Harmonic Functions with Montel’s Normalization

Abstract

In the Geometric Theory of Analytic Functions, classes of functions with several normalizations are considered. We consider the symmetric idea for harmonic functions. Classes of harmonic functions f with normalization $f\left(0\right)=f_{\overline{z}}^{\prime }\left(0\right)=0,$ $f_z^{\prime }\left(0\right)=1$ are usually considered in the geometric theory of harmonic functions. The normalization is called the classical normalization. We can obtain some interesting results by using Montel’s normalization $f\left(0\right)=f_{\overline{z}}^{\prime }\left(0\right)=0,$ $f_z^{\prime }\left(\rho \right)-f_{\overline{z}}^{\prime }\left(\rho \right)=1$, where $\rho \in [0,1).$ In the paper, we consider the class of harmonic functions with Montel’s normalization associated with the generalized hypergeometric function.

1. Introduction

In the Geometric Theory of Analytic Functions, classes of functions with several normalizations are considered. We consider the symmetric idea for 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. Harmonic functions have been studied by differential geometers such as Choquet [1], Kneser [2], Lewy [3], and Rado [4]. Recent interest in harmonic complex functions has been triggered by geometric function theorists Clunie and Sheil-Small [5] (see also [6,7]). We say that a function $f:D\to \mathbb{C}$ is harmonic in a domain $D\subset \mathbb{C}$ if it has continuous second-order formal derivatives and $f_{z\overline{z}}^{″}=0$in$D.$ Let $\mathcal{H}$ denote the class of functions $f,$ which are harmonic in $\mathbb{U}:=\{$$z\in \mathbb{C}$$:\left|z\right|<1\}$ and normalized by $f\left(0\right)=f_{\overline{z}}^{\prime }\left(0\right)=0$, $f_z^{\prime }\left(0\right)>0$. Then, each function $f\in \mathcal{H}$ can be presented in the form

$$f\left(z\right)=a_{1}z+\sum _{n=2}^{\infty }\left(a_n{z}^n+\overline{b_n{z}^n}\right)\left(a_1>0,z\in \mathbb{U}\right).$$

(1)

If $f_z^{\prime }\left(0\right)=a_1=1,$ then we obtain the classical normalization. We can obtain interesting results by using the normalization due to Montel [8] (see also [9,10]). Let $\rho \in [0,1).$ By ${\mathcal{H}}_{\rho }$, we denote the class of function $f\in \mathcal{H}$ with the normalization

$$f\left(0\right)=f_{\overline{z}}^{\prime }\left(0\right)=0, f_z^{\prime }\left(\rho \right)-f_{\overline{z}}^{\prime }\left(\rho \right)=1.$$

(2)

It is clear that for $\rho =0$, we obtain the classical normalization. If $\rho \ne 0,$ then (2) can be written in the form

$$f\left(0\right)=0,D_{\mathcal{H}}f\left(\rho \right)=\rho ,$$

where

$$D_{\mathcal{H}}f\left(z\right):=z{f}_z^{\prime }\left(z\right)-\overline{z}{f}_{\overline{z}}^{\prime }\left(z\right)\left(z\in \mathbb{U}\right).$$

(3)

We introduce some class of harmonic functions with Montel’s normalization associated with the generalized hypergeometric function.

For positive real parameters ${\alpha }_1,\dots ,{\alpha }_s$, ${\beta }_1,\dots ,{\beta }_s$, we define the function

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

(4)

where

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

(5)

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

$${\left(\lambda \right)}_n:={\displaystyle \frac{\Gamma (\lambda +n)}{\Gamma \left(\lambda \right)}}=\lambda (\lambda +1)···(\lambda +n-1).$$

Corresponding to the generalized hypergeometric function, Dziok and Srivastava [11] introduced the linear operator $H_s$ defined on the space $\mathcal{A}$ of the analytic function by the convolution $H_{s}f:=F_s\ast f.$The linear operator $H_s$ includes such other linear operators, which were considered in earlier works, as (for example) the Hohlov operator, Carlson–Shaffer operator, Ruscheweyh derivatives, generalized Bernardi–Libera–Livingston integral operator, and Srivastava and Owa fractional derivative operator (for details see [11]).

Al-Kharsani and Al-Khal [12] (see also [13,14]) considered a harmonic generalization of the operator $H_s$. Let $\tau \in \left\{-1,1\right\}$, ${\mathbb{N}}_n:=\left\{n,n+1,\dots \right\}$ and

$${\Phi }_{\tau }^s\left(z\right):=F_s\left(z\right)+\overline{\tau F_s\left(z\right)}\left(z\in \mathbb{U}\right).$$

(6)

Due to Dziok and Srivastava [11], we define the operator $H_{\tau }^s=H_{\tau }^s({\alpha }_1,\dots ,{\alpha }_s;{\beta }_1,\dots ,{\beta }_s):\mathcal{H}\to \mathcal{H}$ by the convolution formula

$$H_{\tau }^{s}f:={\Phi }_{\tau }^s\ast f.$$

In particular, we have

$$H_1^1(1;1)f=f,H_{-1}^1(2;1)f=D_{\mathcal{H}}f,$$

and

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

The operator

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

is related to the Ruscheweyh derivatives (see [15]), and the operator

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

is related to the Sălăgean operator [16] (see also [17]).

To define the main class of harmonic functions, we need the definition of the weak subordination due to Mauir [18].

Definition 1.

A complex-valued function f in $\mathbb{U}$ is said to be weakly subordinate to a complex-valued function $\Phi$ in $\mathbb{U}$, and we write $f\left(z\right)⪯F\left(z\right)$ (or simply $f⪯F$), if there exists a complex-valued function $\omega$which maps $\mathbb{U}$ into oneself such that $f\left(z\right)=F\left(\omega \left(z\right)\right)\left(z\in \mathbb{U}\right).$

It is clear that

$$f⪯F\iff f\left(\mathbb{U}\right)\subset F\left(\mathbb{U}\right).$$

Definition 2.

By ${\mathcal{W}}_{\rho }^{s,\tau }={\mathcal{W}}_{\rho }^{s,\tau }\left({\alpha }_1,\dots ,{\alpha }_s;{\beta }_1,\dots ,{\beta }_s;A,B\right),$$-1\le -B\le A<B<1$ we denote the class of functions $f\in {\mathcal{H}}_{\rho }$ of the form

$$f\left(z\right)=a_{1}z-\sum _{n=2}^{\infty }\left(\left|a_n\right|z^n-\tau \left|b_n\right|\overline{z^n}\right)\left(z\in \mathbb{U}\right),$$

(7)

such that

$${\displaystyle \frac{H_{\tau }^{s+1}f\left(z\right)}{H_{\tau }^{s}f\left(z\right)}}⪯{\displaystyle \frac{1+Az}{1+Bz}}\left({\alpha }_{s+1}=2,{\beta }_{s+1}=1\right)$$

(8)

or equivalently

$${\displaystyle \frac{D_{\mathcal{M}}\left(H_{\tau }^{s}f\right)\left(z\right)}{H_{\tau }^{s}f\left(z\right)}}⪯{\displaystyle \frac{1+Az}{1+Bz}}.$$

We observe that ${\mathcal{W}}_{\rho }^{\ast }:={\mathcal{W}}_0^{1,1}(1,1;-1,1)$ and ${\mathcal{W}}_{\rho }^c:={\mathcal{W}}_0^{1,-1}(2,1;-1,1)$ are the classes of functions $f\in \mathcal{H}$ which are starlike in $\mathbb{U}\left(r\right)$ or convex in $\mathbb{U}\left(r\right),$ respectively, for all $r\in \left(0,1\right.⟩.$

In the paper, we obtain some coefficient formulas, distortion theorems, partial sums, convexity, and extreme points for the defined class of functions. The radii of convexity and starlikeness are also considered.

2. Coefficients Conditions

We start with the result, which will be basic in our investigations.

Theorem 1.

Let $f\in {\mathcal{H}}_{\rho }$ be of the form (7). Then, $f\in {\mathcal{W}}_{\rho }^{s,\tau }$ if and only if

$$\sum _{n=2}^{\infty }\left(\left(u_n-n{\rho }^{n-1}\right)\left|a_n\right|+\left(v_n-\tau n{\rho }^{n-1}\right)\left|b_n\right|\right)\le 1,$$

(9)

where

$$\begin{array}{c}\hfill l{u}_n:={\displaystyle \frac{{\left({\alpha }_1\right)}_{n-1}\cdots {\left({\alpha }_s\right)}_{n-1}}{{\left({\beta }_1\right)}_{n-1}\cdots {\left({\beta }_s\right)}_{n-1}}}\left\{n\left(1+B\right)-\left(1+A\right)\right\},\\ \hfill v_n:={\displaystyle \frac{{\left({\alpha }_1\right)}_{n-1}\cdots {\left({\alpha }_s\right)}_{n-1}}{{\left({\beta }_1\right)}_{n-1}\cdots {\left({\beta }_s\right)}_{n-1}}}\left\{n\left(1+B\right)+\left(1+A\right)\right\}.\end{array}$$

(10)

Proof. 

$\left(\Leftarrow \right)$ Let $f\in {\mathcal{H}}_{\rho }$ of the form (7) satisfies the condition (9). Then, $f\in {\mathcal{W}}_{\rho }^{s,\tau }$ if and only if there exists a function $\omega :\mathbb{U}\to \mathbb{U}$ such that

$${\displaystyle \frac{D_{\mathcal{H}}{H}_{\tau }^{s}f\left(z\right)}{H_{\tau }^{s}f\left(z\right)}}={\displaystyle \frac{1+A\omega \left(z\right)}{1+B\omega \left(z\right)}}\left(z\in \mathbb{U}\right),$$

or equivalently

$$\left|{\displaystyle \frac{D_{\mathcal{H}}{H}_{\tau }^{s}f\left(z\right)-H_{\tau }^{s}f\left(z\right)}{B{D}_{\mathcal{H}}{H}_{\tau }^{s}f\left(z\right)-A{H}_{\tau }^{s}f\left(z\right)}}\right|<1\left(z\in \mathbb{U}\right).$$

(11)

By (2) we have

$$a_1=1+\sum _{n=2}^{\infty }n\left(\left|a_n\right|+\tau \left|b_n\right|\right){\rho }^{n-1}.$$

(12)

Thus, for $\left|z\right|=r<1$, we obtain

$$\left|D_{\mathcal{H}}{H}_{\tau }^{s}f\left(z\right)-H_{\tau }^{s}f\left(z\right)\right|-\left|B{D}_{\mathcal{H}}{H}_{\tau }^{s}f\left(z\right)-A{H}_{\tau }^{s}f\left(z\right)\right|=$$

$$\begin{array}{cc}\hfill & {\displaystyle =\left|\sum _{n=2}^{\infty }{\Gamma }_n\left\{\left(n-1\right)\left|a_n\right|z^n-\left(n+1\right)\left|b_n\right|{\overline{z}}^n\right\}\right|}\hfill \\ \hfill & -\left|\left(B-A\right)a_{1}z+\sum _{n=2}^{\infty }{\Gamma }_n\left\{\left(Bn-A\right)\left|a_n\right|z^n-\left(Bn+A\right)\left|b_n\right|{\overline{z}}^n\right\}\right|\hfill \\ \hfill & \le \sum _{n=2}^{\infty }{\Gamma }_n\left\{\left(n-1\right)\left|a_n\right|+\left(n+1\right)\left|b_n\right|\right\}{r}^n-\left(B-A\right)a_{1}r\hfill \\ \hfill & +\sum _{n=2}^{\infty }{\Gamma }_n\left\{\left(Bn-A\right)\left|a_n\right|+\left(Bn+A\right)\left|b_n\right|\right\}{r}^n\hfill \\ \hfill & =r\sum _{n=2}^{\infty }{\Gamma }_n\left\{\left[n\left(1+B\right)-\left(1+A\right)\right]\left|a_n\right|+\left[n\left(1+B\right)+\left(1+A\right)\right]\left|b_n\right|\right\}{r}^{n-1}\hfill \\ \hfill & -\left(B-A\right)r-\left(B-A\right)r\sum _{n=2}^{\infty }n\left(\left|a_n\right|+\tau \left|b_n\right|\right){\rho }^{n-1}\hfill \\ \hfill & =\left(B-A\right)r\left\{\sum _{n=2}^{\infty }\left\{\left(u_n-n{\rho }^{n-1}\right)\left|a_n\right|+\left(v_n-\tau n{\rho }^{n-1}\right)\left|b_n\right|\right\}{r}^{n-1}-1\right\}<0.\hfill \end{array}$$

Thus, we obtain (11).

$\left(\Rightarrow \right)$Let now $f\in {\mathcal{W}}_{\rho }^{s,\tau }.$ Then, it satisfies (11) and (12). Thus, by (7), we can write

$$\left|{\displaystyle \frac{{\displaystyle \sum _{n=2}^{\infty }}{\Gamma }_n\left\{\left(n-1\right)\left|a_n\right|z^n+\left(n+1\right)\left|b_n\right|{\overline{z}}^n\right\}}{\left(B-A\right)a_{1}z-{\displaystyle \sum _{n=2}^{\infty }}{\Gamma }_n\left\{\left(Bn-A\right)\left|a_n\right|z^n+\left(Bn+A\right)\left|b_n\right|{\overline{z}}^n\right\}}}\right|<1(z\in \mathbb{U}).$$

Therefore, putting $z=r(0\le r<1)$, we obtain

$${\displaystyle \frac{\sum _{n=2}^{\infty }{\Gamma }_n\left\{\left(n-1\right)\left|a_n\right|+\left(n+1\right)\left|b_n\right|\right\}{r}^{n-1}}{\left(B-A\right)a_1-\sum _{n=2}^{\infty }{\Gamma }_n\left\{\left(Bn-A\right)\left|a_n\right|+\left(Bn+A\right)\left|b_n\right|\right\}{r}^{n-1}}}<1.$$

(13)

It is clear that the denominator of the left-hand side cannot vanish for $r\in \left.⟨0,1\right).$ Moreover, it is positive for $r=0,$ and in consequence for $r\in \left.⟨0,1\right).$ Thus, by (13), we have

$$\sum _{n=2}^{\infty }\left(u_n\left|a_n\right|+v_n\left|b_n\right|\right)r^{n-1}<a_1.$$

Moreover, using (12), we obtain

$$\sum _{n=2}^{\infty }\left(\left(u_n-n{\rho }^{n-1}\right)\left|a_n\right|+\left(v_n-\tau n{\rho }^{n-1}\right)\left|b_n\right|\right)r^{n-1}<1(0\le r<1),$$

which yields the assertion (9). □

By applying Theorem 1, we obtain the following corollaries.

Corollary 1.

Let a function f of the form (7) belongs to the class ${\mathcal{W}}_{\rho }^{s,\tau }$ and let $\left(u_n\right),\left(v_n\right)$ be defined by (10). If

$$u_n-n{\rho }^{n-1}>0,v_n-\tau n{\rho }^{n-1}>0\left(n\in {\mathbb{N}}_2\right),$$

(14)

then

$$\left|a_n\right|\le {\displaystyle \frac{1}{u_n-n{\rho }^{n-1}}},\left|b_n\right|\le {\displaystyle \frac{1}{v_n-\tau n{\rho }^{n-1}}}\left(n\in {\mathbb{N}}_2\right).$$

(15)

The result is sharp, and the functions $h_n$ and $g_n$ of the form

$$h_n\left(z\right)={\displaystyle \frac{u_{n}z-z^n}{u_n-n{\rho }^{n-1}}},g_n\left(z\right)={\displaystyle \frac{v_{n}z+\tau {\overline{z}}^n}{v_n-\tau n{\rho }^{n-1}}}(z\in \mathbb{U};n\in {\mathbb{N}}_2)$$

(16)

are the extremal functions.

Corollary 2.

Let a function f of the form (7) belong to the class ${\mathcal{W}}_{\rho }^{s,\tau }$ and let $\left(u_n\right),\left(v_n\right)$ be defined by (10). Then, all of the coefficients $a_n$ in the class ${\mathcal{W}}_{\rho }^{s,\tau }$ for which $u_n-n{\rho }^{n-1}=0,$ and all of the coefficients $b_n$ in the class ${\mathcal{W}}_{\rho }^{s,\tau }$ for which $v_n-\tau n{\rho }^{n-1}=0$ are unbounded. Moreover, if there exists $n\in {\mathbb{N}}_2=\left\{2,3,\dots \right\}$such that

$$u_n-n{\rho }^{n-1}<0\mathit{or}{v}_n-\tau n{\rho }^{n-1}<0$$

then all of the coefficients in the class ${\mathcal{W}}_{\rho }^{s,\tau }$ are unbounded.

Proof. 

If

$$u_{n_0}-n_0{\rho }^{n_0-1}\le 0$$

for some $n_0\in {\mathbb{N}}_2,$ then the function

$$f_{n_0}\left(z\right)=\left(1+a{n}_0{\rho }^{n_0-1}\right)z-a{z}^{n_0}(z\in \mathbb{U})$$

belongs to the class ${\mathcal{W}}_{\rho }^{s,\tau }$ for all positive real numbers $a.$ Thus, the coefficient $a_{n_0}$ in the class ${\mathcal{W}}_{\rho }^{s,\tau }$ is unbounded. If there exists $n_0\in {\mathbb{N}}_2$such that

$$u_{n_0}-n_0{\rho }^{n_0-1}<0,$$

(17)

then for any $n\in {\mathbb{N}}_2$ such that

$$u_n-n{\rho }^{n-1}>0,$$

(18)

the function

$$f_n\left(z\right)=\left(1+a{n}_0{\rho }^{n_0-1}+bn{z}^{n-1}\right)z-a{z}^{n_0}-b{z}^n,$$

where

$$b={\displaystyle \frac{1+\left(n_0{\rho }^{n_0-1}-u_n\right)a}{u_n-n{\rho }^{n-1}}},$$

belong to the class ${\mathcal{W}}_{\rho }^{s,\tau }.$ Since b can be any positive real number, the coefficient $a_n$ in the class ${\mathcal{W}}_{\rho }^{s,\tau }$ is unbounded. Analogously, if

$$v_{n_0}-\tau n_0{\rho }^{n_0-1}\le 0$$

for some $n_0\in {\mathbb{N}}_2,$ then the function

$$f_{n_0}\left(z\right)=\left(1+\tau a{n}_0{\rho }^{n_0-1}\right)z-\tau a\overline{z^{n_0}}(z\in \mathbb{U})$$

belongs to the class ${\mathcal{W}}_{\rho }^{s,\tau }$ for all positive real numbers $a.$ Thus, the coefficient $b_{n_0}$ in the class ${\mathcal{W}}_{\rho }^{s,\tau }$ is unbounded. If there exists an integer $n_0\in {\mathbb{N}}_2$ such that

$$v_{n_0}-\tau n_0{\rho }^{n_0-1}<0$$

then for $n\in {\mathbb{N}}_2$ such that

$$v_n-\tau n{\rho }^{n-1}>0,$$

the function

$$f_n\left(z\right)=\left(1+\tau a{n}_0{\rho }^{n_0-1}+\tau bn{\rho }^{n-1}\right)z+\tau a{\overline{z}}^{n_0}+\tau b{\overline{z}}^n,$$

where

$$b={\displaystyle \frac{1-\left(v_{n_0}-\tau n_0{\rho }^{n_0-1}\right)a}{v_n-\tau n{\rho }^{n-1}}},$$

belong to the class ${\mathcal{W}}_{\rho }^{s,\tau }.$ Since b can be any positive real number, the coefficient $b_n$ in the class ${\mathcal{W}}_{\rho }^{s,\tau }$ is unbounded, and the proof is competed. □

3. Distortion Theorems

From Theorem 1, we have the following lemma.

Lemma 1.

Let a function f of the form (7) belong to the class ${\mathcal{W}}_{\rho }^{s,\tau }.$ If the sequences $\left(u_n\right)$, $\left(v_n\right)$ defined by (10) satisfy the inequalities

$$0<{\displaystyle \frac{u_2-2\rho }{2}}\le {\displaystyle \frac{u_n-n{\rho }^{n-1}}{n}},0<{\displaystyle \frac{v_2-2\tau \rho }{2}}\le {\displaystyle \frac{v_n-\tau n{\rho }^{n-1}}{2}}\left(n\in {\mathbb{N}}_2\right),$$

(19)

then

$$\sum _{n=2}^{\infty }n\left|a_n\right|\le {\displaystyle \frac{2}{u_2-2\rho }},\sum _{n=2}^{\infty }n\left|b_n\right|\le {\displaystyle \frac{2}{v_2-2\tau \rho }}.$$

(20)

Remark 1.

Lemma 1 can be rewritten in terms of σ-neighborhood $N_{\sigma }$ defined by

$$N_{\sigma }=\left\{f\left(z\right)=a_{1}z+\sum _{n=2}^{\infty }\left(a_n{z}^n+\overline{b_n{z}^n}\right)\in {\mathcal{H}}_{\rho }:\sum _{n=2}^{\infty }n\left(\left|a_n\right|+\left|b_n\right|\right)\le \sigma \right\}$$

in the following form:

$${\mathcal{W}}_{\rho }^{s,\tau }\subset N_{\sigma },$$

where

$$\delta ={\displaystyle \frac{2}{u_2-2\rho }}+{\displaystyle \frac{2}{v_2-2\tau \rho }}.$$

Theorem 2.

Let a function f belong to the class ${\mathcal{W}}_{\rho }^{s,\tau }$ and let $\left|z\right|=r<1.$ If the sequences $\left(u_n\right)$, $\left(v_n\right)$ defined by (10) satisfy (19), then

$$f_1\left(r\right)\le \left|D_{\mathcal{H}}f\left(z\right)\right|\le f_2\left(r\right),$$

(21)

where

$$\begin{array}{cc}\hfill f_1\left(r\right)& :=\left\{\begin{array}{cc}r& \left(\tau =1,r\le \rho \right)\\ {\displaystyle \frac{v_{2}r-r^2}{v_2+2\rho }}& \left(\tau =-1,r\le \rho \right)\\ {\displaystyle \frac{\left(u_2{v}_2-4\tau {\rho }^2\right)r-2\left(u_2+v_2+2\left(1+\tau \right)\rho \right)r^2}{\left(u_2-2\rho \right)\left(v_2-2\tau \rho \right)}}& \left(r>\rho \right)\end{array}\right.,\hfill \\ \hfill f_2\left(r\right)& :=\left\{\begin{array}{cc}{\displaystyle \frac{u_{2}r+r^2}{u_2-2\rho }}& \left(r+\tau \rho \le 0\right)\\ {\displaystyle \frac{\left(u_2{v}_2-4\tau {\rho }^2\right)r+2\left(u_2+v_2-2\left(1+\tau \right)\rho \right)r^2}{\left(u_2-2\rho \right)\left(v_2-2\tau \rho \right)}}& \left(r+\tau \rho >0\right)\end{array}\right..\hfill \end{array}$$

(22)

The result is sharp, with the extremal functions $h_n, g_n$ of the form (16) and the functions $h_1,$, $f_{\tau }$ of the form

$$h_1\left(z\right)=z,f_{\tau }\left(z\right)={\displaystyle \frac{u_2{v}_2-4\tau {\rho }^2}{\left(u_2-2\rho \right)\left(v_2-2\tau \rho \right)}}z-{\displaystyle \frac{z^2}{u_2-2\rho }}+{\displaystyle \frac{\tau {\overline{z}}^2}{v_2-2\tau \rho }}(z\in \mathbb{U}).$$

(23)

Proof. 

First we observe that the sequence $\left\{\left(r^{n-1}+{\rho }^{n-1}\right)\right\}$ is decreasing and positive. Also, if $r>\rho ,$ then the sequence $\left\{\left(r^{n-1}-{\rho }^{n-1}\right)\right\}$ is decreasing and positive. Thus, we obtain

$$\begin{array}{c}\hfill 0<r+\rho \le r^{n-1}+{\rho }^{n-1}\left(r,\rho \in \left(0,1\right),n\in {\mathbb{N}}_2\right),\\ \hfill 0<r-\rho \le r^{n-1}-{\rho }^{n-1}\left(0<\rho <r<1,n\in {\mathbb{N}}_2\right).\end{array}$$

(24)

Moreover, by Lemma 1, we have (19). Let $f\in {\mathcal{W}}_{\rho }^{s,\tau }$ be of the form (7). Then,

$$\begin{array}{cc}|D_{\mathcal{H}}f\left(z\right)|\hfill & =\left|a_{1}z-{\displaystyle \sum _{n=2}^{\infty }}n\left(a_n{z}^n+\overline{\tau b_n{z}^n}\right)\right|\le r\left(a_1+{\displaystyle \sum _{n=2}^{\infty }}n\left(\left|a_n\right|+\left|b_n\right|\right)r^{n-1}\right)\hfill \\ \hfill & \le r\left(1+{\displaystyle \sum _{n=2}^{\infty }}n\left(\left|a_n\right|+\tau \left|b_n\right|\right){\rho }^{n-1}+{\displaystyle \sum _{n=2}^{\infty }}n\left(\left|a_n\right|+\left|b_n\right|\right)r^{n-1}\right)\hfill \\ \hfill & \le r\left(1+{\displaystyle \sum _{n=2}^{\infty }}(r^{n-1}+{\rho }^{n-1})n\left|a_n\right|+{\displaystyle \sum _{n=2}^{\infty }}\left(r^{n-1}+\tau {\rho }^{n-1}\right)n\left|b_n\right|\right).\hfill \end{array}$$

If $r+\tau \rho >0,$ then by (24), we have

$$\begin{array}{ccc}\hfill \left|D_{\mathcal{H}}f\left(z\right)\right|& \le & r\left(1+(r+\rho )\sum _{n=2}^{\infty }n\left|a_n\right|+\left(r+\tau \rho \right)\sum _{n=2}^{\infty }n\left|b_n\right|\right)\hfill \\ & \le & r\left(1+{\displaystyle \frac{2(r+\rho )}{u_2-2\rho }}+{\displaystyle \frac{2\left(r+\tau \rho \right)}{v_2-2\tau \rho }}\right)\hfill \\ & =& {\displaystyle \frac{\left(u_2{v}_2-4\tau {\rho }^2\right)r+2\left(u_2+v_2-2\left(1+\tau \right)\rho \right)r^2}{\left(u_2-2\rho \right)\left(v_2-2\tau \rho \right)}}.\hfill \end{array}$$

Also, if $r+\tau \rho \le 0,$ then

$$\begin{array}{ccc}\hfill \left|D_{\mathcal{H}}f\left(z\right)\right|& \le & r\left(1+\sum _{n=2}^{\infty }n(r^{n-1}+{\rho }^{n-1})\left|a_n\right|\right)\le r\left(1+(r+\rho )\sum _{n=2}^{\infty }n\left|a_n\right|\right)\hfill \\ & \le & 1+(r+\rho ){\displaystyle \frac{2}{u_2-2\rho }}={\displaystyle \frac{u_{2}r+2r^2}{u_2-2\rho }}.\hfill \end{array}$$

Analogously, we obtain

$$\begin{array}{cc}|D_{\mathcal{H}}f\left(z\right)|\hfill & =\left|a_{1}z-{\displaystyle \sum _{n=2}^{\infty }}n\left(a_n{z}^n+\overline{\tau b_n{z}^n}\right)\right|\ge r\left(a_1-{\displaystyle \sum _{n=2}^{\infty }}n\left(\left|a_n\right|+\left|b_n\right|\right)r^{n-1}\right)\hfill \\ \hfill & =r\left(1+{\displaystyle \sum _{n=2}^{\infty }}\left(\left|a_n\right|+\tau \left|b_n\right|\right)n{\rho }^{n-1}-{\displaystyle \sum _{n=2}^{\infty }}\left(\left|a_n\right|+\left|b_n\right|\right)n{r}^{n-1}\right)\hfill \\ \hfill & =r\left(1-{\displaystyle \sum _{n=2}^{\infty }}(r^{n-1}-{\rho }^{n-1})n\left|a_n\right|-{\displaystyle \sum _{n=2}^{\infty }}\left(r^{n-1}-\tau {\rho }^{n-1}\right)n\left|b_n\right|\right).\hfill \end{array}$$

If $\tau =1$ and $r\le \rho ,$ then we obtain $\left|D_{\mathcal{H}}f\left(z\right)\right|\geqq r.$ If $\tau =-1$ and $r\le \rho ,$ then by (24), we have

$$\begin{array}{ccc}\hfill \left|D_{\mathcal{H}}f\left(z\right)\right|& \ge & r\left(1-\sum _{n=2}^{\infty }\left(r^{n-1}+{\rho }^{n-1}\right)n\left|b_n\right|\right)\ge r\left(1-(r+\rho )\sum _{n=2}^{\infty }n\left|b_n\right|\right)\hfill \\ & \ge & r\left(1-{\displaystyle \frac{2\left(r+\rho \right)}{v_2+2\rho }}\right)={\displaystyle \frac{v_{2}r-2r^2}{v_2+2\rho }}.\hfill \end{array}$$

If $r>\rho ,$ then by (24), we obtain

$$\begin{array}{ccc}\hfill \left|D_{\mathcal{H}}f\left(z\right)\right|& \ge & r\left(1-(r-\rho )\sum _{n=2}^{\infty }n\left|a_n\right|-(r-\tau \rho )\sum _{n=2}^{\infty }n\left|b_n\right|\right)\hfill \\ & \ge & r\left(1-2{\displaystyle \frac{r-\rho }{u_2-2\rho }}-2{\displaystyle \frac{r-\tau \rho }{v_2-2\tau \rho }}\right)\hfill \\ & =& {\displaystyle \frac{\left(u_2{v}_2-4\tau {\rho }^2\right)r-2\left(u_2+v_2+2\left(1+\tau \right)\rho \right)r^2}{\left(u_2-2\rho \right)\left(v_2-2\tau \rho \right)}},\hfill \end{array}$$

and this complete the assertion (21). □

By Corollary 2, we have the following complementary result.

Corollary 3.

Let $\left(u_n\right),\left(v_n\right)$ be defined by (10). If there exists $n\in {\mathbb{N}}_2$ such that

$$u_n-n{\rho }^{n-1}\le 0\mathit{or}{v}_n-\tau n{\rho }^{n-1}\le 0,$$

then for each $r\in \left(0,1\right)$, the set

$$A_r=\left\{\left|D_{\mathcal{H}}f\left(z\right)\right|:f\in {\mathcal{W}}_{\rho }^{s,\tau },\left|z\right|=r\right\}$$

is unbounded.

Putting $\tau =1$ in Theorem 2, we have the following corollary.

Corollary 4.

Let a function f belong to the class ${\mathcal{W}}_{\rho }^{s,1}$ and let $\left|z\right|=r<1.$ If the sequences $\left(u_n\right),$$\left(v_n\right)$ defined by (10) satisfy (19), then

$$f_1\left(r\right)\le \left|D_{\mathcal{H}}f\left(z\right)\right|\le {\displaystyle \frac{\left(u_2{v}_2-4{\rho }^2\right)r+2\left(u_2+v_2-4\rho \right)r^2}{\left(u_2-2\rho \right)\left(v_2-2\rho \right)}},$$

(25)

where

$$f_1\left(r\right):=\left\{\begin{array}{cc}r& \left(r\le \rho \right)\\ {\displaystyle \frac{\left(u_2{v}_2-4\tau {\rho }^2\right)r-2\left(u_2+v_2+2\left(1+\tau \right)\rho \right)r^2}{\left(u_2-2\rho \right)\left(v_2-2\tau \rho \right)}}& \left(r>\rho \right)\end{array}\right..$$

(26)

The result is sharp, with the extremal functions $h_1,$, $f_1$ of the form (23).

Moreover, if ${\alpha }_n\ge {\beta }_n+1\left(n=1,2,\dots ,s\right),$ then the sequences $\left(u_n\right),$ $\left(v_n\right)$ defined by (10) satisfy (19). Thus, by Theorem 2, we have the following corollary.

Corollary 5.

Let a function f belong to the class ${\mathcal{W}}_{\rho }^{s,\tau },$ and let $\left|\zeta \right|=r<1.$ If ${\alpha }_n\ge {\beta }_n+1\left(n=1,2,\dots ,s\right),$ then the estimations (21) hold true. The result is sharp, with the extremal functions of the form (16) and (23).

4. Partial Sums

Let f be a function of the form (7). Due to Silvia [19], we investigate the partial sums $f_m$ of the function f defined by

$$f_1\left(z\right)=a_{1}z;f_m\left(z\right)=a_{1}z-\sum _{n=2}^m\left(\left|a_n\right|z^n-\tau \left|b_n\right|\overline{z^n}\right),(m\in {\mathbb{N}}_2)$$

(27)

In this section, we consider partial sums of functions in the class ${\mathcal{W}}_{\rho }^{s,\tau }$ and obtain sharp lower bounds for the ratios of the real part of f to $f_m.$

Theorem 3.

Let the sequences

$$d_n:=\left(u_n-n{\rho }^{n-1}\right),e_n:=\left(v_n-\tau n{\rho }^{n-1}\right)$$

be increasing and not less than 1. If $f\in {\mathcal{W}}_{\rho }^{s,\tau },$ then

$$Re\left\{{\displaystyle \frac{f\left(z\right)}{f_m\left(z\right)}}\right\}\ge 1-{\displaystyle \frac{1}{d_{m+1}}}\left(z\in \mathbb{U},m\in {\mathbb{N}}_2\right)$$

(28)

and

$$Re\left\{{\displaystyle \frac{f_m\left(z\right)}{f\left(z\right)}}\right\}\ge {\displaystyle \frac{d_{m+1}}{B-A+d_{m+1}}}\left(z\in \mathbb{U},m\in {\mathbb{N}}_2\right).$$

(29)

Proof. 

Since

$$e_{n+1}\ge d_{n+1}>d_n>1\left(n\in {\mathbb{N}}_2\right),$$

by Theorem 1, we have

$$\sum _{n=2}^m\left(\left|a_n\right|+\left|b_n\right|\right)+d_{n+1}\sum _{n=m+1}^{\infty }\left(\left|a_n\right|+\left|b_n\right|\right)\le \sum _{n=2}^{\infty }\left(d_n\left|a_n\right|+e_n\left|b_n\right|\right)\le 1.$$

(30)

Let

$$g\left(z\right):=d_{m+1}\left\{{\displaystyle \frac{f\left(z\right)}{f_m\left(z\right)}}-\left(1-\frac{1}{d_{m+1}}\right)\right\}=1+{\displaystyle \frac{d_{m+1}{\displaystyle \sum _{n=m+1}^{\infty }}{a}_n{z}^{n-p}}{a_1+{\displaystyle \sum _{n=2}^m}{a}_n{z}^{n-p}}}\left(z\in \mathbb{U}\right).$$

(31)

Applying (30), we find that

$$\left|{\displaystyle \frac{g\left(z\right)-1}{g\left(z\right)+1}}\right|\le {\displaystyle \frac{d_{m+1}{\displaystyle \sum _{n.=m+1}^{\infty }}\left|a_n\right|}{2a_1-2\sum _{n=2}^n|a_n|-d_{m+1}{\displaystyle \sum _{n=m+1}^{\infty }}\left|a_n\right|}}\le 1\left(z\in \mathbb{U}\right).$$

Thus, we have $\Re \left[g\left(z\right)\right]\ge 0\left(z\in \mathcal{D}\right)$, and by (31), we have the assertion (28) of Theorem 3. Similarly, if we take

$$h\left(z\right)=\left(1+{\displaystyle \frac{d_{m+1}}{B-A}}\right)\left\{{\displaystyle \frac{f_m\left(z\right)}{f\left(z\right)}}-{\displaystyle \frac{d_{m+1}}{1+d_{m+1}}}\right\}\left(z\in \mathbb{U}\right)$$

and making use of (30), we can deduce that

$$\left|{\displaystyle \frac{h\left(z\right)-1}{h\left(z\right)+1}}\right|\le {\displaystyle \frac{\left(1+d_{m+1}\right){\displaystyle \sum _{n=m+1}^{\infty }}\left|a_n\right|}{2a_p-2{\displaystyle \sum _{n=2}^m}|a_n|-\left(d_{m+1}-1\right){\displaystyle \sum _{n=m+1}^{\infty }}\left|a_n\right|}}\le 1\left(z\in \mathbb{U}\right),$$

which leads us immediately to the assertion (29) of Theorem 3. □

5. Convexity and Extreme Points

Theorem 4.

Let $h_1\left(z\right)=z$ and let $h_n, g_n$ be defined by (16) and let $\left(u_n\right),\left(v_n\right)$ defined by (10) satisfy (14). Then, $f\in$ ${\mathcal{W}}_{\rho }^{s,\tau }$ if and only if

$$f={\gamma }_1{h}_1+\sum _{n=2}^{\infty }\left({\gamma }_n{h}_n+{\delta }_n{g}_n\right),$$

(32)

where

$${\gamma }_1+\sum _{n=2}^{\infty }\left({{\gamma }_n+\delta }_n\right)={1,{\gamma }_n,\delta }_n\ge 0.$$

(33)

Proof. 

$\left(\Rightarrow \right)$ Let f of the form (7) belong to the class ${\mathcal{W}}_{\rho }^{s,\tau }.$ If we put

$${\gamma }_n:=\left(u_n-n{\rho }^{n-1}\right)\left|a_n\right|\ge 0,{\delta }_n:=\left(v_n-\tau n{\rho }^{n-1}\right)\left|b_n\right|\ge 0$$

and

$${\gamma }_1:=1-\sum _{n=2}^{\infty }\left({{\gamma }_n+\delta }_n\right),$$

then by (9), we have ${\gamma }_1\ge 0,$ i.e., (33) holds. Moreover, we have

$$\begin{array}{ccc}& & \left({\gamma }_1{h}_1+\sum _{n=2}^{\infty }\left({\gamma }_n{h}_n+{\delta }_n{g}_n\right)\right)\left(z\right)=\left(1-\sum _{n=2}^{\infty }\left({{\gamma }_n+\delta }_n\right)\right)z\hfill \\ & & +\sum _{n=2}^{\infty }\left(\left(u_n-n{\rho }^{n-1}\right)\left|a_n\right|{\displaystyle \frac{u_{n}z-z^n}{u_n-n{\rho }^{n-1}}}+\left(v_n-\tau n{\rho }^{n-1}\right)\left|b_n\right|{\displaystyle \frac{v_{n}z+\tau {\overline{z}}^n}{v_n-\tau n{\rho }^{n-1}}}\right)\hfill \\ & =& \left(1-\sum _{n=2}^{\infty }\left(\left(u_n-n{\rho }^{n-1}\right)|a_n|+\left(v_n-\tau n{\rho }^{n-1}\right)\left|b_n\right|\right)\right)z\hfill \\ & & +z\sum _{n=2}^{\infty }\left(u_n\left|a_n\right|+v_n\left|b_n\right|\right)-\sum _{n=2}^{\infty }\left(|a_n|z^n-\tau \left|b_n\right|{\overline{z}}^n\right)\hfill \\ & =& \left(1+\sum _{n=2}^{\infty }n\left({\rho }^{n-1}\left|a_n\right|+\tau {\rho }^{n-1}\left|b_n\right|\right)\right)z-\sum _{n=2}^{\infty }\left(|a_n|z^n-\tau \left|b_n\right|{\overline{z}}^n\right)=f\left(z\right).\hfill \end{array}$$

and the condition (32) follows.

$(\Leftarrow )$. Let now a function f of the form (7) satisfy (32). Thus,

$$\begin{array}{ccc}\hfill f\left(z\right)& =& {\gamma }_1{h}_1\left(z\right)+\sum _{n=2}^{\infty }\left({\gamma }_n{h}_n+{\delta }_n{g}_n\right)\left(z\right)\hfill \\ & =& \left(1-\sum _{n=2}^{\infty }\left({{\gamma }_n+\delta }_n\right)\right)z+\sum _{n=2}^{\infty }\left({\gamma }_n{\displaystyle \frac{u_{n}z-z^n}{u_n-n{\rho }^{n-1}}}+{\delta }_n{\displaystyle \frac{v_{n}z+\tau {\overline{z}}^n}{v_n-\tau n{\rho }^{n-1}}}\right)\hfill \\ & =& a_{1}z-\sum _{n=2}^{\infty }\left(\left|a_n\right|z^n-\tau \left|b_n\right|\overline{z^n}\right),\hfill \end{array}$$

where

$$\left|a_n\right|={\displaystyle \frac{{\gamma }_n}{u_n-n{\rho }^{n-1}}},\left|b_n\right|={\displaystyle \frac{{\delta }_n}{v_n-\tau n{\rho }^{n-1}}},$$

and

$$a_1=1+\sum _{n=2}^{\infty }\left(\left|a_n\right|+\tau \left|b_n\right|\right)n{\rho }^{n-1}.$$

Thus, the function f is of the form (7), and

$$\sum _{n=2}^{\infty }\left(\left(u_n-n{\rho }^{n-1}\right)\left|a_n\right|+\left(v_n-\tau n{\rho }^{n-1}\right)\left|b_n\right|\right)=\sum _{n=2}^{\infty }\left({\gamma }_n+{\delta }_n\right)\le 1.$$

Finally, we have $f\in {\mathcal{W}}_{\rho }^{s,\tau }$, which ends the proof. □

Let $\mathcal{F}$ be a subclass of the class $\mathcal{H}$. A function $f\in \mathcal{F}$ is called an extreme point of $\mathcal{F}$ if the condition

$$f=\gamma f_1+\left(1-\gamma \right)f_2\left(f_1,f_2\in \mathcal{F},0<\gamma <1\right)$$

implies $f_1=f_2=f.$ We shall use the notation $E\mathcal{F}$ to denote the set of all extreme points of $\mathcal{F}$.

We say that a class $\mathcal{F}$ is convex if

$$\gamma f+(1-\gamma )g\in \mathcal{F}(f,g\in \mathcal{F},0\le \gamma \le 1).$$

From Theorem 4, we obtain the following corollary.

Corollary 6.

The class ${\mathcal{W}}_{\rho }^{s,\tau }$ is convex. Moreover,

$$E{\mathcal{W}}_{\rho }^{s,\tau }=\left\{h_n:n\in \mathbb{N}\right\}\cup \left\{g_n:n\in {\mathbb{N}}_2\right\},$$

where $h_1\left(z\right)=z$ and $h_n, g_n$ are the functions of the form (16).

6. The Radii of Starlikeness and Convexity

We say that a function $f\in \mathcal{H}$ is harmonic starlike in$\mathbb{U}\left(r\right)$ if$f$ maps $\mathbb{U}\left(r\right)$ onto a domain starlike with respect to the origin (see [20,21]).

Lemma 2.

A function $f\in {\mathcal{H}}_{\rho }$ of the form (7) is starlike in $\mathbb{U}\left(r\right)$ if and only if it satisfies the condition

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

(34)

Proof. 

A function $f\in {\mathcal{H}}_{\rho }$ is starlike in $\mathbb{U}\left(r\right)$ if and only if it maps the circle $\partial \mathbb{U}\left(r\right)$ onto a closed curve that is starlike with respect to the origin i.e.,

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

(35)

It is easy to verify that the condition (35) can be written as

$$\mathrm{Re}{\displaystyle \frac{D_{\mathcal{H}}f\left(\zeta \right)}{f\left(\zeta \right)}}>0(\zeta \in \mathbb{U}\left(r\right)),$$

or equivalently

$$\left|{\displaystyle \frac{D_{\mathcal{H}}f\left(\zeta \right)-f\left(\zeta \right)}{D_{\mathcal{H}}f\left(\zeta \right)+f\left(\zeta \right)}}\right|<1(\zeta \in \mathbb{U}\left(r\right)).$$

(36)

Since for $\left|\zeta \right|=r<1$ we have

$$\begin{array}{cc}\hfill \left|{\displaystyle \frac{D_{\mathcal{H}}f\left(\zeta \right)-f\left(\zeta \right)}{D_{\mathcal{H}}f\left(\zeta \right)+f\left(\zeta \right)}}\right|& =\left|{\displaystyle \frac{\sum _{n=2}^{\infty }\left((n-1)\left|a_n\right|{\zeta }^n-(n+1)\left|b_n\right|{\overline{\zeta }}^n\right)}{2a_1\zeta +\sum _{n=2}^{\infty }\left((n+1)\left|a_n\right|{\zeta }^n-(n-1)\left|b_n\right|{\overline{\zeta }}^n\right)}}\right|\hfill \\ \hfill & \le {\displaystyle \frac{\sum _{n=2}^{\infty }\left((n-1)\left|a_n\right|+\left(n+1\right)\left|b_n\right|\right)r^{n-1}}{2a_1-\sum _{n=2}^{\infty }\left((n+1)\left|a_n\right|+(n-1)\left|b_n\right|\right)r^{n-1}}}.\hfill \end{array}$$

the condition (36) is true if and only if

$$\sum _{n=2}^{\infty }n\left(\left|a_n\right|+\left|b_n\right|\right)r^{n-1}\le a_1,.$$

(37)

Since

$$a_1=1+\sum _{n=2}^{\infty }n\left(\left|a_n\right|+\tau \left|b_n\right|\right){\rho }^{n-1},$$

(38)

the condition (37) is equivalent to (34).

□

Analogously, we say that a function $f\in \mathcal{H}$ is harmonic convex in$\mathbb{U}\left(r\right)$ if$f$ maps $\mathbb{U}\left(r\right)$ onto a convex domain, i.e.,

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

It is clear that any function convex in $\mathbb{U}\left(r\right)$ is also starlike in$\mathbb{U}\left(r\right).$ Moreover, we have the following equivalence.

Corollary 7.

A function $f\in \mathcal{H}$ is convex in$\mathbb{U}\left(r\right)$ if and only if the function $g=D_{\mathcal{H}}f$ is starlike in$\mathbb{U}\left(r\right).$

Definition 3.

We say that the number $R^{*}\left(f\right)\in \left[0,1\right]$ is the radius of the starlikeness of the function $f\in \mathcal{H}$ if f is starlike in$\mathbb{U}\left(r\right)$ for all $r\le R^{*}\left(f\right).$ Similarly, we say that the number $R^c\left(f\right)\in \left[0,1\right]$ is the radius of convexity of the function $f\in \mathcal{H}$ if f is convex in$\mathbb{U}\left(r\right)$ for all $r\le R^c\left(f\right).$

Definition 4.

We say that the number $R^{*}\left(\mathcal{B}\right)\in \left[0,1\right]$ is the radius of the starlikeness of the class $\mathcal{B}\subset \mathcal{H}$ if each function $f\in \mathcal{B}$ is starlike in$\mathbb{U}\left(r\right)$, and we say that the number $R^c\left(\mathcal{B}\right)\in \left[0,1\right]$ is the radius of convexity of the class $\mathcal{B}$ if each function $f\in \mathcal{B}$ is convex in$\mathbb{U}\left(r\right)$.

From the definitions, we have

$$R_{\alpha }^{\ast }\left(\mathcal{B}\right):=inf\left\{R^{\ast }\left(f\right):f\in \mathcal{B}\right\},R_{\alpha }^c\left(\mathcal{B}\right):=inf\left\{R^c\left(f\right):f\in \mathcal{B}\right\}.$$

Corollary 8.

Let $\mathcal{B}\subset {\mathcal{H}}_{\rho }$. If $R^{*}\left(\mathcal{B}\right)>0,$ then all of the coefficients of the class $\mathcal{B}$ are bounded, i.e., for each $n\in {\mathbb{N}}_2$ there exists $M_n>0$ such that

$$\left|a_n\right|\le M_n,\left|b_n\right|\le M_n\left(f\in \mathcal{B}\right),$$

where f is of the form (7).

Proof. 

Let $f\in {\mathcal{H}}_{\rho }$ be of the form (7) and $r:=R^{*}\left(\mathcal{B}\right)>0$. Then, by Lemma 2, we have

$$\begin{array}{ccc}\hfill \left|a_n\right|& \le & {\displaystyle \frac{1}{n\left(1-{\rho }^{n-1}\right)r^{n-1}}}=:M_n,\hfill \\ \hfill \left|b_n\right|& \le & {\displaystyle \frac{1}{n\left(1-\tau {\rho }^{n-1}\right)r^{n-1}}}\le M_n.\hfill \end{array}$$

Thus, all of the coefficients of the class $\mathcal{B}$ are bounded. □

Corollary 8 can be written in the following form.

Corollary 9.

If there exists an unbounded coefficient of the class $\mathcal{B}\subset {\mathcal{H}}_{\rho }$, then $R^{*}\left(\mathcal{B}\right)=0$.

Thus, by Corollary 2, we have the following corollary.

Corollary 10.

Let $\left(u_n\right),\left(v_n\right)$ be defined by (10). If there exists $n\in {\mathbb{N}}_2$ such that

$$u_n-n{\rho }^{n-1}\le 0\mathrm{or}{v}_n-\tau n{\rho }^{n-1}\le 0,$$

then

$$R^{*}\left({\mathcal{W}}_{\rho }^{s,\tau }\right)=0.$$

Theorem 5.

Let $\left(u_n\right),\left(v_n\right)$ be defined by (10). If

$$u_n-n{\rho }^{n-1}>0,v_n-\tau n{\rho }^{n-1}>0\left(n\in {\mathbb{N}}_2\right),$$

then

$$R^{*}\left({\mathcal{W}}_{\rho }^{s,\tau }\right)=\underset{n\in {\mathbb{N}}_2}{inf}{\left({\displaystyle \frac{1}{n}}min\left\{{\displaystyle \frac{u_n-n{\rho }^{n-1}}{1-{\rho }^{n-1}}},{\displaystyle \frac{v_n-\tau n{\rho }^{n-1}}{1-\tau {\rho }^{n-1}}}\right\}\right)}^{\frac{1}{n-1}}.$$

(39)

Proof. 

Let $f\in {\mathcal{W}}_{\rho }^{s,\tau }$ be of the form (7) with (2). By Theorem 1, we have

$$\sum _{n=2}^{\infty }\left(\left(u_n-n{\rho }^{n-1}\right)\left|a_n\right|+\left(v_n-\tau n{\rho }^{n-1}\right)\left|b_n\right|\right)\le 1,$$

Thus, the condition (34) is true if

$$n\left(1-{\rho }^{n-1}\right)r^{n-1}\le u_n-n{\rho }^{n-1},n\left(1-\tau {\rho }^{n-1}\right)r^{n-1}\le v_n-\tau n{\rho }^{n-1}\left(n\in {\mathbb{N}}_2\right),$$

or equivalently

$$r\le {\left(\frac{1}{n}min\left\{{\displaystyle \frac{u_n-n{\rho }^{n-1}}{1-{\rho }^{n-1}}},{\displaystyle \frac{v_n-\tau {\rho }^{n-1}}{1-\tau {\rho }^{n-1}}}\right\}\right)}^{{\displaystyle \frac{1}{n-1}}}\left(n\in {\mathbb{N}}_2\right).$$

(40)

It follows that $R^{*}\left(f\right)\ge r^{*}$, where

$$r^{*}:=\underset{n\in {\mathbb{N}}_2}{inf}{\left({\displaystyle \frac{1}{n}}min\left\{{\displaystyle \frac{u_n-{\rho }^{n-1}}{1-{\rho }^{n-1}}},{\displaystyle \frac{v_n-\tau {\rho }^{n-1}}{1-\tau {\rho }^{n-1}}}\right\}\right)}^{{\displaystyle \frac{1}{n-1}}},$$

and in consequence ${R^{*}\left({\mathcal{W}}_{\rho }^{s,\tau }\right)\ge r}^{*}.$ Since for the functions $h_n, g_n$ of the form (16) we have

$$R^{*}\left(h_n\right)={\left({\displaystyle \frac{u_n-n{\rho }^{n-1}}{n-n{\rho }^{n-1}}}\right)}^{{\displaystyle \frac{1}{n-1}}},R^{*}\left(g_n\right)={\left({\displaystyle \frac{v_n-\tau n{\rho }^{n-1}}{n-\tau n{\rho }^{n-1}}}\right)}^{{\displaystyle \frac{1}{n-1}}},$$

the radius $R^{*}\left({\mathcal{W}}_{\rho }^{s,\tau }\right)$ cannot be larger than $r^{*}$. Thus, we have (39). □

From Theorem 5 and Corollary 7, we have the following result.

Theorem 6.

Let $\left(u_n\right), \left(v_n\right)$ be defined by (10). If

$$u_n-n{\rho }^{n-1}>0,v_n-\tau n{\rho }^{n-1}>0\left(n\in {\mathbb{N}}_2\right),$$

then

$$R^c\left({\mathcal{W}}_{\rho }^{s,\tau }\right)=\underset{n\in {\mathbb{N}}_2}{inf}{\left({\displaystyle \frac{1}{n^2}}min\left\{{\displaystyle \frac{u_n-n{\rho }^{n-1}}{1-{\rho }^{n-1}}},{\displaystyle \frac{v_n-\tau n{\rho }^{n-1}}{1-\tau {\rho }^{n-1}}}\right\}\right)}^{\frac{1}{n-1}}.$$

(41)

Let

$$d_n:={\displaystyle \frac{u_n-n{\rho }^{n-1}}{n-n{\rho }^{n-1}}},e_n:={\displaystyle \frac{v_n-\tau n{\rho }^{n-1}}{n-\tau n{\rho }^{n-1}}}.$$

Then, for ${\alpha }_n\ge {\beta }_n+1\left(n=1,2,\dots ,s\right)$, we have

$$d_n>1,e_n>1\left(n\in {\mathbb{N}}_2\right).$$

Moreover, if ${\alpha }_n\ge {\beta }_n+2\left(n=1,2,\dots ,s\right),$ then

$${\displaystyle \frac{d_n}{n}}>1,{\displaystyle \frac{e_n}{n}}>1\left(n\in {\mathbb{N}}_2\right).$$

Thus, by Theorems 5 and 6, we have the following two corollaries.

Corollary 11.

If ${\alpha }_n\ge {\beta }_n+1\left(n=1,2,\dots ,s\right),$ then $R^{*}\left({\mathcal{W}}_{\rho }^{s,\tau }\right)=1.$It means that ${\mathcal{W}}_{\rho }^{s,\tau }\subset {\mathcal{W}}_{\rho }^{*}.$

Corollary 12.

If ${\alpha }_n\ge {\beta }_n+2\left(n=1,2,\dots ,s\right),$ then $R^c\left({\mathcal{W}}_{\rho }^{s,\tau }\right)=1.$ It means that ${\mathcal{W}}_{\rho }^{s,\tau }\subset {\mathcal{W}}_{\rho }^c.$

7. Conclusions and Declarations

In the paper, we consider class ${\mathcal{W}}_{\rho }^{s,\tau }$ of functions with Montel’s normalization. The normalization provides additional possibilities. First of all, it gives more general results than classical normalization. Namely, if we put $\rho =0,$ then we obtain the class ${\mathcal{W}}^{s,\tau }:={\mathcal{W}}_0^{s,\tau }$ of harmonic functions with classical normalization. Therefore, putting $\rho =0$ in the obtained results, we obtain related results for the class ${\mathcal{W}}^{s,\tau }.$ Moreover, we obtain results that cannot be obtained for classical normalization. On the other hand, by choosing the parameters of the defined class of functions, we can obtain several new and also well-known results (see for example [9,10,11,12,13,14,15,19,20,21,22,23]).