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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

  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].

  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 [6].

  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 [6].

  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 [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]. □

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Polatog̃lu, Y., Kahramaner, Y. & Aydogan, M. Harmonic mappings for which co-analytic part is a close-to-convex function of order b . J Inequal Appl 2015, 18 (2015). https://doi.org/10.1186/s13660-014-0543-x

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