# Harmonic mappings for which co-analytic part is a close-to-convex function of order b

## Abstract

In the present paper we investigate a class of harmonic mappings for which the second dilatation is a close-to-convex function of complex order b, $$b\in\mathbb{C} / \{ 0 \}$$ (Lashin in Indian J. Pure Appl. Math. 34(7):1101-1108, 2003).

## 1 Introduction

A planar harmonic mapping in the open unit disc $$\mathbb{D}= \{z | |z|<1\}$$ is a complex-valued harmonic function f which maps $$\mathbb{D}$$ onto some planar domain $$f(\mathbb{D})$$. Since $$\mathbb{D}$$ is a simply connected domain, the mapping f has a canonical decomposition $$f=h(z)+ \overline{g(z)}$$, where $$h(z)$$ and $$g(z)$$ are analytic in $$\mathbb{D}$$ and have the following power series expansions:

$$h(z) = \sum_{n=0}^{\infty} a_{n}z^{n}, \qquad g(z) = \sum_{n=0}^{\infty}b_{n} z^{n},\quad z \in\mathbb{D},$$

where $$a_{n}, b_{n} \in\mathbb{C}$$, $$n=0,1,2,\ldots$$ . As usual, we call $$h(z)$$ analytic part and $$g(z)$$ co-analytic part of f, respectively. An elegant and complete account of the theory of planar harmonic mappings is given in Duren’s monograph .

Lewy  proved in 1936 that the harmonic mapping f is locally univalent in $$\mathbb{D}$$ if and only if its Jacobian $$J_{f} = |h'(z)|^{2} - |g'(z)|^{2}$$ is different from zero in $$\mathbb{D}$$. In view of this result, locally univalent harmonic mappings in the open unit disc are either sense-reversing if $$|g'(z)| > |h'(z)|$$ or sense-preserving if $$|g'(z)| < |h'(z)|$$ in $$\mathbb{D}$$. Throughout this paper, we restrict ourselves to the study of sense-preserving harmonic mappings. We also note that $$f=h(z)+ \overline{g(z)}$$ is sense-preserving in $$\mathbb{D}$$ if and only if $$h'(z)$$ does not vanish in the unit disc $$\mathbb{D}$$, and the second dilatation $$w(z) = g'(z) / h'(z)$$ has the property $$|w(z)|<1$$ in $$\mathbb{D}$$.

The class of all sense-preserving harmonic mappings in the open unit disc $$\mathbb{D}$$ with $$a_{0}=b_{0}=0$$ and $$a_{1}=1$$ is denoted by $$\mathcal {S}_{\mathcal{H}}$$. Thus $$\mathcal{S}_{\mathcal{H}}$$ contains the standard class $$\mathcal{S}$$ of analytic univalent functions.

The family of all mappings $$f \in\mathcal{S}_{\mathcal{H}}$$ with the additional property that $$g'(0)=0$$, i.e., $$b_{1}=0$$, is denoted by $$\mathcal{S}_{\mathcal{H}}^{0}$$. Thus it is clear that $$\mathcal{S} \subset\mathcal{S}_{\mathcal{H}}^{0} \subset\mathcal{S}_{\mathcal {H}}$$ . Let Ω be the family of functions $$\phi(z)$$ regular in the open unit disc $$\mathbb{D}$$ and satisfying the conditions $$\phi(0)=0$$, $$|\phi(z)|<1$$ for all $$z\in\mathbb{D}$$. We denote by $$\mathcal{P}$$ the family of functions $$p(z)=1+p_{1} z+p_{2}z^{2}+\cdots$$ regular in $$\mathbb{D}$$ such that $$p(z)$$ in $$\mathcal{P}$$ if and only if

$$p(z) = \frac{1+\phi(z)}{1-\phi(z)}$$
(1.1)

for some $$\phi(z)\in\Omega$$ and every $$z\in\mathbb{D}$$.

Let $$s_{1}(z)=z+c_{2}z^{2}+c_{3}z^{3}+\cdots$$ and $$s_{2}(z)=z+d_{2}z^{2}+d_{3}z^{3}+\cdots$$ be analytic functions in $$\mathbb{D}$$. If there exists a function $$\phi(z)\in\Omega$$ such that $$s_{1}(z)=s_{2}(\phi(z))$$ for every $$z\in\mathbb{D}$$, then we say that $$s_{1}(z)$$ is subordinate to $$s_{2}(z)$$ and we write $$s_{1}\prec s_{2}$$. We also note that if $$s_{1}\prec s_{2}$$, then $$s_{1}(\mathbb{D})\subset s_{2}(\mathbb{D})$$ [3, 4].

Next, let $$\mathcal{A}$$ be the class of functions $$s(z)=z+e_{2}z^{2}+\cdots$$ which are analytic in $$\mathbb{D}$$. A function $$s(z)$$ in $$\mathcal{A}$$ is said to be a convex function of complex order b, $$b\in\mathbb{C} / \{ 0 \}$$, that is, $$s(z) \in\mathcal {C}(b)$$ if and only if $$s'(z) \neq0$$, and

$$\operatorname{Re} \biggl( 1 + \frac{1}{b} z \frac{s''(z)}{s'(z)} \biggr) > 0 \quad (z \in\mathbb{D}).$$
(1.2)

We denote by $$\mathcal{S}^{*}(1 - b)$$ the class of $$\mathcal{A}$$ consisting of functions which are starlike of complex order b, that is,

$$\operatorname{Re} \biggl( 1 + \frac{1}{b} \biggl( z \frac{s''(z)}{s'(z)} - 1 \biggr) \biggr) > 0\quad (z \in \mathbb{D}).$$
(1.3)

Moreover, let $$s(z)$$ be an element of $$\mathcal{A}$$, then $$s(z)$$ is said to be close-to-convex of complex order b, $$b\in\mathbb{C} / \{ 0 \}$$ if and only if there exists a function $$\varphi(z)\in\mathcal {C}(b)$$ satisfying the condition

$$\operatorname{Re} \biggl( 1 + \frac{1}{b} \biggl( \frac{s'(z)}{\varphi'(z)} - 1 \biggr) \biggr) > 0\quad (z \in \mathbb{D}).$$
(1.4)

The class of such functions is denoted by $$\mathcal{CC}(b)$$.

The classes $$\mathcal{C}(b)$$ and $$\mathcal{S}^{*}(1 - b)$$ were introduced and studied by Nasr and Aouf [5, 6], and the class $$\mathcal{CC}(b)$$ was introduced by Lashin .

### Remark 1.1

1. (i)

For $$b=1$$ we obtain $$\mathcal{S}^{*}(0) = \mathcal{S}^{*}$$, $$\mathcal{C}(1)=\mathcal{C}$$, and $$\mathcal{CC}(1)=\mathcal{CC}$$ are well-known classes of starlike, convex and close-to-convex functions, respectively .

2. (ii)

$$\mathcal{S}^{*}(1 - (1-\alpha)) = \mathcal{S}^{*}(\alpha )$$, $$\mathcal{C}(1-\alpha)$$, and $$\mathcal{CC}(1-\alpha)$$, $$0 \leq \alpha< 1$$, are the classes of starlike, convex and close-to-convex functions of order α, respectively .

3. (iii)

If we take $$b=e^{-i\lambda} \cos\lambda$$, $$|\lambda| <\pi/2$$, we obtain the following classes: λ-spirallike, analytic functions for which $$zf'(z)$$ is λ-spirallike and λ-spirallike and λ-spiral close-to-convex functions .

4. (iv)

$$\mathcal{S}^{*}(1 - (1-\alpha)e^{-i\lambda} \cos\lambda )$$, $$\mathcal{C}^{*}((1-\alpha)e^{-i\lambda} \cos\lambda)$$, $$\mathcal{CC}^{*}((1-\alpha)e^{-i\lambda} \cos\lambda)$$, $$0 \leq \alpha< 1$$, $$|\lambda|<\pi/2$$, are the classes of λ-spirallike functions of order α, analytic functions for which $$zf'(z)$$ is λ-spirallike of order α and λ-spiral close-to-convex functions of order α, respectively .

Finally, the aim of this investigation is to obtain some properties of the class of harmonic functions defined by

\begin{aligned} \begin{aligned} \mathcal{S}_{\mathcal{HCC}(b)} ={}& \biggl\{ f = h(z) + \overline{g(z)} \Big| w(z) = \frac{g'(z)}{h'(z)} \prec b_{1} \frac{1+(2b-1)z}{1-z} \\ & \Leftrightarrow \operatorname{Re} \biggl[ 1 + \frac{1}{b} \biggl( \frac{g'(z)}{h'(z)} - b_{1} \biggr) \biggr]>0, b, b_{1} \in \mathbb{C} / \{0\}, h(z)\in\mathcal {C}(b) \biggr\} \end{aligned} \end{aligned}

for all z in $$\mathbb{D}$$.

For the purpose of this paper, we need the following lemma and theorem.

### Lemma 1.2



Let $$\phi(z)$$ be regular in the unit disc $$\mathbb{D}$$ with $$\phi(0)=0$$. If the maximum value of $$|\phi(z)|$$ on the circle $$|z|=r<1$$ is attained at point $$z_{1}$$, then we have $$z_{1} \phi'(z_{1}) = k \phi(z_{1})$$ for some $$k \geq1$$.

### Theorem 1.3



If $$s(z) \in\mathcal{C}(b)$$, then

$$2 \biggl[ 1 + \frac{1}{b} \biggl( z\frac{s'(z)}{s(z)} - 1 \biggr) \biggr] - 1 =p(z) = \frac{1+\phi(z)}{1-\phi(z)}$$

for some $$\phi(z)\in\Omega$$ and every z in $$\mathbb{D}$$, and

$$\int_{0}^{2\pi}\operatorname{Re} \biggl( z \frac {s'(z)}{s(z)} \biggr)\, d \theta= 2pn\pi$$
(1.5)

for every $$z\in\mathbb{D}$$. A member of $$\mathcal{S}^{*}(p,n)$$ is called p-valent starlike function in the unit disc $$\mathbb{D}$$.

Finally, a planar harmonic mapping in the open unit disc $$\mathbb{D}$$ is a complex-valued harmonic function f, which maps $$\mathbb{D}$$ onto some planar domain $$f(\mathbb{D})$$. Since $$\mathbb{D}$$ is a simply connected domain, the mapping f has a canonical decomposition $$f= h+ \overline{g}$$, where $$h(z)$$ and $$g(z)$$ are analytic in $$\mathbb {D}$$ and have the following power series expansion:

$$h(z) = z^{p} + a_{np+1} z^{np+1} + a_{np+2} z^{np+2} + \cdots+ a_{np+m} z^{np+m} + \cdots$$

and

$$g(z) = b_{np} z^{np} + b_{np+1} z^{np+1} + b_{np+2} z^{np+2} + \cdots+ b_{np+m} z^{np+m} + \cdots,$$

where $$|b_{np}|<1$$, $$p\geq1$$ and $$n\geq1$$ are integers, $$a_{np+m}, b_{np+m} \in\mathbb{C}$$ and every $$z\in\mathbb{D}$$. As usual, we call $$h(z)$$ the analytic part and $$g(z)$$ the co-analytic part of f, respectively, and let the class of such harmonic mappings be denoted by $$\mathcal{SH}(p, n)$$. Lewy  proved in 1936 that the harmonic mapping f is locally univalent in $$\mathbb{D}$$ if and only if its Jacobian $$J_{f} = |h'(z)|^{2} - |g'(z)|^{2}$$ is strictly positive in $$\mathbb{D}$$. In view of this result, locally univalent harmonic mappings in the open unit disc are either sense-reversing if $$|g'(z)| > |h'(z)|$$ or sense-preserving if $$|g'(z)| < |h'(z)|$$ in $$\mathbb{D}$$. Throughout this paper, we restrict ourselves to the study of sense-preserving harmonic mappings. We also note that an elegant and complete treatment theory of the harmonic mapping is given in Duren’s monograph .

The main aim of this paper is to investigate some properties of the following class:

\begin{aligned} \mathcal{S^{*}H}(p, n) =& \biggl\{ f = h + \overline{g} \in\mathcal {SH}(p, n) \Big| w(z) = \frac{g'(z)}{h'(z)} \prec b_{np} \frac{1+\phi (z)}{1-\phi(z)}, \\ & \phi(z)=z^{n} \psi(z), \psi(z)\in\Omega_{1}, h(z)\in \mathcal {S}^{*}(p, n), z\in\mathbb{D} \biggr\} \end{aligned}

and for this aim we need the following lemma.

### Lemma 1.4



Let $$w(z)=a_{n}z^{n} + a_{n+1}z^{n+1} + a_{n+1}z^{n+2} + \cdots$$ ($$a_{n}\neq0$$, $$n\geq1$$) be analytic in $$\mathbb{D}$$. If the maximum value of $$|w(z)|$$ on the circle $$|z|=r<1$$ is attained at $$z=z_{0}$$, then we have $$z_{0}w'(z_{0})=p w(z_{0})$$, where $$p\geq n$$ and every $$z\in\mathbb{D}$$.

## 2 Main results

### Lemma 2.1

Let $$h(z)$$ be an element of $$\mathcal {C}(b)$$, then

$$\mathcal{F}_{1} \biggl( \frac{1}{2}|b|, \frac{1}{2} \operatorname{Re} b, -r \biggr) \leq\bigl\vert h(z)\bigr\vert \leq\mathcal{F}_{1} \biggl( \frac{1}{2}|b|, \frac{1}{2} \operatorname{Re} b, r \biggr)$$
(2.1)

and

$$\mathcal{F}_{2} \bigl( \vert b\vert , \operatorname{Re} b, -r \bigr) \leq\bigl\vert h'(z)\bigr\vert \leq\mathcal{F}_{2} \bigl( \vert b\vert , \operatorname{Re} b, r \bigr),$$
(2.2)

where

$$\mathcal{F}_{1} \biggl( \frac{1}{2}|b|, \frac{1}{2} \operatorname{Re} b, -r \biggr) = \frac{(1+r)^{|b| - \operatorname {Re}b}}{(1-r)^{|b| + \operatorname{Re}b}}$$
(2.3)

and

$$\mathcal{F}_{2} \bigl( \vert b\vert , \operatorname{Re} b, r \bigr) = \frac{(1+r)^{\frac{1}{2}|b| - \frac{1}{2} \operatorname {Re}b}}{(1-r)^{\frac{1}{2} |b| + \frac{1}{2} \operatorname{Re}b}}.$$
(2.4)

These inequalities are sharp because the extremal function is $$h(z) = \frac{1}{(1-z)^{b}}$$ with $$z = \frac{r ( r - \frac{\overline{ b}}{b} )^{1/2}}{1 - r (\frac{\overline{b}}{b} )^{1/2}}$$.

### Proof

Using Theorem 1.3, the definition of class $$\mathcal{C}(b)$$ and the definition of the subordination principle, we obtain

$$z\frac{h'(z)}{h(z)} = \frac{1 + (b-1)\phi (z)}{1-\phi(z)} \quad \Rightarrow\quad z \frac{h'(z)}{h(z)} \prec\frac{1 + (b-1)z}{1-z}$$

or

$$\biggl\vert z\frac{h'(z)}{h(z)} - \frac {br^{2}}{1-r^{2}} \biggr\vert \leq\frac{|b|r}{1-r^{2}},$$
(2.5)

and similarly

$$z\frac{h''(z)}{h'(z)} = \frac{2b\phi (z)}{1-\phi(z)} \quad \Rightarrow \quad z \frac{h''(z)}{h'(z)} \prec\frac {2bz}{1-z}$$

or

$$\biggl\vert z\frac{h''(z)}{h'(z)} - \frac {2br^{2}}{1-r^{2}} \biggr\vert \leq\frac{2|b|r}{1-r^{2}}.$$
(2.6)

Using (2.5) and (2.6), we get (2.1) and (2.2), respectively. □

### Theorem 2.2

Let $$f=h(z)+\overline{g(z)}$$ be an element of $$\mathcal{S}_{\mathcal{HCC}(b)}$$, then

$$\frac{g(z)}{h(z)} \prec b_{1} \frac{1+(2b-1)z}{1-z}\quad (z\in \mathbb{D}).$$

### Proof

Since $$f=h(z)+\overline{g(z)}$$ is an element of $$\mathcal{S}_{\mathcal{HCC}(b)}$$, then we have

$$\frac{g'(z)}{h'(z)} \prec b_{1} \frac{1+(2b-1)z}{1-z} \quad \Leftrightarrow \quad \operatorname{Re} \biggl[ 1 + \frac{1}{b} \biggl( \frac{g'(z)}{h'(z)} - b_{1} \biggr) \biggr]> 0,$$

so

$$\frac{g'(z)}{h'(z)} = b_{1} \frac {1+(2b-1)\phi(z)}{1-\phi(z)}$$
(2.7)

for some $$\phi(z)\in\Omega$$ and every z in $$\mathbb{D}$$. Now, we define the function $$\phi(z)$$ by

$$\frac{g(z)}{h(z)} = b_{1} \frac{1+\phi(z)}{1-\phi(z)} \quad (z\in\mathbb{D}),$$

then $$\phi(z)$$ is analytic in $$\mathbb{D}$$ and $$\frac {g(z)}{h(z)}|_{z=0} = b_{1} = b_{1} \frac{1+\phi(0)}{1-\phi (0)}$$, then $$\phi(0)=0$$ and

$$w(z) = \frac{g'(z)}{h'(z)} = b_{1} \biggl( \frac{1+\phi(z)}{1-\phi(z)} + \frac{2z\phi(z)}{1-\phi(z)} \cdot \frac{1}{1+(b-1)\phi(z)} \biggr)\quad (z\in\mathbb{D}).$$

Now it is easy to realize that the subordination $$\frac {g'(z)}{h'(z)}\prec b_{1} \frac{1+(2b-1)z}{1-z}$$ is equivalent to $$|\phi (z)|<1$$ for all $$z\in\mathbb{D}$$. Indeed, assume to the contrary, that there exists $$z_{1}\in\mathbb{D}$$ such that $$|\phi(z_{1})| = 1$$. Then by Jack’s lemma (Lemma 1.4), $$z_{1} \phi'(z_{1}) = k\phi (z_{1})$$, $$k\geq1$$, for such $$z_{1} \in\mathbb{D}$$, we have

$$w(z_{1})=\frac{g'(z_{1})}{h'(z_{1})} = b_{1} \biggl( \frac{1+\phi (z_{1})}{1-\phi(z_{1})} + \frac{2k\phi(z_{1})}{1-\phi(z_{1})} \cdot\frac {1}{1+(b-1)\phi(z_{1})} \biggr) = w\bigl( \phi(z_{1})\bigr) \notin w(\mathbb{D})$$

because $$|\phi(z_{1})|=1$$ and $$k\geq1$$. But this is a contradiction to the condition $$\frac{g'(z)}{h'(z)}\prec b_{1}\frac{1+(2b-1)z}{1-z}$$, and so assumption is wrong, i.e., $$|\phi(z)|<1$$ for all $$z\in\mathbb{D}$$. □

### Corollary 2.3

Let $$f=h(z)+\overline{g(z)}$$ be an element of $$\mathcal{S}_{\mathcal{HCC}(b)}$$, then

\begin{aligned}& \mathcal{F}_{1} \biggl( \frac{1}{2} |b|, \frac{1}{2} \operatorname{Re}b,-r \biggr) \frac{|b_{1}|-2|b|r-|b_{1}-2b|r^{2}}{1-r^{2}} \\& \quad \leq \bigl\vert g(z)\bigr\vert \leq\mathcal{F}_{1} \biggl( \frac{1}{2} |b|, \frac{1}{2} \operatorname{Re}b, r \biggr) \frac{|b_{1}|+2|b|r+|b_{1}-2b|r^{2}}{1-r^{2}} \end{aligned}
(2.8)

and

\begin{aligned}& \mathcal{F}_{2} \bigl(|b|, \operatorname{Re}b,-r \bigr) \frac{|b_{1}|-2|b|r-|b_{1}-2b|r^{2}}{1-r^{2}} \\& \quad \leq\bigl\vert g'(z)\bigr\vert \leq\mathcal{F}_{2} \bigl(|b|, \operatorname{Re}b, r \bigr) \frac {|b_{1}|+2|b|r+|b_{1}-2b|r^{2}}{1-r^{2}} \end{aligned}
(2.9)

for all $$|z|=r<1$$, where $$\mathcal{F}_{1}$$ and $$\mathcal{F}_{2}$$ are defined by (2.3) and (2.4), respectively.

### Proof

Since $$f=h(z)+\overline{g(z)} \in\mathcal {S}_{\mathcal{HCC}(b)}$$, we have

$$\operatorname{Re} \biggl[ 1 + \frac{1}{b} \biggl( \frac{g'(z)}{h'(z)} - b_{1} \biggr) \biggr] > 0\quad \Leftrightarrow\quad \frac{g'(z)}{h'(z)} \prec b_{1} \frac{1 + (2b-1)z}{1-z}$$

or

$$\biggl\vert \frac{g'(z)}{h'(z)} - \frac{b_{1}+(2b-b_{1})r^{2}}{1-r^{2}} \biggr\vert \leq \frac{2|b|r}{1-r^{2}},$$

then

$$\frac{|b_{1}|-2|b|r-|b_{1}-2b|r^{2}}{1-r^{2}} \leq\frac{|g'(z)|}{|h'(z)|} \leq\frac{|b_{1}|+2|b|r+|b_{1}-2b|r^{2}}{1-r^{2}},$$
(2.10)

and using Theorem 2.2 we obtain

$$\biggl\vert \frac{g(z)}{h(z)} - \frac{b_{1}+(2b-b_{1})r^{2}}{1-r^{2}} \biggr\vert \leq \frac{2|b|}{1-r^{2}}$$

or

$$\frac{|b_{1}|-2|b|r-|b_{1}-2b|r^{2}}{1-r^{2}} \leq\frac{|g(z)|}{|h(z)|} \leq\frac{|b_{1}|+2|b|r+|b_{1}-2b|r^{2}}{1-r^{2}}$$
(2.11)

for all $$|z|=r<1$$. Considering Lemma 2.1, (2.10) and (2.11) together, we obtain (2.8) and (2.9). □

### Lemma 2.4

If $$f=h(z)+ \overline{g(z)} \in\mathcal {S}_{\mathcal{HCC}(b)}$$, then

\begin{aligned}& \frac{|b_{1}|-r}{1+|b_{1}|r} \leq\bigl\vert w(z)\bigr\vert \leq \frac{|b_{1}|+r}{1+|b_{1}|r}, \end{aligned}
(2.12)
\begin{aligned}& \frac{(1-r^{2})(1-|b_{1}|^{2})}{(1+|b_{1}|r)^{2}} \leq1 - \bigl\vert w(z)\bigr\vert ^{2} \leq\frac{(1-r^{2})(1-|b_{1}|^{2})}{(1-|b_{1}|r)^{2}}, \end{aligned}
(2.13)
\begin{aligned}& \frac{(1-r)(1+|b_{1}|)}{1-|b_{1}|r} \leq1 + \bigl\vert w(z)\bigr\vert \leq \frac{(1+r)(1+|b_{1}|)}{1+|b_{1}|r} \end{aligned}
(2.14)

and

$$\frac{(1-r)(1-|b_{1}|)}{1+|b_{1}|r} \leq1 - \bigl\vert w(z)\bigr\vert \leq \frac{(1+r)(1-|b_{1}|)}{1-|b_{1}|r}$$
(2.15)

for all $$|z|=r<1$$.

### Proof

Since $$f=h(z)+ \overline{g(z)} \in\mathcal {S}_{\mathcal{HCC}(b)}$$, it follows that

$$w(z) = \frac{g'(z)}{h'(z)} = \frac{b_{1} + 2 b_{2} z + \cdots}{1 + 2 a_{2} z + \cdots}\quad \text{so } w(0) = b_{1} \text{ and } \bigl\vert w(z)\bigr\vert <1.$$

So, the function

$$\phi(z) = \frac{w(z)-w(0)}{1-\overline{w(0)}w(z)} = \frac {w(z)-b_{1}}{1- \overline{b}_{1} w(z)} \quad (z\in\mathbb{D})$$

satisfies the conditions of Schwarz lemma. Therefore, we have

$$w(z) = \frac{b_{1} + \phi(z)}{1+\overline {b}_{1}\phi(z)} \quad \text{if and only if}\quad w(z) \prec \frac{b_{1} + z}{1+\overline{b}_{1}z}\quad (z\in\mathbb{D}).$$

On the other hand, the linear transformation $$\frac{b_{1}+z}{1+\overline {b}_{1}z}$$ maps $$|z|=r$$ onto the disc with the center $$C(r) = ( \frac{(1-r^{2}) \operatorname{Re}b_{1}}{1-|b_{1}|^{2}r^{2}}, \frac {(1-r^{2}) \operatorname{Im}b_{1}}{1-|b_{1}|^{2}r^{2}} )$$ and the radius $$\rho (r)=\frac{(1-|b_{1}|^{2})r}{1-|b_{1}|r^{2}}$$. Then we have (2.12), which gives (2.13), (2.14) and (2.15). □

### Corollary 2.5

Let $$f(z)$$ be an element of $$\mathcal {S}_{\mathcal{HCC}(b)}$$, then

$$\frac{(1-r^{2})(1-|b_{1}|)^{2}}{(1+|b_{1}|r)^{2}} \bigl(\mathcal{F}_{2} \bigl( \vert b\vert , \operatorname{Re} b, -r \bigr)\bigr)^{2} \leq J_{f} \leq \frac{(1-r^{2})(1-|b_{1}|)^{2}}{(1-|b_{1}|r)^{2}} \bigl(\mathcal{F}_{2} \bigl( \vert b\vert , \operatorname{Re} b, r \bigr)\bigr)^{2}$$

and

\begin{aligned}& \bigl(1-|b_{1}|\bigr) \int_{0}^{r} \frac{1-\rho }{1+|b_{1}|\rho} \mathcal{F}_{2} \bigl( \vert b\vert , \operatorname{Re} b, -\rho \bigr)\, d\rho \\& \quad \leq|f| \leq\bigl(1+|b_{1}|\bigr) \int _{0}^{r} \frac{1+\rho}{1+|b_{1}|\rho} \mathcal{F}_{2} \bigl( \vert b\vert , \operatorname{Re} b, \rho \bigr)\, d\rho \end{aligned}

for all $$|z|=r<1$$, where $$\mathcal{F}_{2}$$ is defined by (2.4).

### Proof

Since

$$\bigl(1-\bigl\vert w(z)\bigr\vert ^{2}\bigr)\bigl\vert h'(z)\bigr\vert ^{2} \leq J_{j} \leq\bigl(1+ \bigl\vert w(z)\bigr\vert ^{2}\bigr)\bigl\vert h'(z) \bigr\vert$$

and

$$\bigl(1-\bigl\vert w(z)\bigr\vert \bigr)\bigl\vert h'(z)\bigr\vert |dz| \leq|df| \leq\bigl(1+\bigl\vert w(z)\bigr\vert \bigr)\bigl\vert h'(z)\bigr\vert |dz|,$$

thus using Lemma 2.1 and Lemma 2.4 in the last two inequalities we obtain the desired result. □

### Theorem 2.6

Let $$f(z)$$ be an element of $$\mathcal {S}_{\mathcal{HCC}(b)}$$, then

$$\sum_{k=2}^{n} |b_{k}-b_{1}a_{k}|^{2} \leq\sum_{k=1}^{n-1} \bigl\vert b_{k} + b_{1} (2b-1)a_{k}\bigr\vert ^{2}.$$

### Proof

Using Theorem 2.2, we obtain the following relation:

$$\frac{g(z)}{h(z)} \prec b_{1} \frac{1+(2b-1)z}{1-z} \quad \Rightarrow \quad \frac {g(z)}{h(z)} = \frac{b_{1}+b_{1}(2b-1)\phi(z)}{1-\phi(z)}$$

or

$$g(z) - b_{1} h(z) = \bigl(g(z)+b_{1}(2b-1)h(z) \bigr)\phi(z) \quad \bigl(z\in\mathbb{D}, \phi(z)\in\Omega\bigr).$$
(2.16)

Equality (2.16) can be written in the following form:

$$\sum_{k=2}^{n} (b_{k}-b_{1}a_{k})z^{k} + \sum _{k=n+1}^{\infty}d_{k}z^{k} = \Biggl( \sum_{k=1}^{n-1} \bigl(b_{k} + b_{1} (2b-1)a_{k}\bigr)z^{k} \Biggr) \phi(z)\quad (z\in\mathbb{D}).$$
(2.17)

Since the last equality has the form $$f_{1}(z) = f_{2}(z) \phi(z)$$ with $$|\phi(z)|<1$$, it follows that

$$\frac{1}{2\pi} \int_{0}^{2\pi} \bigl\vert f_{1}\bigl(re^{i\theta}\bigr) \bigr\vert ^{2}\, d\theta\leq \frac{1}{2\pi} \int_{0}^{2\pi} \bigl\vert f_{2}\bigl(re^{i\theta}\bigr) \bigr\vert ^{2} \, d\theta$$
(2.18)

for each r ($$0< r<1$$). Expressing (2.18) in terms of the coefficients in (2.17), we obtain the inequality

$$\sum_{k=2}^{n} |b_{k} - b_{1}a_{k}|^{2}r^{2k}+ \sum_{k=n+1}^{\infty}|d_{k}|^{2}r^{2k} \leq\sum_{k=1}^{n-1}\bigl\vert b_{k} + b_{1}(2b-1)a_{k}\bigr\vert ^{2}r^{2k},$$
(2.19)

where $$d_{k}= (b_{k}-b_{1}a_{k}) - (b_{k}+b_{1}(2b-1)a_{k}) \phi(z)$$. By letting $$r\rightarrow1^{-}$$ in (2.19) we obtain the desired result. The proof of this method is due to Clunie . □

## References

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Correspondence to Melike Aydogan.

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