On a Harmonic Univalent Subclass of Functions Involving a Generalized Linear Operator

Abstract

In this paper, a subclass of complex-valued harmonic univalent functions defined by a generalized linear operator is introduced. Some interesting results such as coefficient bounds, compactness, and other properties of this class are obtained.

1. Introduction

Let $H$ represent the continuous harmonic functions which are harmonic in the open unit disk $U=\left\{z:\hspace{0.33em}z\in ℂ,\left|z\right|<1\right\}$ and let $A$ be a subclass of $H$ which represents the functions which are analytic in $U$. A harmonic function in $U$ could be expressed as $f=h+\overline{g}$, where $h$ and $g$ are in $A$, $h$ is the analytic part of $f$, $g$ is the co-analytic part of $f$ and $\left|h\prime \left(z\right)\right|>\left|g\prime \left(z\right)\right|$ is a necessary and sufficient condition for $f$ to be locally univalent and sense-preserving in $U$ (see Clunie and Sheil-Small [1]). Now we write,

$$h\left(z\right)=z+{\displaystyle \sum }_{n=2}^{\infty }{a}_n{z}^n, g\left(z\right)={\displaystyle \sum }_{n=2}^{\infty }{b}_n{z}^n.$$

(1)

Let $SH$ represents the functions of the form $f=h+\overline{g}$ which are harmonic and univalent in $U$, which normalized by the condition $f\left(0\right)=f_z\left(0\right)-1=0$. The subclass $S{H}^0$ of $SH$ consists of all functions in $SH$ which have the additional property $f_{\overline{z}}\left(0\right)=0$. The class $SH$ was investigated by Clunie and Sheil-Smallas [1]. Since then, many researchers have studied the class $SH$ and even investigated some subclasses of it. Also, we observe that the class $SH$ reduces to the class $S$ of normalized analytic univalent functions in $U$, if the co-analytic part of $f$ is equal to zero. For $f\in S$, the Salagean differential operator $D^n\left(n\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}\right)$ was defined by Salagean [2]. For $f=h+\overline{g}$ given by (1), Jahangiri et al. [3] defined the modified Salagean operator of $f$ as

$$D^{m}f\left(z\right)=D^{m}h\left(z\right)+{\left(-1\right)}^m\overline{D^{m}g\left(z\right)},$$

where

$$D^{m}h\left(z\right)=z+{\displaystyle \sum }_{n=2}^{\infty }{n}^m{a}_n{z}^n, D^{m}g\left(z\right)={\displaystyle \sum }_{n=2}^{\infty }{n}^m{b}_n{z}^n.$$

Next, for functions $f\in A$, For $n\in {\mathbb{N}}_0,\beta \ge \gamma \ge 0$, Yalçın and Altınkaya [4] defined the differential operator of $I_{\gamma ,\beta }^{m}f:S{H}^0\to S{H}^0$. Now we define our differential operator:

$$\begin{gathered} I_{\delta ,\mu ,\lambda ,\eta ,\varsigma ,\tau }^{0}f\left(z\right)=h\left(z\right)+\overline{g\left(z\right)} \\ {I}_{\delta ,\mu ,\lambda ,\varsigma ,\tau }^{1}f\left(z\right)=z+{\displaystyle \sum }_{n=2}^{\infty }\left(\frac{\mu +\lambda -\left(\delta -\varsigma \right)\left(\lambda -\tau \right)D^{0}f\left(z\right)+\left(\delta -\varsigma \right)\left(\lambda -\tau \right)D^{1}f\left(z\right)}{\mu +\lambda }\right) \\ =\frac{\mu +\lambda -\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(h\left(z\right)+\overline{g\left(z\right)}\right)+\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(zh\left(z\right)+z\overline{g^{\prime }\left(z\right)}\right)}{\mu +\lambda } \end{gathered}$$

(2)

$$I_{\delta ,\mu ,\lambda ,\varsigma ,\tau }^{m}f\left(z\right)=I_{\delta ,\mu ,\lambda ,\varsigma ,\tau }^1\left(I_{\delta ,\mu ,\lambda ,\varsigma ,\tau }^{m-1}f\left(z\right)\right).$$

(3)

If $f$ is given by (1), then from (2) and (3), we get (see [5])

$$\begin{gathered} \underset{\delta ,\mu ,\lambda ,\varsigma ,\tau }{\overset{m}{I}}f\left(z\right)=z+{\displaystyle \sum }_{n=2}^{\infty }{\left(\frac{\mu +\lambda +\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }\right)}^m{a}_n{z}^n \\ +{\left(-1\right)}^m{\displaystyle \sum }_{n=2}^{\infty }{\left(\frac{\mu +\lambda +\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }\right)}^m\overline{b_n{z}^n}. \end{gathered}$$

(4)

The operator $I_{\delta ,\mu ,\lambda ,\varsigma ,\tau }^{m}f\left(z\right)$ generalizes the following differential operators:

If $f\in A,$ then when we take $\mu =1, \lambda =0, \delta =0, \tau =1, \varsigma =1$ we obtain $I_{0,\tau , \delta ,\varsigma }^{m}f\left(z\right)$ was introduced and studied by Ramadan and Darus [6]. By taking different choices of $\mu , \lambda , \delta , \tau and \varsigma$ we get $I_{1-\lambda ,\tau , 0,\varsigma }^{m}f\left(z\right)$ was introduced and studied by Darus and Ibrahim [7], $I_{\mu , \lambda ,0, 1,0}^{m}f\left(z\right)$ was introduced and studied by Swamy [8], $I_{1-\lambda ,0, \mathrm{1,0}}^{m}f\left(z\right)$ was introduced and studied by Al-Oboudi [9] and $I_{0, 0,\mathrm{}, 1,0}^{m}f\left(z\right)$ was introduced and studied by Salagean [2].

If $f\in H,$ then $I_{\mu , \lambda ,0, 1,0}^{m}f\left(z\right)$ becomes the modified Salagean operator introduced by Yasar and Yalçin [10].

A function $f:U\to C$ is subordinate to the function $g:U\to C$ denoted by $f\left(z\right)\prec g\left(z\right)$, if there exists an analytic function $w:U\to U$ with $w\left(0\right)=0$ such that

$$f\left(z\right)=g\left(w\left(z\right)\right),\left(z\in U\right).$$

Moreover, if the function $g$ is univalent in $U$, then we have (see [11,12]):

$$f\left(z\right)\prec g\left(z\right) \mathrm{if} \mathrm{and} \mathrm{only} \mathrm{if} f\left(0\right)=g\left(0\right), f\left(U\right)\subset g\left(U\right).$$

Denote by $S{H}^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right)$ the subclass of $S{H}^0$ consisting of functions of the form (1) that satisfy the condition

$$\frac{I_{\delta ,\mu ,\lambda ,\varsigma ,\tau }^{m+1}f\left(z\right)}{I_{\delta ,\mu ,\lambda ,\varsigma ,\tau }^{m}f\left(z\right)}\prec \frac{1+Az}{1+Bz},-1\le A<B\le 1$$

(5)

where $I_{\delta ,\mu ,\lambda ,\eta ,\varsigma ,\tau }^{m}f\left(z\right)$ is defined by (4). For relevant and recent references related to this work, we refer the reader to [13,14,15,16,17,18,19,20].

In this paper we use the same techniques that have been used earlier by Dziok [21] and Dziok et al. [22], to investigate coefficient bound, distortion bounds, and some other properties for the class $S{H}^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right)$.

2. Coefficient Bounds

In this section we find the coefficient bound for the class $S{H}^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right).$

Theorem 1.

Let the function$f\left(z\right)=h+\overline{g}$be defined by (1). Then$f\in S{H}^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right)$ if

$${\displaystyle \sum }_{n=2}^{\infty }\left(C_n\left|a_n\right|+D_n\left|b_n\right|\right)\le B-A$$

(6)

where

$$C_n={\left(\frac{\mu +\lambda +\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }\right)}^m\left\{\frac{\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)\left[B+1\right]-\left(\mu +\lambda \right)\left(B-A\right)}{\mu +\lambda }\right\}$$

(7)

and

$$D_n={\left(\frac{\mu +\lambda +\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }\right)}^m\left\{\frac{\left[A+B\left(2+\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)\right)\right]\left(\mu +\lambda \right)}{\mu +\lambda }\right\}.$$

(8)

Proof. 

Let $a_n\ne 0$ or $b_n\ne 0$ for $n\ge 2$. Since $C_n, D_n\ge n\left(B-A\right)$ by (6), we obtain

$$\begin{gathered} \left|h^{\prime }\left(z\right)\right|-\left|g^{\prime }\left(z\right)\right|\ge 1-{\displaystyle \sum }_{n=2}^{\infty }n\left|a_n\right|{\left|z\right|}^{n-1}-{\displaystyle \sum }_{n=2}^{\infty }n\left|b_n\right|{\left|z\right|}^{n-1} \\ \ge 1-\left|z\right|{\displaystyle \sum }_{n=2}^{\infty }\left(n\left|a_n\right|+n\left|b_n\right|\right) \\ \ge 1-\frac{\left|z\right|}{B-A}{\displaystyle \sum }_{n=2}^{\infty }\left(C_n\left|a_n\right|+D_n\left|b_n\right|\right) \\ \ge 1-\left|z\right|>0 \end{gathered}$$

.

Therefore, $f$ is univalent in $U$. To ensure the univalence condition, consider $z_1,z_2\in U$ so that $z_1\ne z_2$. Then

$$\left|\frac{z_1^n-z_2^n}{z_1-z_2}\right|=\left|{\displaystyle \sum }_{m=1}^n{z}_1^{m-1}-z_2^{n-m}\right|\le {\displaystyle \sum }_{m=1}^n\left|z_1^{m-1}\right|\left|z_2^{n-m}\right|<n , n\ge 2$$

.

So, we have

$$\begin{gathered} \left|\frac{f\left(z_1\right)-f\left(z_2\right)}{h\left(z_1\right)-h\left(z_2\right)}\right|\ge 1-\left|\frac{g\left(z_1\right)-g\left(z_2\right)}{h\left(z_1\right)-h\left(z_2\right)}\right|=1-\left|\frac{{{\displaystyle \sum }}_{n=2}^{\infty }{b}_k\left(z_1^n-z_2^n\right)}{z_1-z_2+{{\displaystyle \sum }}_{n=2}^{\infty }{a}_n\left(z_1^n-z_2^n\right)}\right| \\ >1-\frac{{{\displaystyle \sum }}_{n=2}^{\infty }n\left|b_n\right|}{1-{{\displaystyle \sum }}_{n=2}^{\infty }n\left|a_n\right|}\ge 1-\frac{{{\displaystyle \sum }}_{n=2}^{\infty }\frac{D_n}{B-A}\left|b_n\right|}{{{\displaystyle \sum }}_{n=2}^{\infty }\frac{C_n}{B-A}\left|a_n\right|}\ge 0, \end{gathered}$$

which proves univalences.

On the other hand, $f\in S{H}^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right)$ if and only if there exists a function $w$; with $w\left(0\right)=0$, and $\left|w\left(z\right)\right|<1\left(z\in U\right)$ such that

$$\frac{I_{\delta ,\mu ,\lambda ,\varsigma ,\tau }^{m+1}f\left(z\right)}{I_{\delta ,\mu ,\lambda ,\varsigma ,\tau }^{m}f\left(z\right)}\prec \frac{1+Az}{1+Bz}$$

or

$$\frac{I_{\delta ,\mu ,\lambda ,\varsigma ,\tau }^{m+1}f\left(z\right)-I_{\delta ,\mu ,\lambda ,\varsigma ,\tau }^{m}f\left(z\right)}{B{I}_{\delta ,\mu ,\lambda ,\varsigma ,\tau }^{m+1}f\left(z\right)-A{I}_{\delta ,\mu ,\lambda ,\varsigma ,\tau }^{m}f\left(z\right)}<1, \left(z\in U\right).$$

(9)

The above inequality (9) holds, since for $\left|z\right|=r (0<r<1)$ we obtain

$$\begin{gathered} \left|I_{\delta ,\mu ,\lambda ,\varsigma ,\tau }^{m+1}f\left(z\right)-I_{\delta ,\mu ,\lambda ,\varsigma ,\tau }^{m}f\left(z\right)\right|-\left|B{I}_{\delta ,\mu ,\lambda ,\varsigma ,\tau }^{m+1}f\left(z\right)-A{I}_{\delta ,\mu ,\lambda ,\varsigma ,\tau }^{m}f\left(z\right)\right| \\ =\left|{\displaystyle \sum }_{n=2}^{\infty }{\left(\frac{\mu +\lambda +\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }\right)}^m\frac{\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }{a}_n{z}^n\right. \\ \left.+{\left(-1\right)}^m{\displaystyle \sum }_{n=2}^{\infty }{\left(\frac{\mu +\lambda +\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }\right)}^m\frac{2\left(\mu +\lambda \right)+\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }\overline{b_n{z}^n}\right| \\ -\hspace{0.33em}\left|\left(B-A\right)z+{\displaystyle \sum }_{n=2}^{\infty }{\left(\frac{\mu +\lambda +\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }\right)}^m\left(B\frac{\mu +\lambda +\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }-A\right)a_n{z}^n\right. \\ -\left.{\left(-1\right)}^m{\displaystyle \sum }_{n=2}^{\infty }{\left(\frac{\mu +\lambda +\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }\right)}^m\left(B,\frac{2\left(\mu +\lambda \right)+\delta \left(-\varsigma \right)\left(\lambda -\tau \right)\left(1-n\right)}{\mu +\lambda }+A\right)\overline{b_n{z}^n}\right| \\ \le {\displaystyle \sum }_{n=2}^{\infty }{\left(\frac{\mu +\lambda +\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }\right)}^m\frac{\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }\left|a_n\right|r^n+ \\ {\displaystyle \sum }_{n=2}^{\infty }{\left(\frac{\mu +\lambda +\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }\right)}^m\frac{2\left(\mu +\lambda \right)+\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(1-n\right)}{\mu +\lambda }\left|b_n\right|r^n-\left(B-A\right)r \\ +{\displaystyle \sum }_{n=2}^{\infty }{\left(\frac{\mu +\lambda +\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }\right)}^m\left(B\frac{\mu +\lambda +\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)+A}{\mu +\lambda }-A\right)\left|a_n\right|r^n \\ +{\displaystyle \sum }_{n=2}^{\infty }{\left(\frac{\mu +\lambda +\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }\right)}^m\left(B\frac{2\left(\mu +\lambda \right)+\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }+A\right)\left|b_n\right|r^n \\ \le r\left\{{\displaystyle \sum }_{n=2}^{\infty }\left(C_n\left|a_n\right|+D_n\left|b_n\right|\right)r^{n-1}-\left(B-A\right)\right\}<0. \end{gathered}$$

Therefore, $f\in S{H}^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right)$, and so the proof is completed.

Next we show that the condition (6) is also necessary for the functions $f\in H$ to be in the class $S{H}_T^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right)=T^m\cap S{H}^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right)$ where $T^m$ is the class of functions $f=h+\overline{g}\in S{H}^0$ so that

$$f=h+\overline{g}=z-{\displaystyle \sum }_{n=2}^{\infty }{a}_n{z}^n+{\left(-1\right)}^m{\displaystyle \sum }_{n=2}^{\infty }\left|b_n\right|\overline{z^n}\left(z\in U\right).$$

(10)

□

Theorem 2.

Let$f=h+\overline{g}$be defined by (10). Then$f\in S{H}_T^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right)$if and only if the condition (6) holds.

Proof. 

For this proof, we let the fractions $\frac{\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }=L$ and $\frac{2\left(\mu +\lambda \right)+\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }=K$. The first part “if statement” follows from Theorem 1. Conversely, we suppose that $f\in S{H}_T^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right)$, then by (9) we have

$$\left|\frac{{\displaystyle \sum }_{n=2}^{\infty }\left[{\left(L\right)}^m\frac{\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }\left|a_n\right|z^n+{\left(K\right)}^m\frac{2\left(\mu +\lambda \right)+\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }\overline{\left|b_n\right|z^n}\right]}{\left(B-A\right)z-{\displaystyle \sum }_{n=2}^{\infty }\left[{\left(L\right)}^m\left(BL-A\right)\left|a_n\right|z^n+{\left(K\right)}^m\left(BK+A\right)\left|b_n\right|z^n\right]}\right|<1.$$

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

$$\frac{{\displaystyle \sum }_{n=2}^{\infty }\left[{\left(L\right)}^m\frac{\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }\left|a_n\right|+{\left(K\right)}^m\frac{2\left(\mu +\lambda \right)+\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }\overline{\left|b_n\right|}\right]r^{n-1}}{\left(B-A\right)-{\displaystyle \sum }_{n=2}^{\infty }\left[{\left(L\right)}^m\left(BL-A\right)\left|a_n\right|+{\left(K\right)}^m\left(BK+A\right)\left|b_n\right|\right]r^{n-1}}<1.$$

Thus, for $C_n$ and $D_n$ as defined by (7) and (8), we have

$${\displaystyle \sum }_{n=2}^{\infty }\left[C_n\left|a_n\right|+D_n\left|b_n\right|\right]r^{n-1}<B-A\left(0\le r<1\right).\left(11\right)$$

(11)

Let $\left\{{\rho }_n\right\}$ be the sequence of partial sums of the series

$${\displaystyle \sum }_{k=2}^n\left[C_k\left|a_k\right|+D_k\left|b_k\right|\right].$$

Then $\left\{{\rho }_n\right\}$ is a non-decreasing sequence and by (11) it is bounded above by $B-A$. Thus, it is convergent and

$${\displaystyle \sum }_{n=2}^{\infty }\left[C_n\left|a_n\right|+D_n\left|b_n\right|\right]=\underset{n\to +\infty }{\lim }{\rho }_n\le B-A.$$

This gives us the condition (6).□

3. Compactness and Convex

In this section we obtain the compactness and the convex relation for the class $S{H}^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right).$

Theorem 3.

The class$S{H}_T^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right)$is convex and compact subset of$SH.$

Proof. 

Let $f_t\in S{H}_T^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right),$ where

$$f_t\left(z\right)=z-{\displaystyle \sum }_{n=2}^{\infty }\left|a_{t,n}\right|z^n+{\left(-1\right)}^m{\displaystyle \sum }_{n=2}^{\infty }\left|b_{t,n}\right|\overline{z^n}\left(z\in U, t\in \mathbb{N}\right).$$

(12)

Then for $0\le \psi \le 1$, let $f_1,f_2\in S{H}_T^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right)$ be defined by (12). Then

$$\begin{gathered} \xi \left(z\right)=\psi f_1\left(z\right)+\left(1-\psi \right)f_2\left(z\right) \\ =z-{\displaystyle \sum }_{n=2}^{\infty }\left(\psi \left|a_{1,n}\right|+\left(1-\psi \right)\left|a_{2,n}\right|\right)z^n+{(-1)}^m{\displaystyle \sum }_{n=2}^{\infty }\left(\psi \left|b_{1,n}\right|+\left(1-\psi \right)\left|b_{2,n}\right|\right)\overline{z^n} \end{gathered}$$

and

$$\begin{gathered} {\displaystyle \sum }_{n=2}^{\infty }\left\{C_n\left(\psi \left|a_{1,n}\right|+\left(1-\psi \right)\left|a_{2,n}\right|\right)+D_n\left(\psi \left|b_{1,n}\right|+\left(1-\psi \right)\left|b_{2,n}\right|\right)\right\} \\ =\psi {\displaystyle \sum }_{n=2}^{\infty }\left\{C_n\left|a_{1,n}\right|+D_n\left|b_{1,n}\right|\right\}+\left(1-\psi \right){\displaystyle \sum }_{n=2}^{\infty }\left\{C_n\left|a_{2,n}\right|+D_n\left|b_{2,n}\right|\right\} \\ \le \psi \left(B-A\right)+\left(1-\psi \right)\left(B-A\right)=B-A. \end{gathered}$$

Thus, the function $\xi =\psi f_1\left(z\right)+\left(1-\psi \right)f_2\left(z\right)$ is in the class $S{H}_T^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right)$. This implies that $S{H}_T^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right)$ is convex.

For $f_t\in S{H}_T^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right),t\in \mathbb{N}$ and $\left|z\right|\le r (0<r<1),$ then we have

$$\begin{gathered} \left|f_t\left(z\right)\right|\le r+{\displaystyle \sum }_{n=2}^{\infty }\left\{\left|a_{t,n}\right|+\left|b_{t,n}\right|\right\}{r}^n \\ \le r+{\displaystyle \sum }_{n=2}^{\infty }\left\{C_n\left|a_{t,n}\right|+D_n\left|b_{t,n}\right|\right\}{r}^n \\ \le r+\left(B-A\right)r^2. \end{gathered}$$

Therefore, $S{H}_T^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right)$ is uniformly bounded. Let

$$f_t\left(z\right)=z-{\displaystyle \sum }_{n=2}^{\infty }\left|a_{t,n}\right|z^n+{\left(-1\right)}^m{\displaystyle \sum }_{n=2}^{\infty }\left|b_{t,n}\right|\overline{z^n}\left(z\in U, t\in \mathbb{N}\right).$$

also, let $f=h+\overline{g}$ where $h$ and $g$ are given by (1). Then by Theorem 2 we get

$${\displaystyle \sum }_{n=2}^{\infty }\left\{C_n\left|a_n\right|+D_n\left|b_{t,n}\right|\right\}\le B-A.$$

(13)

If we assume $f_t\to f,$ then we get that $\left|a_{t,n}\right|\to \left|a_n\right|$ and $\left|b_{t,n}\right|\to \left|b_n\right|$ as $n\to +\infty \left(t\in \mathbb{N}\right).$ Let $\left\{{\rho }_n\right\}$ be the sequence of partial sums of the series $\sum _{n=2}^{\infty }\left\{C_n\left|a_{t,n}\right|+D_n\left|b_{t,n}\right|\right\}$. Then $\left\{{\rho }_n\right\}$ is a non-decreasing sequence and by (13) it is bounded above by $B-A$. Thus, it is convergent and

$${\displaystyle \sum }_{n=2}^{\infty }\left\{C_n\left|a_{t,n}\right|+D_n\left|b_{t,n}\right|\right\}=\underset{n\to \infty }{\lim }{\rho }_n\le B-A.$$

Therefore, $f\in S{H}_T^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right)$ and therefore the class $S{H}_T^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right)$ is closed. As a result, the class is closed, and the class $S{H}_T^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right)$ is also compact subset of $SH,$ which completes the proof.□

Lemma 1

[23]. Let $f=h+\overline{g}$ be so that $h$ and $g$ are given by (1). Furthermore, let

$${\displaystyle \sum }_{n=2}^{\infty }\left\{\frac{n-\alpha }{1-\alpha }\left|a_n\right|+\frac{n+\alpha }{1-\alpha }\left|b_n\right|\right\}\le 1\left(z\in U\right)$$

where $0\le \alpha <1$. Then $f$ is harmonic, orientation preserving, univalent in $U$ and $f$ is starlike of order $\alpha$.

Theorem 4.

Let$0\le \alpha <1,C_n$and$D_n$be defined by (7) and (8). Then

$$r_{\alpha }^{*}\left(S{H}_T^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,n,A,B\right)\right)=\underset{n\ge 2}{\inf }{\left[\frac{1-\alpha }{B-A}\min \left\{\frac{C_n}{n+\alpha },\frac{D_n}{n+\alpha }\right\}\right]}^{\frac{1}{n-1}},$$

(14)

where$r_{\alpha }^{*}$is the radius of starlikeness of order$\alpha .$

Proof. 

Let $f\in S{H}_T^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right)$ be of the form (10). Then, for $\left|z\right|=r<1$, we get

$$\begin{gathered} \left|\frac{I_{0,\eta }f\left(z\right)-\left(1+\alpha \right)f\left(z\right)}{I_{0,\eta }f\left(z\right)+\left(1+\alpha \right)f\left(z\right)}\right| \\ =\left|\frac{-\alpha z-{\displaystyle \sum }_{n=2}^{\infty }\left(n-1-\alpha \right)\left|a_n\right|z^n-{\left(-1\right)}^m{\displaystyle \sum }_{n=2}^{\infty }\left(n+1+\alpha \right)\left|b_n\right|\overline{z^n}}{\left(2-\alpha \right)z-{\displaystyle \sum }_{n=2}^{\infty }\left(n-1-\alpha \right)\left|a_n\right|z^n-{\left(-1\right)}^m{\displaystyle \sum }_{n=2}^{\infty }\left(n-1+\alpha \right)\left|b_n\right|\overline{z^n}}\right| \\ \le \frac{\alpha -{\displaystyle \sum }_{n=2}^{\infty }\left\{\left(n-1-\alpha \right)\left|a_n\right|-{\left(-1\right)}^m{\displaystyle \sum }_{n=2}^{\infty }\left(n+1+\alpha \right)\left|b_n\right|\right\}{r}^{n-1}}{2-\alpha -{\displaystyle \sum }_{n=2}^{\infty }\left\{\left(n-1-\alpha \right)\left|a_n\right|-{\left(-1\right)}^m{\displaystyle \sum }_{n=2}^{\infty }\left(n-1+\alpha \right)\left|b_n\right|\right\}}. \end{gathered}$$

By using Lemma 1, we observe that f is starlike of order $\alpha$ in $U_r$ if and only if

$$\left|\frac{I_{0,\eta }f\left(z\right)-\left(1+\alpha \right)f\left(z\right)}{I_{0,\eta }f\left(z\right)+\left(1+\alpha \right)f\left(z\right)}\right|<1,z\in U_r$$

or

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

(15)

Furthermore, by using Theorem 2, we get

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

Condition (15) is true if

$$\frac{n-\alpha }{1-\alpha }{r}^{n-1}\le \frac{C_n}{B-A}{r}^{n-1}.$$

This proves

$$\frac{n+\alpha }{1-\alpha }{r}^{n-1}\le \frac{D_n}{B-A}{r}^{n-1}\left(n=2,3\dots \right).$$

So, the function $f$ is starlike of order $\alpha$ in the disk $U_{r_{\alpha }}^{*}$ where

$$r_{\alpha }^{*}=\underset{n\ge 2}{\inf }{\left[\frac{1-\alpha }{B-A}\min \left\{\frac{C_n}{n+\alpha },\frac{D_n}{n+\alpha }\right\}\right]}^{\frac{1}{n-1}},$$

and the function

$$f_n\left(z\right)=h_n\left(z\right)+\overline{g_n\left(z\right)}=z-\frac{B-A}{C_n}{z}^n+{(-1)}^m\frac{B-A}{D_n}\overline{z^n}.$$

So, the radius $r_{\alpha }^{*}$ cannot be larger. Then we get (14).□

4. Extreme Points

In this section we find the extreme points for the class $S{H}^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right).$

Theorem 5.

The extreme points of$S{H}_T^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right)$are the functions$f$of the form (1) where$h=h_k$and$g=g_k$are of the form

$$\begin{gathered} h_1\left(z\right)=z, \\ {h}_n\left(z\right)=z-\frac{B-A}{C_n}{z}^n, \\ {g}_n\left(z\right)={\left(-1\right)}^m\frac{B-A}{D_n}\overline{z^n}, \left(z\in U, n\ge 2\right). \end{gathered}$$

(16)

Proof. 

Suppose that $g_n=\psi f_1+\left(1-\psi \right)f_2$ where $0<\psi <1$ and $f_1,f_2\in S{H}_T^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right)$ are written in the form

$$f_t\left(z\right)=z-{\displaystyle \sum }_{n=2}^{\infty }\left|a_{t,n}\right|z^n+{\left(-1\right)}^m{\displaystyle \sum }_{n=2}^{\infty }\left|b_{t,n}\right|\overline{z^n}\left(z\in U, t\in \left\{\mathrm{1,2}\right\}\right).$$

Then, by (16), we get

$$\left|b_{1,n}\right|=\left|b_{2,n}\right|=\frac{B-A}{D_n},$$

and $a_{1,t}=a_{2,t}=0$ for $t\in \left\{2,3\dots \right\}$ and $b_{1,t}=b_{2,t}=0$ for $t\in \left\{2,3\dots \right\}\setminus \left\{n\right\}$. It follows that $g_n\left(z\right)=f_1\left(z\right)=f_2\left(z\right)$ and $g_n$ are in the class of extreme points of the class $S{H}_T^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right)$. We also can ensure that the functions $h_n\left(z\right)$ are the extreme points of the class $S{H}_T^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right).$ Now, assume that a function $f$ of the form (1) is in the class of the extreme points of the class $S{H}_T^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right)$ and $f$ is not of the form (16). Then there exists $k\in \left\{2,3\dots \right\}$ such that

$$0<\left|a_k\right|<\frac{B-A}{{\left(\frac{\mu +\lambda +\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(k-1\right)}{\mu +\lambda }\right)}^m\left\{\frac{\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left[\left(k-1\right)\left(B+1\right)\right]+\left(\mu +\lambda \right)\left(B-A\right)}{\mu +\lambda }\right\}}$$

or

$$0<\left|b_k\right|<\frac{B-A}{{\left(\frac{\mu +\lambda -\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n+1\right)}{\mu +\lambda }\right)}^m\left\{\frac{\left[A+B\left(2+\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)\right)\right]\left(\mu +\lambda \right)}{\mu +\lambda }\right\}}.$$

If

$$0<\left|a_k\right|<\frac{B-A}{{\left(\frac{\mu +\lambda +\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }\right)}^m\left\{\frac{\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)\left[B+1\right]-\left(\mu +\lambda \right)\left(B-A\right)}{\mu +\lambda }\right\}}$$

then putting

$$\psi =\frac{\left|a_k\right|\left[{\left(\frac{\mu +\lambda +\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }\right)}^m\left\{\frac{\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)\left[B+1\right]-\left(\mu +\lambda \right)\left(B-A\right)}{\mu +\lambda }\right\}\right]}{B-A}$$

and

$$\chi =\frac{f-\psi h_k}{1-\psi },$$

we have $0<\psi <1$, $h_k\ne \chi$. Therefore, $f$ is not in the class of the extreme points of the class $S{H}_T^0\left(\delta ,\mu ,\lambda ,\eta ,\varsigma ,\tau ,m,A,B\right)$. Similarly, if

$$0<\left|b_k\right|<\frac{B-A}{{\left(\frac{\mu +\lambda +\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }\right)}^m\left\{\frac{\left[A+B\left(2+\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)\right)\right]\left(\mu +\lambda \right)}{\mu +\lambda }\right\}}$$

then putting

$$\psi =\frac{\left|b_k\right|{\left(\frac{\mu +\lambda +\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)}{\mu +\lambda }\right)}^m\left\{\frac{\left[A+B\left(2+\left(\delta -\varsigma \right)\left(\lambda -\tau \right)\left(n-1\right)\right)\right]\left(\mu +\lambda \right)}{\mu +\lambda }\right\}}{B-A}$$

and

$$\chi =\frac{f-\psi g_k}{1-\psi },$$

we have $0<\psi <1$, $g_k\ne \chi$. It follows that $f$ is not in the family of extreme points of the class $S{H}_T^0\left(\delta ,\mu ,\lambda ,\varsigma ,\tau ,m,A,B\right)$ and so the proof is completed.□