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 [1].

Lewy [2] 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 $\mathcal{S} \subset\mathcal{S}_{\mathcal{H}}^{0} \subset\mathcal{S}_{\mathcal {H}}$ [1]. 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 [7].

Remark 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 [6].

• (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 [6].

• (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 [6].

• (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 [6].

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

[8]

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

[9]

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 [2] 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 [1].

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

[1]

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 [10]. □