${\mathcal{R}}_{\mathcal{M}}={\left\{{\phi }_{\zeta }:\left|\zeta \right|=1\right\}}^{\ast },$$

where

$${\phi }_{\zeta }\left(\varsigma \right):=\varsigma +\sum _{n=1}^{\infty }{\displaystyle \frac{N+\zeta }{N-M}}\left({\mathrm{\Gamma }}_n{\varsigma }^{-n}+\overline{{\mathrm{\Gamma }}_n{\varsigma }^{-n}}\right)\left(\varsigma \in \mathbb{D}\right).$$

(12)

Proof. 

Similarly to the proof of Theorem, we obtain that $f\in {\mathcal{W}}_{\mathcal{M}}$ if and only if

$$\left(N+\zeta \right)H_{\mathcal{M}}^{s,\tau }f\left(\varsigma \right)-\left(M+\zeta \right)\varsigma \ne 0\left(\varsigma \in \mathbb{D},\left|\zeta \right|=1\right).$$

Since

$$\begin{array}{cc}\hfill & \left(N+\zeta \right)H_{\mathcal{M}}^{s,\tau }f\left(\varsigma \right)-\left(M+\zeta \right)\varsigma \hfill \\ \hfill & =\left(N+\zeta \right)H_{\mathcal{M}}^{s,\tau }f\left(\varsigma \right)\ast \left({\displaystyle \frac{1}{\varsigma }}+\sum _{n=1}^{\infty }\left({\varsigma }^{-n}+\overline{{\varsigma }^{-n}}\right)\right)-\left(M+\zeta \right)H_{\mathcal{M}}^{s,\tau }f\left(\varsigma \right)\ast \varsigma \hfill \\ \hfill & =H_{\mathcal{M}}^{s,\tau }f\left(\varsigma \right)\ast \left(\left(N-M\right)\varsigma +\left(N+\zeta \right)\sum _{n=1}^{\infty }\left({\varsigma }^{-n}+\overline{{\varsigma }^{-n}}\right)\right)\hfill \\ \hfill & =\left(N-M\right)\left\{H_{\mathcal{M}}^{s,\tau }f\left(\varsigma \right)\ast \left(\varsigma +\sum _{n=1}^{\infty }{\displaystyle \frac{N+\zeta }{N-M}}\left({\varsigma }^{-n}-\overline{{\varsigma }^{-n}}\right)\right)\right\}\hfill \\ \hfill & =\left(N-M\right)\zeta \left\{f\left(\varsigma \right)\ast {\phi }_{\zeta }\left(\varsigma \right)\right\}\left(\varsigma \in \mathbb{D},\left|\zeta \right|=1\right),\hfill \end{array}$$

we have $f\in {\mathcal{R}}_{\mathcal{M}}$ if and only if $f\left(\varsigma \right)\ast {\phi }_{\zeta }\left(\varsigma \right)\ne 0$ for $\varsigma \in \mathbb{D}$ and $\left|\zeta \right|=1,$ i.e., $f\in {\left\{{\varphi }_{\zeta }:\left|\zeta \right|=1\right\}}^{\ast }.$ □

Next, we construct the alternative definitions for the main classes of functions with correlated coefficients. Let $\phi \in {\mathcal{M}}_{\mathcal{H}}$ be the function of the form

$$\phi =\gamma +\overline{\delta },\gamma \left(\varsigma \right)=\varsigma +\sum _{n=1}^{\infty }{\gamma }_n{\varsigma }^{-n},\delta \left(\varsigma \right)=\sum _{n=1}^{\infty }{\delta }_n{\varsigma }^{-n}\left(\varsigma \in \mathbb{E}\right).$$

(13)

If a function $f\in {\mathcal{M}}_{\mathcal{H}}$ of the form (2) satisfies

$${\gamma }_n{a}_n=-\left|{\gamma }_n\right|\left|a_n\right|,{\delta }_n{b}_n=\left|{\delta }_n\right|\left|b_n\right|\left(n\in \mathbb{N}\right),$$

(14)

then we say that it has coefficients correlated with the function $\phi .$ In particular, we obtain functions with varying coefficients, as introduced by Jahangiri and Silverman [24] (see also [15,22]), by setting

$$\phi \left(\varsigma \right)=\varsigma +2e^{i\eta }\mathrm{Re}{\displaystyle \frac{e^{i\eta }\varsigma }{1-e^{i\eta }\varsigma }}=\varsigma +\sum _{n=1}^{\infty }\left(e^{i\left(n-1\right)\eta }{\varsigma }^{-n}+\overline{e^{i\left(n+1\right)\eta }{\overline{\varsigma }}^n}\right)\left(\varsigma \in \mathbb{E}\right),$$

for some real number $\eta$. Moreover, if we set

$$\phi \left(\varsigma \right)=\varsigma +2\mathrm{Re}{\displaystyle \frac{\varsigma }{1-\varsigma }}=\varsigma +\sum _{n=1}^{\infty }\left({\varsigma }^{-n}+{\overline{\varsigma }}^n\right)\left(\varsigma \in \mathbb{E}\right),$$

then we obtain functions with negative coefficients, as defined by Silverman [25] (see also [20,26]).

We denote by $\mathcal{T}\left(\eta \right)$ the class of functions $f\in \mathcal{M}$ with coefficients correlated with the function

$$\phi \left(\varsigma \right)=e^{-i\eta }{\mathrm{\Phi }}_{\mathcal{M}}^{s,\tau }\left(e^{i\eta }\varsigma \right)\left(\varsigma \in \mathbb{E}\right),$$

(15)

where ${\mathrm{\Phi }}_{\mathcal{M}}^{s,\tau }$ is defined by (6). Moreover, let us define

$${\mathcal{W}}_{\mathcal{M}}^{\eta }:=\mathcal{T}\left(\eta \right)\cap {\mathcal{W}}_{\mathcal{M}},{\mathcal{R}}_{\mathcal{M}}^{\eta }:=\mathcal{T}\left(\eta \right)\cap {\mathcal{R}}_{\mathcal{M}}.$$

If a function$f\in \mathcal{T}\left(\eta \right)$ is of the form (2), then by (14) and (15), we have

$$a_n=-{\displaystyle \frac{\left|{\mathrm{\Gamma }}_n\right|}{{\mathrm{\Gamma }}_n}}{e}^{i\left(n+1\right)\eta }\left|a_n\right|,b_n={\displaystyle \frac{\left|{\mathrm{\Gamma }}_n\right|}{\tau {\mathrm{\Gamma }}_n}}\left|b_n\right|e^{i\left(n-1\right)\eta }\left(n\in \mathbb{N}\right),$$

(16)

or, equivalently,

$$f\left(\varsigma \right)=\varsigma -\sum _{n=1}^{\infty }\left(\left|a_n\right|e^{i{\zeta }_n}{\varsigma }^{-n}-\left|b_n\right|e^{-i{\delta }_n}\overline{{\varsigma }^{-n}}\right)\left(\varsigma \in \mathbb{E}\right),$$

where

$${\zeta }_n=\left(n+1\right)\eta -arg{\mathrm{\Gamma }}_n,{\delta }_n=\left(n-1\right)\eta -arg\left(\tau {\mathrm{\Gamma }}_n\right).$$

Theorem 3. 

Let ${\mathrm{\Gamma }}_n$ be defined by (5) and

$$u_n={\mathrm{\Gamma }}_n{\displaystyle \frac{n\left(1+N\right)-\left(1+M\right)}{N-M}},v_n={\mathrm{\Gamma }}_n{\displaystyle \frac{n\left(1+N\right)+\left(1+M\right)}{N-M}}.$$

(17)

If a function $f\in {\mathcal{M}}_{\mathcal{H}}$of the form (2) satisfies the following coefficient condition:

$$\sum _{n=1}^{\infty }\left(\left|u_n\right|\left|a_n\right|+\left|v_n\right|\left|b_n\right|\right)\le 1,$$

(18)

then $f\in {\mathcal{W}}_{\mathcal{M}}$.

Proof. 

Let $f\in {\mathcal{M}}_{\mathcal{H}}$ be of the form (2). Then by (7), we have

$$\left|u_n\right|\ge n,\left|v_n\right|\ge n,n\in \mathbb{N}.$$

(19)

Moreover, by (18), we obtain

$$\sum _{n=1}^{\infty }\left(n\left|a_n\right|+n\left|b_n\right|\right)\le 1.$$

(20)

It is easy to show that the Jacobian of the function$f$ is given by

$$J_f\left(\varsigma \right)={\left|h^{\prime }\left(\varsigma \right)\right|}^2-{\left|g^{\prime }\left(\varsigma \right)\right|}^2\left(\varsigma \in \mathbb{E}\right).$$

A function $f$ is sense-preserving and locally univalent if the Jacobian of f is positive in$\mathbb{E}$. Lewy [4] proved that the converse theorem holds for harmonic mappings. Since

$$\begin{array}{cc}\hfill \left|h^{\prime }\left(\varsigma \right)\right|-\left|g^{\prime }\left(\varsigma \right)\right|& \ge 1-\sum _{n=1}^{\infty }n\left|a_n\right|{\left|\varsigma \right|}^{-\left(n+1\right)}-\sum _{n=1}^{\infty }n\left|b_n\right|{\left|\varsigma \right|}^{-\left(n+1\right)}\hfill \\ \hfill & \ge 1-\left|\varsigma \right|\sum _{n=1}^{\infty }\left(n\left|a_n\right|+n\left|b_n\right|\right)\ge 1-\left|\varsigma \right|>0\left(\varsigma \in \mathbb{E}\right),\hfill \end{array}$$

we have that$f$ is locally univalent and sense-preserving in$\mathbb{E}.$ To obtain the univalence of f, we assume that $z_1,z_2\in \mathbb{E},$ $z_1\ne z_2.$ If we set $w_1={\left(z_1\right)}^{-1},$ $w_2={\left(z_2\right)}^{-1}$, by (20) and the inequality

$$\left|{\displaystyle \frac{w_1^n-w_2^n}{w_1-w_2}}\right|=\left|\sum _{l=1}^n{w}_1^{l-1}{w}_2^{n-l}\right|\le \sum _{l=1}^n{\left|w_1\right|}^{l-1}{\left|w_2\right|}^{n-l}<n\left(n\in \mathbb{N}\right),$$

we obtain

$$\begin{array}{cc}\hfill \left|f\left(z_1\right)-f\left(z_2\right)\right|& \ge \left|h\left(z_1\right)-h\left(z_2\right)\right|-\left|g\left(z_1\right)-g\left(z_2\right)\right|\hfill \\ \hfill & =\left|{\displaystyle \frac{1}{w_1}}-{\displaystyle \frac{1}{w_2}}-\sum _{n=1}^{\infty }{a}_n\left(w_1^n-w_2^n\right)\right|-\left|\sum _{n=1}^{\infty }\overline{b_n\left(w_1^n-w_2^n\right)}\right|\hfill \\ \hfill & \ge {\displaystyle \frac{\left|w_1-w_2\right|}{\left|w_1{w}_2\right|}}-\sum _{n=1}^{\infty }\left|a_n\right|\left|w_1^n-w_2^{-n}\right|-\sum _{n=1}^{\infty }\left|b_n\right|\left|w_1^n-w_2^n\right|\hfill \\ \hfill & =\left|w_1-w_2\right|\left({\displaystyle \frac{1}{\left|w_1{w}_2\right|}}-\sum _{n=1}^{\infty }\left|a_n\right|\left|{\displaystyle \frac{w_1^n-w_2^n}{w_1-w_2}}\right|-\sum _{n=1}^{\infty }\left|b_n\right|\left|{\displaystyle \frac{w_1^n-w_2^n}{w_1-w_2}}\right|\right)\hfill \\ \hfill & >\left|w_1-w_2\right|\left(1-\sum _{n=1}^{\infty }n\left|a_n\right|-\sum _{n=1}^{\infty }n\left|b_n\right|\right)\ge 0.\hfill \end{array}$$

Thus, $f\in {\mathcal{M}}_{\mathcal{H}}.$ Moreover, for functions ${\varphi }_{\zeta }$ of the form (10), $\left|\zeta \right|=1$ and $\left|\varsigma \right|=r>1,$ we have

$$\begin{array}{cc}\hfill \left|f\left(\varsigma \right)\ast {\varphi }_{\zeta }\left(\varsigma \right)\right|& =\left|\varsigma +\sum _{n=1}^{\infty }\left({\displaystyle \frac{nN-M+\left(n-1\right)\zeta }{N-M}}{\mathrm{\Gamma }}_n{\varsigma }^{-n}-{\displaystyle \frac{nN+M+\left(n+1\right)\zeta }{N-M}}\overline{\tau {\mathrm{\Gamma }}_n}\overline{{\varsigma }^{-n}}\right)\right|\hfill \\ \hfill & \ge r-\sum _{n=1}^{\infty }\left({\displaystyle \frac{nN-M+n-1}{N-M}}\left|a_n\right|+{\displaystyle \frac{nN+M+n+1}{N-M}}\left|b_n\right|\right)\left|{\mathrm{\Gamma }}_n\right|r^n\hfill \\ \hfill & \ge {\displaystyle \frac{1}{r}}-\sum _{n=1}^{\infty }\left({\displaystyle \frac{n\left(1+N\right)-\left(1+M\right)}{N-M}}\left|a_n\right|+{\displaystyle \frac{n\left(1+N\right)+\left(1+M\right)}{N-M}}\left|b_n\right|\right)\left|{\mathrm{\Gamma }}_n\right|r^n\hfill \\ \hfill & \ge {\displaystyle \frac{1}{r}}\left(1-\sum _{n=1}^{\infty }\left(\left|u_n\right|\left|a_n\right|+\left|v_n\right|\left|b_n\right|\right)r^{n+1}\right)>0.\hfill \end{array}$$

This means that

$$f\left(\varsigma \right)\ast {\varphi }_{\zeta }\left(\varsigma \right)\ne 0\left(\left|\zeta \right|=1,\varsigma \in \mathbb{E}\right),$$

and, by Theorem 3, we conclude that $f\in {\left\{{\varphi }_{\zeta }:\left|\zeta \right|=1\right\}}^{\ast }={\mathcal{W}}_{\mathcal{M}}.$ □

The sufficient coefficient bound given in Theorem 3 becomes the definition of the class ${\mathcal{W}}_{\mathcal{M}}^{\eta }$ of functions with correlated coefficients, as stated in the following theorem.

Theorem 4. 

Let $f\in \mathcal{T}\left(\eta \right)$ be a function of the form (2). Then, $f\in {\mathcal{W}}_{\mathcal{M}}^{\eta }$ if and only if the condition (18) holds true.

Proof. 

By Theorem 3, we need to prove the “only if” part. Let $f\in {\mathcal{W}}_{\mathcal{M}}^{\eta }$ be of the form (2) and let

$$\psi \left(\varsigma \right):={\displaystyle \frac{D_{\mathcal{M}}\left(H_{\mathcal{M}}^{s\tau }f\right)\left(\varsigma \right)}{H_{\mathcal{M}}^{s\tau }f\left(\varsigma \right)}},J\left(\varsigma \right):={\displaystyle \frac{M+\varsigma }{N+\varsigma }}\left(\varsigma \in \mathbb{E}\right).$$

Since $\psi ⪯J$, i.e., $\psi \left(\mathbb{E}\right)\subset J\left(\mathbb{E}\right)$, and J is univalent in $\mathbb{E}$, we can define the function

$$\omega \left(\varsigma \right)=J^{-1}\left(\psi \left(\varsigma \right)\right)\left(\varsigma \in \mathbb{E}\right).$$

Then, $\left|\omega \left(\varsigma \right)\right|>1$ and $\psi \left(\varsigma \right)=J\left(\omega \left(\varsigma \right)\right)$ for $\varsigma \in \mathbb{E}$, i.e.,

$${\displaystyle \frac{D_{\mathcal{M}}\left(H_{\mathcal{M}}^{s\tau }f\right)\left(\varsigma \right)}{H_{\mathcal{M}}^{s\tau }f\left(\varsigma \right)}}={\displaystyle \frac{M+\omega \left(\varsigma \right)}{N+\omega \left(\varsigma \right)}}\left(\varsigma \in \mathbb{E}\right).$$

Thus,

$$\left|{\displaystyle \frac{D_{\mathcal{M}}\left(H_{\mathcal{M}}^{s\tau }f\right)\left(\varsigma \right)-H_{\mathcal{M}}^{s\tau }f\left(\varsigma \right)}{N·D_{\mathcal{M}}\left(H_{\mathcal{M}}^{s\tau }f\right)\left(\varsigma \right)-M·H_{\mathcal{M}}^{s\tau }f\left(\varsigma \right)}}\right|={\displaystyle \frac{1}{\left|\omega \left(\varsigma \right)\right|}}<1\left(\varsigma \in \mathbb{E}\right).$$

(21)

Since

$$H_{\mathcal{M}}^{s\tau }f\left(\varsigma \right)=\varsigma +\sum _{n=1}^{\infty }\left({\mathrm{\Gamma }}_n{a}_n{\varsigma }^{-n}+\overline{\tau {\mathrm{\Gamma }}_n{b}_n{\varsigma }^{-n}}\right)\left(\varsigma \in \mathbb{E}\right),$$

we obtain

$$\left|{\displaystyle \frac{{\displaystyle \sum _{n=1}^{\infty }}\left(\left(n-1\right){\mathrm{\Gamma }}_n{a}_n{\varsigma }^{-n-1}-\left(n+1\right)\overline{\tau {\mathrm{\Gamma }}_n{b}_n{\varsigma }^{-n}}{\varsigma }^{-1}\right)}{\left(N-M\right)+{\displaystyle \sum _{n=1}^{\infty }}\left(\left(Nn-M\right){\mathrm{\Gamma }}_n{a}_n{\varsigma }^{-n-1}-\left(Nn+M\right)\overline{\tau {\mathrm{\Gamma }}_n{b}_n{\varsigma }^{-n}}{\varsigma }^{-1}\right)}}\right|<1(\varsigma \in \mathbb{E}).$$

Thus, for $\varsigma =r{e}^{i\eta }(r>1),$ we have

$$\left|{\displaystyle \frac{{\displaystyle \sum _{n=1}^{\infty }}\left(\left(n-1\right){\mathrm{\Gamma }}_n{e}^{-i\left(n+1\right)\eta }{a}_n{r}^{-n-1}-\left(n+1\right)\overline{\tau {\mathrm{\Gamma }}_n{e}^{-i\left(n-1\right)\eta }{b}_n}{r}^{-n-1}\right)}{\left(N-M\right)+{\displaystyle \sum _{n=1}^{\infty }}\left(\left(Nn-M\right){\mathrm{\Gamma }}_n{e}^{-i\left(n+1\right)\eta }{a}_n{r}^{-n-1}-\left(Nn+M\right)\overline{\tau {\mathrm{\Gamma }}_n{e}^{-i\left(n-1\right)\eta }{b}_n}{r}^{-n-1}\right)}}\right|<1,$$

which, by (16), gives

$${\displaystyle \frac{{\displaystyle \sum _{n=1}^{\infty }}\left|{\mathrm{\Gamma }}_n\right|\left\{\left(n-1\right)\left|a_n\right|+\left(n+1\right)\left|b_n\right|\right\}{r}^{-n-1}}{\left(N-M\right)-{\displaystyle \sum _{n=1}^{\infty }}\left|{\mathrm{\Gamma }}_n\right|\left\{\left(Nn-M\right)\left|a_n\right|+\left(Nn+M\right)\left|b_n\right|\right\}{r}^{-n-1}}}<1.$$

(22)

We note that the denominator on the left-hand side cannot be zero for $r>1,$ and it is positive for $r=\infty .$ Thus, it is positive for $r>1$. Consequently, by (22), we obtain

$$\sum _{n=1}^{\infty }\left(\left|u_n\right|\left|a_n\right|+\left|v_n\right|\left|b_n\right|\right)r^{-n-1}<1(r>1).$$

(23)

Let $\left\{S_n\right\}$ denote the sequence of partial sums associated with the series ${\displaystyle \sum _{n=1}^{\infty }}\left(\left|u_n\right|\left|a_n\right|+\left|v_n\right|\left|b_n\right|\right).$ It is clear that this is a non-decreasing sequence and bounded by $1.$ Thus, the sequence is convergent and

$$\sum _{n=1}^{\infty }\left(\left|u_n\right|\left|a_n\right|+\left|v_n\right|\left|b_n\right|\right)=\underset{n\to \infty }{lim}{S}_n\le 1,$$

which gives (18). □

In the same way, we obtain the following theorem.

Theorem 5. 

A function $f\in \mathcal{T}\left(\eta \right)$ of the form (2) belongs to the class ${\mathcal{R}}_{\mathcal{T}}^{\eta }$ if and only if

$$\sum _{n=1}^{\infty }\left|{\mathrm{\Gamma }}_n\right|\left(\left|a_n\right|+\left|b_n\right|\right)\le {\displaystyle \frac{N-M}{1+N}}.$$

By Theorems 4 and 5, we have the following corollary.

Corollary 1. 

If $a={\displaystyle \frac{1+M}{1+N}}$ and

$$\begin{array}{cc}\hfill \varphi \left(\varsigma \right)& =\varsigma +\sum _{n=1}^{\infty }\left({\displaystyle \frac{1}{n-a}}{\varsigma }^{-n}+{\displaystyle \frac{1}{n+a}}{\overline{\varsigma }}^n\right)\left(\varsigma \in \mathbb{E}\right),\hfill \\ \hfill \omega \left(\varsigma \right)& =\varsigma +\sum _{n=1}^{\infty }\left(\left(n-a\right){\varsigma }^{-n}+\left(n+a\right){\overline{\varsigma }}^n\right)\left(\varsigma \in \mathbb{E}\right),\hfill \end{array}$$

then

$$f\in {\mathcal{R}}_{\mathcal{M}}^{\eta }\iff f\ast \varphi \in {\mathcal{W}}_{\mathcal{M}}^{\eta },f\in {\mathcal{W}}_{\mathcal{M}}^{\eta }\iff f\ast \omega \in {\mathcal{R}}_{\mathcal{M}}^{\eta }.$$

3. Radii of Meromorphic Starlikeness and Convexity

A function $f\in \mathcal{M}$ is called meromorphically starlike of order $\alpha$ in$\mathbb{E}\left(r\right)$ if

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

(24)

Similarly, if

$${\displaystyle \frac{\partial }{\partial t}}\left(arg{\displaystyle \frac{\partial }{\partial t}}f\left(\rho e^{it}\right)\right)>\alpha ,0\le t\le 2\pi ,\rho >r,$$

then f is called convex of order $\alpha$ in$\mathbb{E}\left(r\right).$ Simple calculations show that a function $f\in \mathcal{T}\left(\eta \right)$ is meromorphically starlike of order $\alpha$ in$\mathbb{E}\left(r\right)$ if and only if

$$\mathrm{Re}{\displaystyle \frac{D_{\mathcal{M}}f\left(\varsigma \right)}{f\left(\varsigma \right)}}>\alpha (\varsigma \in \mathbb{E}\left(r\right)),$$

or

$$\left|{\displaystyle \frac{D_{\mathcal{M}}f\left(\varsigma \right)-\left(1+\alpha \right)f\left(\varsigma \right)}{D_{\mathcal{M}}f\left(\varsigma \right)-\left(1-\alpha \right)f\left(\varsigma \right)}}\right|<1(\varsigma \in \mathbb{E}\left(r\right)).$$

(25)

For the class $\mathcal{B}\subset \mathcal{M}$, we define the radius of starlikeness of order $\alpha$ by

$$R_{\alpha }^{\ast }\left(\mathcal{B}\right):=\underset{f\in \mathcal{B}}{sup}\left(inf\left\{r\in \left(1,\infty \right):f\mathrm{is}\mathrm{starlike}\mathrm{of}\mathrm{order}\alpha \mathrm{in}\mathbb{E}\left(r\right)\right\}\right),$$

and the radius of convexity of order $\alpha$ by

$$R_{\alpha }^c\left(\mathcal{B}\right):=\underset{f\in \mathcal{B}}{sup}\left(inf\left\{r\in \left(1,\infty \right):f\mathrm{is}\mathrm{convex}\mathrm{of}\mathrm{order}\alpha \mathrm{in}\mathbb{E}\left(r\right)\right\}\right).$$

In particular, we obtain the radius of starlikeness $R^{\ast }\left(\mathcal{B}\right):=R_0^{\ast }\left(\mathcal{B}\right)$and the radius of convexity $R^c\left(\mathcal{B}\right):=R_0^c\left(\mathcal{B}\right)$ for the class $\mathcal{B}$.

Theorem 6. 

$$R_{\alpha }^{\ast }\left({\mathcal{W}}_{\mathcal{M}}^{\eta }\right)=\underset{n\in \mathbb{N}}{sup}{\left(max\left\{{\displaystyle \frac{n-\alpha }{\left(1-\alpha \right)\left|u_n\right|}},{\displaystyle \frac{n+\alpha }{\left(1-\alpha \right)\left|v_n\right|}}\right\}\right)}^{{\displaystyle \frac{1}{n+1}}},$$

(26)

where$u_n$ and $v_n$ are given by (17).

Proof. 

For a function $f\in {\mathcal{W}}_{\mathcal{M}}^{\eta }$ of the form (2) and for $\left|\varsigma \right|=r>1$, we have

$$\begin{array}{cc}\hfill \left|{\displaystyle \frac{D_{\mathcal{M}}f\left(\varsigma \right)-\left(1+\alpha \right)f\left(\varsigma \right)}{D_{\mathcal{M}}f\left(\varsigma \right)+\left(1-\alpha \right)f\left(\varsigma \right)}}\right|& =\left|{\displaystyle \frac{-\alpha \varsigma +{\displaystyle \sum _{n=1}^{\infty }}\left((n-1-\alpha )a_n{\varsigma }^{-n}-(n+1+\alpha )\overline{b_n{\overline{\varsigma }}^{-n}}\right)}{\left(2-\alpha \right)\varsigma +{\displaystyle \sum _{n=1}^{\infty }}\left((n+1-\alpha )a_n{\varsigma }^{-n}-(n+1-\alpha )\overline{b_n{\overline{\varsigma }}^{-n}}\right)}}\right|\hfill \\ \hfill & \le {\displaystyle \frac{\alpha +{\displaystyle \sum _{n=1}^{\infty }}\left((n-1-\alpha )\left|a_n\right|+(n+1+\alpha )\left|b_n\right|\right)r^{-\left(n+1\right)}}{2-\alpha -{\displaystyle \sum _{n=1}^{\infty }}\left((n+1-\alpha )\left|a_n\right|+(n+1-\alpha )\left|b_n\right|\right)r^{-\left(n+1\right)}}}.\hfill \end{array}$$

Thus, the condition (25) is true if

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

(27)

By Theorem 1, we have

$$\sum _{n=1}^{\infty }\left(\left|u_n\right|\left|a_n\right|+\left|v_n\right|\left|b_n\right|\right)\le 1,$$

(28)

where $u_n$ and $v_n$ are given by (17). Therefore, the condition (27) is true if

$${\displaystyle \frac{n-\alpha }{1-\alpha }}{r}^{-\left(n+1\right)}\le \left|u_n\right|,{\displaystyle \frac{n+\alpha }{1-\alpha }}{r}^{-\left(n+1\right)}\le \left|v_n\right|(n\in \mathbb{N}),$$

or, equivalently,

$$r\ge {\left(max\left\{{\displaystyle \frac{n-\alpha }{\left(1-\alpha \right)\left|u_n\right|}},{\displaystyle \frac{n+\alpha }{\left(1-\alpha \right)\left|v_n\right|}}\right\}\right)}^{{\displaystyle \frac{1}{n+1}}}(n\in \mathbb{N}).$$

It follows that the function$f$ is meromorphically starlike of order $\alpha$ in the disk $\mathbb{E}\left(r^{\ast }\right)$, where

$$r^{\ast }:=\underset{n\in \mathbb{N}}{sup}{\left(max\left\{{\displaystyle \frac{n-\alpha }{\left(1-\alpha \right)\left|u_n\right|}},{\displaystyle \frac{n+\alpha }{\left(1-\alpha \right)\left|v_n\right|}}\right\}\right)}^{{\displaystyle \frac{1}{n+1}}}.$$

It is easy to verify that the functions $h_n$ and $g_n$ of the form

$$h_0\left(\varsigma \right)=\varsigma ,h_n\left(\varsigma \right)=\varsigma -{\displaystyle \frac{1}{u_n}}{e}^{i\left(n+1\right)\eta }{\varsigma }^{-n},g_n\left(\varsigma \right)=\varsigma +\overline{{\displaystyle \frac{1}{\tau v_n}}{e}^{-i\left(n-1\right)\eta }{\varsigma }^{-n}}(n\in \mathbb{N},\varsigma \in \mathbb{E})$$

(29)

give equality in (28). Thus, we have (26). □

In the same way, we obtain the following two theorems.

Theorem 7. 

$$R_{\alpha }^c\left({\mathcal{W}}_{\mathcal{M}}^{\eta }\right)=\underset{n\in \mathbb{N}}{sup}{\left(max\left\{{\displaystyle \frac{\left(1-\alpha \right)\left|u_n\right|}{n\left(n+\alpha \right)}},{\displaystyle \frac{\left(1-\alpha \right)\left|v_n\right|}{n\left(n-\alpha \right)}}\right\}\right)}^{{\displaystyle \frac{1}{n+1}}},$$

where$u_n$ and $v_n$ are given by (17).

Theorem 8. 

$$\begin{array}{cc}\hfill R_{\alpha }^{\ast }\left({\mathcal{R}}_{\mathcal{M}}^{\eta }\right)& =\underset{n\in \mathbb{N}}{sup}{\left({\displaystyle \frac{\left(1-\alpha \right)\left(1+N\right)}{\left(n+\alpha \right)\left(N-M\right)}}\left|{\mathrm{\Gamma }}_n\right|\right)}^{{\displaystyle \frac{1}{n+1}}},\hfill \\ \hfill R_{\alpha }^c\left({\mathcal{R}}_{\mathcal{M}}^{\eta }\right)& =\underset{n\in \mathbb{N}}{sup}{\left({\displaystyle \frac{\left(1-\alpha \right)\left(1+N\right)}{n\left(n+\alpha \right)\left(N-M\right)}}\left|{\mathrm{\Gamma }}_n\right|\right)}^{{\displaystyle \frac{1}{n+1}}},\hfill \end{array}$$

where${\mathrm{\Gamma }}_n$ is given by (5).

Setting $\alpha =0$ in Theorems 6–8, we obtain the following corollary.

Corollary 2. 

$$\begin{array}{cc}\hfill R^{\ast }\left({\mathcal{W}}_{\mathcal{M}}^{\eta }\right)& =R_{\alpha }^{\ast }\left({\mathcal{R}}_{\mathcal{M}}^{\eta }\right)=1,R^c\left({\mathcal{W}}_{\mathcal{M}}^{\eta }\right)=\underset{n\in \mathbb{N}}{inf}{\left({\displaystyle \frac{\left|v_n\right|}{n^2}}\right)}^{{\displaystyle \frac{1}{n+1}}},\hfill \\ \hfill R_{\alpha }^c\left({\mathcal{R}}_{\mathcal{M}}^{\eta }\right)& =\underset{n\in \mathbb{N}}{inf}{\left({\displaystyle \frac{\left(1-\alpha \right)\left(1+N\right)}{n\left(n+\alpha \right)\left(N-M\right)}}\left|{\mathrm{\Gamma }}_n\right|\right)}^{{\displaystyle \frac{1}{n+1}}}.\hfill \end{array}$$

4. Extreme Points

First, we study the topological and convexity properties of the classes of correlated coefficients. Let us consider thelinear topology on$\mathcal{M}$ given byuniform convergence on a compact subset of $\mathbb{E}$.

We say that aclass $\mathcal{B}\subset \mathcal{M}$ is convex if any convex linear combination of two functions from $\mathcal{B}$ belongs to $\mathcal{B}$. Wedenote by $\overline{co}\mathcal{B}$ the closed convex hull of $\mathcal{B}$, i.e., theintersection of all closed convex subsets of $\mathcal{M}$ that contain $\mathcal{B}$.

A function $f\in \mathcal{B}$ is called an extreme point of $\mathcal{B}$ if it cannot be represented as a non-degenerate convex linear combination of two functions from $\mathcal{B}$. Wedenote by $E\mathcal{B}$ theset of all extreme points of $\mathcal{B}.$ It is clear that $E\mathcal{B}\subset \mathcal{B}.$

We say that a functional $\mathcal{J}:\mathcal{M}\to \mathbb{R}$ is convex on aconvex class $\mathcal{B}\subset \mathcal{M}$ if

$$\mathcal{J}\left(\lambda f+\left(1-\lambda \right)g\right)\le \lambda \mathcal{J}\left(f\right)+\left(1-\lambda \right)\mathcal{J}\left(g\right)\left(f,g\in \mathcal{B},0\le \lambda \le 1\right).$$

Let $a\ge 0,d_n>1\left(n=1,2,...\right)$. We denote by ${\mathcal{A}}_{\mathbb{E}}\left(a\right)$ the class of functions f of the form

$$f\left(\varsigma \right)=a\varsigma +\sum _{n=1}^{\infty }{a}_n{\varsigma }^{-n}\left(\varsigma \in \mathbb{E},a_n\ge 0\right)$$

which are analytic in $\mathbb{E}$. By $\mathcal{G}\left(a,d_n\right)$, we denote class of functions $f\in {\mathcal{A}}_{\mathbb{E}}\left(a\right)$ that satisfy the condition

$$\sum _{n=1}^{\infty }{d}_n\left|a_n\right|\le 1.$$

(30)

From Montel’s theorem, we have the following lemma.

Lemma 1. 

A class $\mathcal{B}\subset {\mathcal{A}}_{\mathbb{E}}\left(a\right)$ is compact if and only if $\mathcal{B}$ is closed and locally uniformly bounded.

Lemma 2. 

The class $\mathcal{G}\left(a,d_n\right)$ is compact.

Proof. 

A class $\mathcal{B}\subset {\mathcal{A}}_{\mathbb{E}}\left(a\right)$ islocally uniformly bounded if for each $r,R,1<r<R,$ there isa real constant $L=L\left(r,R\right)$ such that

$$\left|f\left(\varsigma \right)\right|\le L\left(f\in \mathcal{B},r\le \left|\varsigma \right|\le R\right).$$

Let $f\in \mathcal{G}\left(a,d_n\right),1<r<R.$ Then, by (30), we have

$$\left|f\left(\varsigma \right)\right|\le aR+\sum _{n=1}^{\infty }\left|a_n\right|r^{-n}\le aR+\sum _{n=1}^{\infty }{d}_n\left|a_n\right|\le aR+m=:L.$$

This implies that the class $\mathcal{G}\left(a,d_n\right)$ is locally uniformly bounded. Next, we show that it is a closed set. Let f be a function of the form (2), and let $\left(f_l\right)\subset \mathcal{G}\left(a,d_n\right)$ be a sequence of functions $f_l$ given by

$$f_l\left(\varsigma \right)=a\varsigma +\sum _{n=1}^{\infty }{a}_{l,n}{\varsigma }^{-n}\left(\varsigma \in \mathbb{E},l\in \mathbb{N}\right).$$

By (30), we obtain

$$\sum _{n=1}^{\infty }{d}_n\left|a_{l,n}\right|\le 1\left(l\in \mathbb{N}\right).$$

(31)

If $f_l\to f,$ then we obtain that $a_{l,n}\to a_n$ and $b_{l,n}\to b_n$ as $l\to \infty \left(n\in \mathbb{N}\right)$. Thus, by (31), we have

$$\sum _{n=1}^{\infty }{d}_n\left|a_n\right|\le 1.$$

and consequently, $f\in \mathcal{G}\left(a,d_n\right),$ which completes the proof. □

Theorem 9. 

The class ${\mathcal{W}}_{\mathcal{M}}^{\eta }$ is a compact and convex subclass of $\mathcal{M}$.

Proof. 

Let $0\le \lambda \le 1$ and $f_1,f_2\in {\mathcal{W}}_{\mathcal{M}}^{\eta }$ be functions of the form

$$f_l\left(\varsigma \right)=\varsigma +\sum _{n=1}^{\infty }\left(a_{l,n}{\varsigma }^{-n}+\overline{b_{l,n}{\varsigma }^{-n}}\right)\left(\varsigma \in \mathbb{E},l=1,2\right).$$

(32)

Then, we have

$$\lambda f_1\left(\varsigma \right)+(1-\lambda )f_2\left(\varsigma \right)=\varsigma +\sum _{n=1}^{\infty }\left\{\left(\lambda a_{1,n}+\left(1-\lambda \right)a_{2,n}\right){\varsigma }^{-n}+\overline{\left(\lambda b_{1,n}+\left(1-\lambda \right)b_{2,n}\right){\varsigma }^{-n}}\right\}.$$

Moreover, by Theorem 4, we obtain

$$\begin{array}{cc}\hfill & \sum _{n=1}^{\infty }\left\{\left|u_n\right|\left|\lambda a_{1,n}+\left(1-\lambda \right)a_{2,n}\right|+\left|v_n\right|\left|\lambda b_{1,n}+\left(1-\lambda \right)b_{2,n}\right|\right\}\hfill \\ \hfill & \le \lambda \sum _{n=1}^{\infty }\left\{\left|u_n\right|\left|a_{1,n}\right|+\left|v_n\right|\left|b_{1,n}\right|\right\}+\left(1-\lambda \right)\sum _{n=1}^{\infty }\left\{\left|u_n\right|\left|a_{2,n}\right|+\left|v_n\right|\left|b_{2,n}\right|\right\}\hfill \\ \hfill & \le \lambda ·1+\left(1-\lambda \right)·1=1.\hfill \end{array}$$

Thus, the function $\phi =\lambda f_1+(1-\lambda )f_2$ belongs to the class ${\mathcal{W}}_{\mathcal{M}}^{\eta }$, and consequently, the class is convex.

Let $\left(f_l\right)\subset {\mathcal{W}}_{\mathcal{M}}^{\eta }$ be a sequence of the form $f_l\left(\varsigma \right)=h_l+\overline{g_l}\left(l\in \mathbb{N}\right),$ where

$$h_l\left(\varsigma \right)=\varsigma +\sum _{n=1}^{\infty }{a}_{l,n}{\varsigma }^{-n},g_l\left(\varsigma \right)=\sum _{n=1}^{\infty }{b}_{l,n}{\varsigma }^{-n}\left(\varsigma \in \mathbb{E},l\in \mathbb{N}\right).$$

By Theorem 4, we have

$$\sum _{n=1}^{\infty }\left(\left|u_n\right|\left|a_{l,n}\right|+\left|v_n\right|\left|b_{l,n}\right|\right)\le 1\left(l\in \mathbb{N}\right).$$

Thus,

$$\sum _{n=1}^{\infty }\left|u_n\right|\left|a_{l,n}\right|\le 1,\sum _{n=1}^{\infty }\left|v_n\right|\left|b_{l,n}\right|\le 1\left(l\in \mathbb{N}\right),$$

and consequently,

$$h_l\in \mathcal{G}\left(1,\left|u_n\right|\right),g_l\in \mathcal{G}\left(0,\left|v_n\right|\right)\left(l\in \mathbb{N}\right).$$

By Lemma 2, there exists a subsequence $\left(h_{l_k}\right)$ of the sequence $\left(h_l\right)$ that converges to the function $h\in \mathcal{G}\left(1,\left|u_n\right|\right).$ In the same way, there exists a subsequence $\left(g_{l_{k_j}}\right)$ of the subsequence $\left(g_{l_k}\right)$ that converges to the function $g\in \mathcal{G}\left(0,\left|v_n\right|\right).$ Thus, the subsequence $f_{l_{k_j}}=h_{l_{k_j}}+\overline{g_{l_{k_j}}}$ of the sequence $\left(f_l\right)$ converges to the function $f=h+\overline{g}.$ By Theorem 4, we obtain that $a_{l_{k_j},n}\to a_n$ and $b_{l_{k_j},n}\to b_n$ as $j\to \infty \left(n\in \mathbb{N}\right)$. Thus, by (31), we have

$$\sum _{n=1}^{\infty }\left(\left|u_n\right|\left|a_n\right|+\left|v_n\right|\left|b_n\right|\right)\le 1.$$

and consequently, $f\in {\mathcal{W}}_{\mathcal{M}}^{\eta }.$ Thus, we obtain that the class ${\mathcal{W}}_{\mathcal{M}}^{\eta }$ is compact, which completes the proof. □

Theorem 10. 

The extreme points of the class ${\mathcal{W}}_{\mathcal{M}}^{\eta }$ are the functions $h_n$ and $g_n$ given by (29), i.e.,

$$E{\mathcal{W}}_{\mathcal{M}}^{\eta }=\left\{h_n:n\in {\mathbb{N}}_0\right\}\cup \left\{g_n:n\in \mathbb{N}\right\}.$$

Proof. 

Let $t\in \left(0,1\right)$ and $f_1,f_2\in {\mathcal{W}}_{\mathcal{M}}^{\eta }$ be functions of the form (32). If $g_n=t{f}_1+\left(1-t\right)f_2,$ then by (29), we obtain $\left|b_{1,n}\right|=\left|b_{2,n}\right|=$${\displaystyle \frac{1}{\left|v_n\right|}}.$ Thus, $u_{1,l}=u_{2,l}=0$ for $l\in \mathbb{N}$ and $b_{1,l}=b_{2,l}=0$ for $l\in \mathbb{N}\setminus \left\{n\right\}.$ This means that $g_n=f_1=f_2,$ and consequently, $g_n\in E{\mathcal{W}}_{\mathcal{M}}^{\eta }.$ Analogously, we obtain that the functions $h_n$ of the form (29) are the extreme points of the class ${\mathcal{W}}_{\mathcal{M}}^{\eta }.$ Now, let $f\in E{\mathcal{W}}_{\mathcal{M}}^{\eta }$ be not of the form (29). Then there exists $k\in \mathbb{N}$ such that

$$0<\left|a_k\right|<{\displaystyle \frac{1}{\left|u_k\right|}}\mathrm{or}0<\left|b_k\right|<{\displaystyle \frac{1}{\left|v_k\right|}}.$$

If $0<\left|a_k\right|<{\displaystyle \frac{1}{u_k}}$, then for

$$t={\displaystyle \frac{\left|u_k\right|\left|a_k\right|}{1}},\phi ={\displaystyle \frac{1}{1-t}}\left(f-t{h}_k\right),$$

we have that $0<t<1,h_k\ne \phi$, and $f=t{h}_k+\left(1-t\right)\phi .$ Thus, $f\notin E{\mathcal{W}}_{\mathcal{M}}^{\eta }.$ Analogously, if $0<\left|b_k\right|<{\displaystyle \frac{1}{\left|v_n\right|}}$, then by setting $\varphi ={\displaystyle \frac{1}{1-t}}\left(f-t{g}_k\right)$ with $t=\left|v_k\right|\left|b_k\right|,$ we obtain

$$f=t{g}_k+\left(1-t\right)\varphi ,t\in \left(0,1\right),g_k\ne \varphi .$$

Thus, $f\notin E{\mathcal{W}}_{\mathcal{M}}^{\eta },$ which completes the proof. □

Analogously, we prove the next theorem.

Theorem 11. 

The class ${\mathcal{R}}_{\mathcal{M}}^{\eta }$ is a compact and convex subclass of $\mathcal{M}$. Moreover, the functions $h_n$ and $g_n$ of the form

$$\begin{array}{cc}\hfill h_0\left(\varsigma \right)& =\varsigma ,h_n\left(\varsigma \right)=\varsigma -{\displaystyle \frac{N-M}{\tau \left(1+N\right){\mathrm{\Gamma }}_n}}{e}^{-i\left(n+1\right)\eta }{\varsigma }^{-n},\hfill \\ \hfill g_n\left(\varsigma \right)& =\varsigma +\overline{{\displaystyle \frac{N-M}{\tau \left(1+N\right){\mathrm{\Gamma }}_n}}{e}^{-i\left(n-1\right)\eta }{\varsigma }^{-n}}(n\in \mathbb{N},\varsigma \in \mathbb{E}),\hfill \end{array}$$

(33)

are the extreme points of the class ${\mathcal{R}}_{\mathcal{M}}^{\eta }.$

5.  Applications

From the Krein–Milman theorem (see [27]), we have the following lemma.

Lemma 3. 

Let $\mathcal{B}$ be a non-empty compact convex subclass of the class $\mathcal{M}$, and let $\mathcal{J}:\mathcal{M}\to \mathbb{R}$ be a real-valued, continuous, and convex functional on $\mathcal{B}.$ Then,

$$\mathcal{B}=\overline{co}E\mathcal{B}$$

and

$$max\left\{\mathcal{J}\left(f\right):f\in \mathcal{B}\right\}=max\left\{\mathcal{J}\left(f\right):f\in E\mathcal{B}\right\}.$$

Lemma 4 

([28]). Let $h,g\in \mathcal{A}$. If $h\prec g,$ then

$${\displaystyle \frac{1}{2\pi }}\underset{0}{\overset{2\pi }{\int }}{\left|h\left(r{e}^{i\theta }\right)\right|}^{\lambda }d\theta \le {\displaystyle \frac{1}{2\pi }}\underset{0}{\overset{2\pi }{\int }}{\left|g\left(r{e}^{i\theta }\right)\right|}^{\lambda }d\theta .$$

If the class $\mathcal{B}=\left\{f_n\in \mathcal{M}:n\in \mathbb{N}\right\}$ is locally uniformly bounded, then

$$\overline{co}\mathcal{B}=\left\{\sum _{n=1}^{\infty }{\lambda }_n{f}_n:\sum _{n=1}^{\infty }{\lambda }_n=1,{\lambda }_n\ge 0\left(n\in \mathbb{N}\right)\right\}.$$

Thus, by Lemma 3 and Theorem 10, we obtain

Corollary 3. 

$${\mathcal{W}}_{\mathcal{M}}^{\eta }=\left\{\sum _{n=0}^{\infty }\left({\lambda }_n{h}_n+{\sigma }_n{g}_n\right):\sum _{n=0}^{\infty }\left({\lambda }_n+{\sigma }_n\right)=1\left({\sigma }_0=0,{\lambda }_n,{\sigma }_n\ge 0\right)\right\},$$

where $h_n$ and $g_n$ are given by (29).

Corollary 4. 

$${\mathcal{R}}_{\mathcal{M}}^{\eta }=\left\{\sum _{n=0}^{\infty }\left({\lambda }_n{h}_n+{\sigma }_n{g}_n\right):\sum _{n=0}^{\infty }\left({\lambda }_n+{\sigma }_n\right)=1\left({\sigma }_0=0,{\lambda }_n,{\sigma }_n\ge 0\right)\right\},$$

where $h_n$ and $g_n$ are given by (33).

It is easy to verify that the functionals

$$\begin{array}{ccc}\hfill \mathcal{J}\left(f\right)& =& \left|a_n\right|,\_\mathcal{J}\left(f\right)=\left|b_n\right|,\mathcal{J}\left(f\right)=\left|f\left(\varsigma \right)\right|\mathcal{J}\left(f\right)=\left|D_{\mathcal{M}}f\left(\varsigma \right)\right|\left(n\in \mathbb{N},\varsigma \in \mathbb{E},\right).\hfill \\ \hfill \mathcal{J}\left(f\right)& =& {\left({\displaystyle \frac{1}{2\pi }}\underset{0}{\overset{2\pi }{\int }}{\left|f\left(r{e}^{i\theta }\right)\right|}^{\lambda }d\theta \right)}^{1/\lambda }\left(\lambda \ge 1,r>1\right),\hfill \end{array}$$

are continuous and convex functionals on $\mathcal{M}$. Therefore, by Lemma 3 and Theorem 10, we have the following corollaries.

Corollary 5. 

Let $f\in {\mathcal{W}}_{\mathcal{M}}^{\eta }$ be a function of the form (2) with $r>1$. Then,

$$\begin{array}{cc}\hfill & \left|a_n\right|\le {\displaystyle \frac{1}{\left|u_n\right|}},\left|b_n\right|\le {\displaystyle \frac{1}{\left|v_n\right|}}(n\in \mathbb{N}),\hfill \\ \hfill & \left|f\left(\varsigma \right)\right|\le r+{\displaystyle \frac{1}{r}},\left|D_{\mathcal{M}}f\left(\varsigma \right)\right|\le r+{\displaystyle \frac{1}{r}}\left(\left|\varsigma \right|=r\right),\hfill \end{array}$$

where $u_n$ and $v_n$ are given by (17). The estimates are sharp, with extremal functions $h_n$ and $g_n$ of the form (29).

Corollary 6. 

If $f\in {\mathcal{W}}_{\mathcal{M}}^{\eta }$ and $\lambda \ge 1,$ $r>1,$ then

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2\pi }}\underset{0}{\overset{2\pi }{\int }}{\left|f\left(r{e}^{i\theta }\right)\right|}^{\lambda }d\theta \le {\displaystyle \frac{1}{2\pi }}\underset{0}{\overset{2\pi }{\int }}{\left|g_1\left(r{e}^{i\theta }\right)\right|}^{\lambda }d\theta ,\hfill \\ \hfill & {\displaystyle \frac{1}{2\pi }}\underset{0}{\overset{2\pi }{\int }}{\left|D_{\mathcal{H}}f\left(r{e}^{i\theta }\right)\right|}^{\lambda }d\theta \le {\displaystyle \frac{1}{2\pi }}\underset{0}{\overset{2\pi }{\int }}{\left|D_{\mathcal{H}}{g}_1\left(r{e}^{i\theta }\right)\right|}^{\lambda }d\theta ,\hfill \end{array}$$

where $u_n$ and $v_n$ are given by (17). The estimates are sharp, with extremal functions $h_n$ and $g_n$ of the form (29).

Analogously, we obtain the following results for the class ${\mathcal{R}}_{\mathcal{M}}^{\eta }.$

Corollary 7. 

Let $f\in {\mathcal{W}}_{\mathcal{M}}^{\eta }$ be a function of the form (2) with $\lambda \ge 1$ and $r>1$. Then,

$$\begin{array}{cc}\hfill & \left|a_n\right|\le {\displaystyle \frac{N-M}{\left(1+N\right)\left|{\mathrm{\Gamma }}_n\right|}},\left|b_n\right|\le {\displaystyle \frac{N-M}{\left(1+N\right)\left|{\mathrm{\Gamma }}_n\right|}}(n\in \mathbb{N}),\hfill \\ \hfill & \left|f\left(\varsigma \right)\right|\le r+{\displaystyle \frac{1}{r}},\left|D_{\mathcal{M}}f\left(\varsigma \right)\right|\le r+{\displaystyle \frac{1}{r}}\left(\left|\varsigma \right|=r\right),\hfill \\ \hfill & {\displaystyle \frac{1}{2\pi }}\underset{0}{\overset{2\pi }{\int }}{\left|f\left(r{e}^{i\theta }\right)\right|}^{\lambda }d\theta \le {\displaystyle \frac{1}{2\pi }}\underset{0}{\overset{2\pi }{\int }}{\left|g_1\left(r{e}^{i\theta }\right)\right|}^{\lambda }d\theta ,\hfill \\ \hfill & {\displaystyle \frac{1}{2\pi }}\underset{0}{\overset{2\pi }{\int }}{\left|D_{\mathcal{H}}f\left(r{e}^{i\theta }\right)\right|}^{\lambda }d\theta \le {\displaystyle \frac{1}{2\pi }}\underset{0}{\overset{2\pi }{\int }}{\left|D_{\mathcal{H}}{g}_1\left(r{e}^{i\theta }\right)\right|}^{\lambda }d\theta ,\hfill \end{array}$$

where $u_n$ and $v_n$ are given by (17). The estimates are sharp, with extremal functions $h_n,g_n$ of the form (33).

Remark 1. 

If we choose parameters in the considered classes of functions, we can obtain several additional results. Some of these results were obtained in earlier works (see, for example, [6,7,8,9,10,11]).