Regions of variability for a subclass of analytic functions

1.   Introduction

Let $A$ be the class of functions $f$ analytic in the open unit disc $E = \{z:|z| < 1, \ z\in \mathbb{C} \}$ with the usual normalization $f(0) = f^{\prime }(0)-1 = 0$ and $S$ be the subclass of $A$ consisting of functions which are univalent in $E$. Consider that $A$ as a topological vector space endowed with the topology of uniform convergence over a compact subsets of $E$. Also let $\mathcal{B}$ denote the class of analytic functions $w$ in $E$ such that $|w(z)| < 1$ and $w(0) = 0$. A function $f$ is said to be subordinate to a function $\ g$ written as $f\prec g$, if there exists a Schwarz function $w\in \mathcal{B}$ such that $f\left(z\right) = g\left(w(z)\right)$. In particular if $g$ is univalent in $E$, then $f\left(0\right) = g\left(0\right)$ and $f\left(E\right) \subset g\left(E\right)$.

A set $D$ in the complex plane is said to be starlike with respect to a point $w_{0}$, an interior point of $D$ if the line segment with initial point $w_{0}$ lies entirely in $D$. If a function $f$ maps $E$ onto a domain that is starlike with respect to $w_{0}$, then we say that $f$ is starlike with respect to $w_{0}$. In the special case that $f$ is starlike function with respect to origin. The class of univalent starlike functions with respect to origin is denoted by $S^{\ast }$. The class of starlike functions with respect to the origin have been studied extensively, for some details ^[5,6].

In 1933 Spacek ^[19] extended the idea of starlike functions by using the logarithmic spirals instead of line segment. Let $\alpha \in \left(- \frac{\pi }{2}, \frac{\pi }{2}\right)$. The curve $\gamma _{\alpha }:t\rightarrow \exp \left(te^{i\alpha }\right), \ t\in \mathbb{R}$ and its rotation $e^{i\theta }\gamma _{\alpha }\left(t\right),$ $\theta \in \mathbb{R}$ are called $\alpha$-spirals. A domain $D\subset \mathbb{C}$ is said to be $\alpha$-spirallike with respect to the origin if the spiral with initial point $0$ to every point in $D$ lies in $D.$ A function $f\in A$ is spirallike if it maps $E$ onto a domain which is spirallike with respect to $0.$ The class of spirallike functions is denoted by $S_{\alpha }.$ In otherwise

Kim and Sugawa ^[10] introduced the notion of $\alpha$-argument. Consider that $\theta = \arg _{\alpha }w$ with $w\in e^{i\theta }\gamma _{\alpha }\left(\mathbb{R} \right).$ By using the $\alpha$-argument, it is clear that $f\in S_{\alpha }$ if and only if

For some details about the spirallike functions ^[1,5,6]. A function$\ f$ with $f(0) = f^{\prime }(0)-1 = 0$ is said to be Robertson functions if

The class of Robertson functions is denoted by $C_{\alpha }$ and defined by Robertson ^[17]. It is clear that $C_{0} = C, \ $usual class of convex functions. In the view of above discussions about the spirallike functions and Robertson functions, we have the following relation. Let $f\in A$. Then $f\in C_{\alpha }$ if and only if $zf^{\prime }\in S_{\alpha }$. By using the concept of Janowski ^[8], Sen et al. ^[18] studied the class $C_{\alpha }\left[ A, B\right]$. Let $A\in \mathbb{C},$ $B\in \lbrack -1, 0)\ $and $\alpha \in \left(-\frac{\pi }{2}, \frac{\pi }{2 }\right)$. Then $C_{\alpha }\left[ A, B\right]$ denotes the class of analytic functions $f$ in the open unit disc with $f\left(0\right) = 0 =$ $f^{\prime }\left(0\right) -1$ such that

with

The single-valued branch of logarithms for $f^{\prime }\left(z\right)$ when $f\in C_{\alpha }\left[ A, B\right]$ is denoted by $\log f^{\prime }\left(z\right)$ with $\log f^{\prime }\left(0\right) = 0.\ $Yanagihara ^[21,22] determined the region of variability for the class of convex functions. Ponnusamy et al. ^[13] found the region of variability for some subclasses of univalent functions. For some work on region of variability, ^{[7,11,12,13,14,15,16]} and references therein.

Using the Herglotz representation for Janowski functions, it can be seen that for $f\in C_{\alpha }\left[ A, B\right]$ there exists a unique positive unit measure $\mu$ in $(-\pi, \pi ]$ such that

This shows that

It follows from $\left(1.2\right)$ that for each fixed $z_{0}\in E,$ the region of variability is the set $V_{\lambda }\left[ z_{0}, A, B\right]$ given as

Let

Then there exists a Schwarz function $w\in \mathcal{B},$ such that

Let$\ f\in C_{\alpha }\left[ A, B\right]$. Then by applying Schwarz lemma, that is $\left\vert w_{p}^{\prime }\left(0\right) \right\vert \leq 1$ ^[4], we get

for some $\lambda \in \overline{E}$, Now for $\lambda \in \overline{E} = \left\{ z\in \mathbb{C} :\left\vert z\right\vert \leq 1\right\} \ $and $z_{0}\in E$, we introduce

and

The aim of this article is to investigate the region of variability $V_{\lambda }\left(z_{0}, A, B\right)$ for the class $f\in C_{\alpha }\left[ \lambda, A, B\right]$.

2.   Basic properties of $V_{\lambda }\left(z_{0}, A, B\right)$

We start our investigations by studying certain general properties of the set $V_{\lambda }\left(z_{0}, A, B\right)$ such as compactness and convexity.

Proposition 2.1. (i) $V_{\lambda }\left(z_{0}, A, B\right)$ is a compact subset of $\mathbb{C}$.

(ii) $V_{\lambda }\left(z_{0}, A, B\right)$ is a convex subset of $\mathbb{C}$.

(iii) If $|\lambda | = 1$ or $z_{0} = 0$, then $V_{\lambda }\left(z_{0}, A, B\right) = \left\{ \frac{A-B}{B}e^{-i\alpha }\cos \alpha \log (1+B\lambda z_{0})\right\}$ and if $|\lambda | < 1$ and $z_{0}\neq 0$, then the set $V_{\lambda }\left(z_{0}, A, B\right)$ has $e^{-i\alpha }\cos \alpha \log (1+B\lambda z_{0})$ an interior point.

Proof. (ⅰ) Since $C_{\alpha }\left[ \lambda, A, B\right]$ is a compact subset of $\mathbb{C},$ therefore $V_{\lambda }\left(z_{0}, A, B\right)$ is also compact.

(ⅱ) Let $f_{1}, f_{2}\in C_{\alpha }\left[ \lambda, A, B\right].$ Then for $0\leq t\leq 1$, the function

is also in $C_{\alpha }\left[ \lambda, A, B\right],$ therefore $V_{\lambda }\left(z_{0}, A, B\right)$ is convex because$\ \log f^{\prime }\left(z\right) = \left(1-t\right) \log f_{1}^{\prime }\left(z\right) +t\log f_{2}^{\prime }\left(z\right),$ $t\in \left[ 0, 1\right]$.

(ⅲ)Since $|\lambda | = |w_{f}^{\prime }(0)| = 1,$ then from Schwarz lemma, we obtain $w_{f}(z) = \lambda z,$ which yields $p(z) = \frac{1+\lambda Az }{1+\lambda Bz}$. This implies that

Therefore

This also trivially holds true when $z_{0} = 0$. For $\lambda \in E$ and $\alpha \in \overline{E}$, set

Then $F_{a, \lambda }\left(z\right)$ is in $C_{\alpha }\left[ \lambda, A, B \right]$ and $w_{f}(z) = z\delta (az, \lambda)$. For fixed $\lambda \in E$ and $z_{0}\in E\backslash \{0\}$ the function

is a non-constant analytic function of $a\in E,$ and therefore is an open mapping. Hence $\log F_{0, \lambda }^{\prime }\left(z\right) = \left\{ \frac{ A-B}{B}e^{-i\alpha }\cos \alpha \log (1+B\lambda z_{0})\right\}$ is an interior point of $\left\{ \log F_{a, \lambda }^{\prime }\left(z\right) :a\in E\right\} \subset V_{\lambda }\left(z_{0}, A, B\right)$.

Keeping in view the above proposition, it is sufficient to find $V_{\lambda }\left(z_{0}, A, B\right)$ for $0\leq \lambda < 1$ and $z_{0}\in E\backslash \{0\}$.

3.   Main results

In this section, we state and prove some results which are needed in the proof of our main theorem.

In the following proposition, we prove that for $f\in C_{\alpha }\left[ \lambda, A, B\right],$ the ratio $f^{\prime \prime }\left(z\right) /f^{\prime }\left(z\right)$ is contained in a closed disc with center $q\left(z, \lambda \right)$ and radius $r\left(z, \lambda \right).$

Proposition 3.1. For $C_{\alpha }\left[ \lambda, A, B\right]$, we have

where

Proof. Since $f\in C_{\alpha }\left[ \lambda, A, B\right]$. Then by using Schwarz lemma for $w_{p}\in \mathcal{B}$ with $w_{p}^{\prime }\left(0\right) = \lambda$ such that

Now from $\left(1.3\right)$ this can be written equivalently as

where

This is equivalent to

Now after simple calculations, we have

Also

and

By setting

The relation $\left(3.1\right)$ occurs from $\left(3.6\right)$ and the above relations. Equality is attained in $\left(3.1\right)$ when $f = F_{e^{i\theta }, \lambda }(z),$ for some $z\in E.$ Conversely if equality occurs in $\left(3.1\right)$ for some $z\in$ $E\backslash \{0\}$, then equality must hold in $\left(3.3\right)$. Thus by Schwarz lemma there exists $\theta \in \mathbb{R}$ such that $w_{f}(z) = z\delta (e^{i\theta }z, \lambda)$ for all $z\in E$. This implies $f = F_{e^{i\theta }, \lambda }$.

Geometrically the above proposition means that the functional log$f^{\prime }$ lies in the closed disk centred at $q(z, \lambda)$ with radius $r(z, \lambda)$.

For $\lambda = 0,$ we have the following special result which gives us bounds on pre-Schwarzian norm of locally univalent functions.

Corollary 3.2. Let $f\in C_{\alpha }\left[ 0, A, B\right]$. Then

Therefore

Since $\left\vert B\right\vert \leq 1, \ $so

The pre-Schwarzian norm for locally univalent functions is defined as

It is well-known that $\left\Vert f\right\Vert \leq 6, \ $if $f$ is univalent$.\ $Becker and Pommerenke ^[2] proved that if $\left\Vert f\right\Vert \leq 1,$ then $f$ is univalent in $E$ and this bound is sharp. Yamashita ^[20] proved that if $f$ is convex, then $\left\Vert f\right\Vert \leq 1.$ The norm estimates for some subclasses of univalent functions are studied by many authors. For some details ^[3,9]. From Corollary 3.2, it is evident that for $f^{\prime \prime }\left(0\right)$ and $A = 1, B = -1$, we have $\left\Vert f\right\Vert \leq 2\cos \alpha$ for Robertson functions. This result was proved by Ponnusamy et al. ^[15] also for $\alpha = 0$, we have $\left\Vert f\right\Vert \leq 2$.

In the following result, we prove that the set $V_{\lambda }\left(z_{0}, A, B\right)$ is contained in a closed disc with centre $Q\left(\lambda, r\right)$ and radius $R\left(\lambda, r\right).$

Corollary 3.3. Consider the curve $\gamma :z\left(t\right), \ 0\leq t\leq 1\ $in $E$ with $z\left(0\right) = 0\ $and $z\left(1\right) = z_{0}, \ $then

with

where $q\left(z, \lambda \right)$ and$\ r\left(z, \lambda \right)$ are given in Proposition 3.1.

Proof. Suppose that $f\in C_{\alpha }\left[ \lambda, A, B\right]$, then from proposition 3.1, we get

Now using proposition 3.1, we get

This shows $\log f^{\prime }\left(z_{0}\right) \in \ \overline{D}\left(Q\left(\lambda, r\right), R\left(\lambda, r\right) \right)$. Hence the required result.

We need the following lemma which ensures the existence of normalized starlike function which is useful in the proof of next result.

Lemma 3.4. For $\theta \in \mathbb{R}$ and $|\lambda | < 1$, the function

has zeros of order $2$ at the origin and no zero elsewhere in $E.$ Moreover, there exists a starlike normalized univalent function $G_{0}$ in $E$ such that $G = \frac{1}{2}e^{i\theta }G_{0}^{2}$.

The above lemma is due to Ponnusamy et al. ^[7]. In the below proposition we show that $\log F_{e^{i\theta }, \lambda }^{\prime }\left(z_{0}\right)$ lies on the boundary of the set $V_{\lambda }\left(z_{0}, A, B\right)$.

Proposition 3.5. Let $z_{0}\in E\backslash \left\{ 0\right\}$. Then for $\theta \in (-\pi, \pi ], \ $we have $\log F_{e^{i\theta }, \lambda }^{\prime }\left(z_{0}\right) \in \partial V_{\lambda }\left(z_{0}, A, B\right)$. Further if $\log f^{\prime }\left(z_{0}\right) = \log F_{e^{i\theta }, \lambda }^{^{\prime }}\left(z_{0}\right)$ for $f\in C_{\alpha }\left[ \lambda, A, B \right], \ $then $f = F_{e^{i\theta }, \lambda }$.

Proof. Using 2.1, we have

Therefore

From $\left(3.5\right)$, it follows that

Therefore

Putting $a = e^{i\theta }$, we get

By using Lemma 3.4, we obtain

Using the argument of Lemma 3.4 that $G = 2^{-1}e^{i\theta }G_{0}^{2}$, where $G_{0}$ is starlike in $E$ with $G_{0}\left(0\right) = G_{0}^{\prime }\left(0\right) -1 = 0,$ for any $z_{0}\in E\backslash \{0\}$ the linear segment joining $0$ and $G_{0}(z_{0})$ lies entirely in $G_{0}(E)$. Let $\gamma _{0}$ be the curve defined by

Since $G(z(t)) = 2^{-1}e^{i\theta }\left(G_{0}(z(t))\right) ^{2} = 2^{-1}e^{i\theta }\left(tG_{0}(z_{0})\right) ^{2} = t^{2}G(z_{0}).$ Differentiation w.r.t $t$ gives us

Therefore

This relation together with $(3.7)$, we get

This shows that $\log F_{e^{i\theta }, \lambda }^{\prime }\left(z\right) \in \partial E\left(Q\left(\gamma _{0}, \lambda \right), R\left(\gamma _{0}, r\right) \right), \ $where $Q(\lambda, \gamma _{0})$ and $R(\lambda, \gamma _{0})$ are defined as in Corollary 3.3. Also we have $\log F_{e^{i\theta }, \lambda }^{\prime }\left(z_{0}\right) \in V_{\lambda }\left(z_{0}, A, B\right)$, therefore $\log F_{e^{i\theta }, \lambda }^{\prime }\left(z_{0}\right) \in \partial V_{\lambda }\left(z_{0}, A, B\right)$.

Now we have to prove $\log f^{^{\prime }}\left(z_{0}\right) = \log F^{^{\prime }}\left(z_{0}\right) \ $for some $f\in C_{\alpha }\left[ \lambda, A, B\right]$, we have

where $\gamma _{0}:z(t), \ 0\leq t\leq 1$. Then the function $h$ is continuous and

Using Proposition 3.1, we have

Now using Proposition 3.1, we get $\left\vert h(t)\right\vert \leq r(z(t), \lambda)|z^{\prime }(t)|$. Further from $\left(3.9\right)$, we have From $\left(3.7\right)$ and $\left(3.8\right)$, this implies that $\frac{f^{\prime \prime }\left(z\right) }{f^{\prime }\left(z\right) } = \frac{F_{e^{i\theta }, \lambda }^{\prime \prime }\left(z\right) }{ F_{e^{i\theta }, \lambda }^{\prime }\left(z\right) }$ on $\gamma _{0}$. The identity theorem for analytic functions yields us $f = F_{e^{i\theta }, \lambda },$ $z\in E$.

Main theorem

In our main result, we give precise description of regions of variability for the class $C_{\alpha }\left[ \lambda, A, B\right]$ and show that the boundary $\partial V_{\lambda }(z_{0}, A, B)$ is a Jordan curve.

Theorem 3.6. Let $\lambda \in E$ and $z_{0}\in E\backslash \{0\}.$ Then boundary $\partial V_{\lambda }(z_{0}, A, B)$ is the Jordan curve given by

If $\log f^{\prime }\left(z_{0}\right) = \log F_{e^{i\theta }, \lambda }^{\prime }\left(z_{0}\right)$ for some $f\in C_{\alpha }\left[ \lambda, A, B\right]$ and $\theta \in (-\pi, \pi ]$, then $f\left(z\right) = F_{e^{i\theta }, \lambda }(z)$.

Proof. First we have to show that the curve

is simple. Let us assume that

for some $\theta _{1, }\theta _{2}\in (-\pi, \pi ]$ with $\theta _{1}\neq \theta _{2}.$ Then the use of Proposition 3.5 yield us that $F_{e^{i\theta _{1}}, \lambda }^{\prime }(z_{0}) = F_{e^{i\theta _{2}}, \lambda }^{\prime }(z_{0})$, which further gives the following relation

This implies that

After some simplification, we obtain $ze^{i\theta _{1}} = ze^{i\theta _{2}},$ which leads us to a contradiction. Hence the curve is simple. Since $V_{\lambda }(z_{0}, A, B)$ is compact convex subset of $\mathbb{C}$ and has non-empty interior, therefore the boundary $\partial V_{\lambda }(z_{0}, A, B)$ is a simple closed curve. From Proposition 3.5 the curve $\partial V_{\lambda }(z_{0}, A, B)$ contains the curve $(-\pi, \pi ]\ni \theta \mapsto \log F_{e^{i\theta }, \lambda }\left(z_{0}\right).$ Since a simple closed curve cannot contain any simple closed curve other than itself. Thus $\partial V_{\lambda }(z_{0}, A, B)$ is given by $(-\pi, \pi ]\ni \theta \mapsto \log F_{e^{i\theta }, \lambda }^{\prime }\left(z_{0}\right)$.

Geometric view of theorem

The following figures show us the geometric view of our main theorem with various choices of involved parameters.

Acknowledgments

Authors are thankful to the editor and anonymous referees for their valuable comments and suggestions.

Conflict of interest

The authors declare no conflicts of interest.