On certain subclasses of close-to-convex functions related with the second-order differential subordination

TL;DR

This paper introduces new subclasses of close-to-convex functions defined via second-order differential subordination, analyzing their properties, including real part bounds and univalence radius, expanding understanding of these function classes.

Contribution

It defines and studies new subclasses of close-to-convex functions related to second-order differential subordination, providing bounds and univalence radius results.

Findings

Re{f'(z)} > eta for functions in alR(lpha,eta)

Re{f(z)/z} > eta for functions in alR(lpha,eta)

Univalence radius for 2nd section sum of functions in alR(lpha,eta)

Abstract

Let $\mathcal{A}$ be the family of analytic and normalized functions in the open unit disc $|z|<1$. In this article we consider the following classes \begin{equation*} \mathcal{R}(\alpha,\beta):=\left\{ f\in \mathcal{A}: {\rm Re}\left\{f'(z)+\frac{1+e^{i\alpha}}{2}zf''(z)\right\}>\beta,\, |z|<1\right\} \end{equation*} and \begin{equation*} \mathcal{L}_\alpha(b):=\left\{f\in\mathcal{A}:\left|f'(z) +\frac{1+e^{i\alpha}}{2}zf''(z)-b\right|< b,\, |z|<1 \right\}, \end{equation*} where $-\pi<\alpha\leq \pi$, $0\leq \beta<1$ and $b>1/2$. We show that if $f\in \mathcal{R}(\alpha,\beta)$, then ${\rm Re}\{f'(z)\}$ and ${\rm Re}\{f(z)/z\}$ are greater than $\beta$, and if $f\in\mathcal{L}_\alpha(b)$, then $0<{\rm Re}\{f'(z)\}<2b$. Also, some another interesting properties of the class $\mathcal{L}_\alpha(b)$ are investigated. Finally, the radius of univalence of 2-th section sum of $f\in \mathcal{R}(\alpha,\beta)$ is obtained.

Figures

Equations

$$\mathcal{R}(\alpha ,\beta ):=\left\{f\in \mathcal{A}:\mathrm{Re}\left\{f^{\prime }(z)+\frac{1+e^{i\alpha }}{2}z{f}^{\prime \prime }(z)\right\}>\beta ,|z|<1\right\}$$

$${\mathcal{L}}_{\alpha }(b):=\left\{f\in \mathcal{A}:\left|f^{\prime }(z)+\frac{1+e^{i\alpha }}{2}z{f}^{\prime \prime }(z)-b\right|<b,|z|<1\right\},$$

$$f(z)\prec g(z)\iff f(0)=g(0)\mathrm{and}f(\Delta )\subset g(\Delta ).$$

$$\mathrm{Re}\left\{e^{i\delta }\frac{f^{\prime }(z)}{g^{\prime }(z)}\right\}>0(z\in \Delta ).$$

$$\mathcal{C}:=\left\{f\in \mathcal{A}:\exists \delta \in \mathbb{R};\mathrm{Re}\left\{e^{i\delta }{f}^{\prime }(z)\right\}>0,z\in \Delta \right\}.$$

$${\mathcal{L}}_{\alpha }:=\left\{f\in \mathcal{A}:\mathrm{Re}\left\{f^{\prime }(z)+\frac{1+e^{i\alpha }}{2}z{f}^{\prime \prime }(z)\right\}>0,z\in \Delta \right\}$$

$${\mathcal{L}}_{\alpha }(b):=\left\{f\in \mathcal{A}:\left|f^{\prime }(z)+\frac{1+e^{i\alpha }}{2}z{f}^{\prime \prime }(z)-b\right|<b,z\in \Delta \right\},$$

$${\varphi }_b(z)=\frac{1+z}{1+\left(\frac{1}{b}-1\right)z}(z\in \Delta ,b>1/2),$$

$$f^{\prime }(z)+\frac{1+e^{i\alpha }}{2}z{f}^{\prime \prime }(z)\prec {\varphi }_b(z)(z\in \Delta ),$$

$$f^{\prime }(z)+\frac{1+e^{i\alpha }}{2}z{f}^{\prime \prime }(z)\prec \phi (z)(z\in \Delta ),$$

$$\mathrm{Re}\left\{f^{\prime }(z)+\gamma z{f}^{\prime \prime }(z)\right\}>0(z\in \Delta ),$$

$$R(\beta ,\gamma )=\left\{f\in \mathcal{A}:\mathrm{Re}\left\{f^{\prime }(z)+\gamma z{f}^{\prime \prime }(z)\right\}>\beta ,\gamma >0,\beta <1,z\in \Delta \right\}.$$

$$\mathrm{Re}\left\{f^{\prime }(z)+\frac{1+e^{i\alpha }}{2}z{f}^{\prime \prime }(z)\right\}>\beta (z\in \Delta ),$$

$$\mathcal{C}(\beta ):=\left\{f\in \mathcal{A}:\mathrm{Re}\{f^{\prime }(z)\}>\beta ,z\in \Delta ,0\le \beta <1\right\}.$$

$$f^{\prime }(z)+\frac{1+e^{i\alpha }}{2}z{f}^{\prime \prime }(z)\prec \frac{1+(1-2\beta )z}{1-z}(z\in \Delta ,0\le \beta <1,-\pi <\alpha \le \pi ).$$

$$\psi (i\rho ,\sigma ;z)\notin \Xi$$

$$\psi (p(z),z{p}^{\prime }(z);z)\in \Xi ,$$

$$f_x(z)=z+4(1-\beta )\sum _{n=2}^{\infty }\frac{x^{n-1}}{n[n+1+(n-1)e^{i\alpha }]}{z}^n(|x|=1).$$

$$|a_n|\le \frac{4(1-\beta )}{n|n+1+(n-1)e^{i\alpha }|}=\frac{4(1-\beta )}{n\sqrt{2[n^2+1+(n^2-1)\cos \alpha ]}}(n\ge 2),$$

$$\mathrm{Re}\{f^{\prime }(z)\}>\beta (0\le \beta <1).$$

$$p(z)=\frac{1}{1-\beta }\left(f^{\prime }(z)-\beta \right)(0\le \beta <1).$$

$$f^{\prime }(z)+\frac{1+e^{i\alpha }}{2}z{f}^{\prime \prime }(z)=(1-\beta )[p(z)+(1+e^{i\alpha })z{p}^{\prime }(z)/2]+\beta =\varphi (p(z),z{p}^{\prime }(z);z),$$

$$\{\varphi (p(z),z{p}^{\prime }(z);z):z\in \Delta \}\subset \{w:\mathrm{Re}\{w\}>\beta \}=:{\Omega }_{\beta }.$$

$$\mathrm{Re}\{\varphi (i\rho ,\sigma ;z)\}$$

$$=(1-\beta )(1+\cos \alpha )\sigma /2+\beta$$

$$\le \beta -\frac{(1-\beta )}{4}(1+\cos \alpha )(1+{\rho }^2)$$

$$\le \beta .$$

$$\mathrm{Re}\left\{\frac{f(z)}{z}\right\}>\beta (0\le \beta <1).$$

$$\text{textsl}p(z):=\frac{1}{1-\beta }\left(\frac{f(z)}{z}-\beta \right).$$

$$f^{\prime }(z)=\beta +(1-\beta )\text{textsl}p(z)+(1-\beta )z\text{textsl}{p}^{\prime }(z)$$

$$f^{\prime \prime }(z)=2(1-\beta )\text{textsl}{p}^{\prime }(z)+(1-\beta )z\text{textsl}{p}^{\prime \prime }(z).$$

$$f^{\prime }(z)+\frac{1+e^{i\alpha }}{2}z{f}^{\prime \prime }(z)$$

Full text

[TABLE]

and

[TABLE]

where $-\pi<\alpha\leq\pi$, $0\leq\beta<1$ and $b>1/2$. We show that if $f\in\mathcal{R}(\alpha,\beta)$, then ${\rm Re}\{f^{\prime}(z)\}$ and ${\rm Re}\{f(z)/z\}$ are greater than $\beta$, and if $f\in\mathcal{L}_{\alpha}(b)$, then $0<{\rm Re}\{f^{\prime}(z)\}<2b$. Also, some another interesting properties of the class $\mathcal{L}_{\alpha}(b)$ are investigated. Finally, the radius of univalence of 2-th section sum of $f\in\mathcal{R}(\alpha,\beta)$ is obtained.

*Corresponding Author

1. Introduction

Let $\Delta:=\{z\in\mathbb{C}:|z|<1\}$ where $\mathbb{C}$ is the complex plane. We denote by $\mathcal{B}$ the class of all analytic functions $w(z)$ in $\Delta$ with $w(0)=0$ and $|w(z)|<1$, and denote by $\mathcal{A}$ the class of all functions that are analytic and normalized in $\Delta$. The subclass of $\mathcal{A}$ consisting of univalent functions in $\Delta$ is denoted by $\mathcal{S}$. For functions $f$ and $g$ belonging to the class $\mathcal{A}$, we say that $f$ is subordinate to $g$ in the unit disk $\Delta$, written $f(z)\prec g(z)$ or $f\prec g$, if and only if there exists a function $w\in\mathcal{B}$ such that $f(z)=g(w(z))$ for all $z\in\Delta$. In particular, if $g$ is univalent function in $\Delta$, then we have the following relation

[TABLE]

Denote by $\mathcal{S}^{*}$ and $\mathcal{K}$ the set of all starlike and convex functions in $\Delta$, respectively. A function $f\in\mathcal{A}$ is said to be close-to-convex, if there exists a convex function $g$ and $\delta\in\mathbb{R}$ such that

[TABLE]

The functions class which satisfy the last condition was introduced by Kaplan in [6] and we denote by $\mathcal{CK}$. It is clear that if we take $g(z)\equiv z$ in the class $\mathcal{CK}$, then we have the Noshiro-Warschawski class as follows

[TABLE]

By the basic Noshiro-Warschawski lemma [1, §2.6], we have $\mathcal{C}\subset\mathcal{S}$.

Here, we recall from [15], two certain subclasses of analytic functions as follows

[TABLE]

and

[TABLE]

where $\alpha\in(-\pi,\pi]$ and $b>1/2$. Notice that if $b\rightarrow\infty$, then $\mathcal{L}_{\alpha}(b)\rightarrow\mathcal{L}_{\alpha}$. Also, $\mathcal{L}_{\pi}$ contains $\mathcal{L}_{\alpha}$ for each $\alpha$. On the other hand, Trojnar-Spelina [19] showed that $\mathcal{L}_{\alpha}(b)\subset\mathcal{L}_{\pi}$, for every $\alpha\in(-\pi,\pi]$ and $b\geq 1$.

By definition of subordination and this fact that the image of the function

[TABLE]

is $\{w\in\mathbb{C}:|w-b|<b\}$ (see Figure 1 for $b=3/2$), we have the following lemma.

Lemma 1.1**.**

(see [19])* A necessary and sufficient condition for $f$ to be in the class $\mathcal{L}_{\alpha}(b)$ is*

[TABLE]

where $\phi_{b}$ is given by (1.1).

It is necessary to point out that the class $\mathcal{R}_{\alpha}(\varphi)$ including of all normalized analytic functions in $\Delta$ satisfying the following differential subordination

[TABLE]

was studied extensively by Srivastava $et$ $al.$ (see [16]), where the function $\varphi$ is analytic in the open unit disc $\Delta$ such that $\varphi(0)=1$. Also, Chichra [2] studied the class of all functions whose derivative has positive real part in the unit disc $\Delta$. Indeed, he denoted by $\mathcal{F}_{\gamma}$ the class of functions $f\in\mathcal{A}$ which satisfying the following inequality

[TABLE]

where $\gamma\geq 0$, and showed that $\mathcal{F}_{\gamma}\subset\mathcal{S}$. Also, he proved that if $f\in\mathcal{F}_{\gamma}$ and ${\rm Re}\{\gamma\}\geq 0$, then ${\rm Re}\{f^{\prime}(z)\}>0$ in $\Delta$. Recent result, also was obtained by Lewandowski et al. in [7].

On the other hand, Gao and Zhou [3] considered the class $R(\beta,\gamma)$ as follows:

[TABLE]

They found the extreme points of $R(\beta,\gamma)$, some sharp bounds of certain linear problems, the sharp bounds for ${\rm Re}\{f^{\prime}(z)\}$ and ${\rm Re}\{f(z)/z\}$ and determined the number $\beta(\gamma)$ such that $R(\beta,\gamma)\subset\mathcal{S}^{*}$, where $\gamma$ is certain fixed number in $[1,\infty)$. Also, the class $R(\beta,\gamma)$ was studied by Ponnusamy and Singh when ${\rm Re}\{\gamma\}>0$, see [11].

Motivated by the above classes, we define the class of all functions $f\in\mathcal{A}$, denoted by $\mathcal{R}(\alpha,\beta)$ which satisfy the condition

[TABLE]

where $0\leq\beta<1$ and $-\pi<\alpha\leq\pi$. It is obvious that $\mathcal{R}(\pi,\beta)$ becomes the class $\mathcal{C}(\beta)$, where

[TABLE]

The class $\mathcal{C}(\beta)$ was considered in [4] and $\mathcal{C}(\beta)\subset\mathcal{S}$ when $0\leq\beta<1$. It follows from [2, Theorem 5] that $\mathcal{R}(\alpha,0)\subset\mathcal{R}(\pi,0)\equiv\mathcal{C}(0)\equiv\mathcal{C}$. The class $\mathcal{R}(0,0)$ studied by Singh and Singh [12], and they showed that $\mathcal{R}(0,0)\subset\mathcal{S}^{*}$ [13]. Also, they found for $f\in\mathcal{R}(0,0)$ and $z\in\Delta$ that ${\rm Re}\{f(z)/z\}>1/2$ and $\mathcal{R}(0,\beta)\subset\mathcal{S}^{*}$ for $\beta\geq-1/4$. Silverman in [14] improved this lower bound. He showed that $\mathcal{R}(0,\beta)\subset\mathcal{S}^{*}$ for $\beta\geq-0.2738$ and also found the smallest $\beta$ ($\beta\geq-0.63$) for which $\mathcal{R}(0,\beta)\subset\mathcal{S}$.

Since the function $z\mapsto(1+(1-2\beta)z)/(1-z)$ $(z\in\Delta,0\leq\beta<1)$ is univalent and maps $\Delta$ onto the right half plane, having real part greater than $\beta$, we have the following lemma directly. With the proof easy, the details are omitted.

Lemma 1.2**.**

A function $f\in\mathcal{A}$ belongs to the class $\mathcal{R}(\alpha,\beta)$ if, and only if,

[TABLE]

To prove of our main results we need the following lemma.

Lemma 1.3**.**

[10, p. 35]** Let $\Xi$ be a simply connected domain in the complex plane $\mathbb{C}$ and let $t$ be a complex number such that ${\rm Re}\{t\}>0$. Suppose that a function $\psi:\mathbb{C}^{2}\times\Delta\rightarrow\mathbb{C}$ satisfies the condition

[TABLE]

for all real $\rho,\sigma\leq-\mid t-i\rho\mid^{2}/(2{\rm Re}\,t)$ and all $z\in\Delta$. If the function $p(z)$ defined by $p(z)=t+t_{1}z+t_{2}z^{2}+\cdots$ is analytic in $\Delta$ and if

[TABLE]

then ${\rm Re}\{p(z)\}>0$ in $\Delta$.

This paper is organized as follows. In Section 2 some properties of the classes $\mathcal{R}(\alpha,\beta)$ and $\mathcal{L}_{\alpha}(b)$ are studied. In Section 3 we obtain the radius of univalence of 2-th section sum of $f\in\mathcal{R}(\alpha,\beta)$ and we conjecture that this radius is for every section sum of the function $f$ that belonging to the class $\mathcal{R}(\alpha,\beta)$.

2. On the classes $\mathcal{R}(\alpha,\beta)$ and $\mathcal{L}_{\alpha}(b)$

At first, applying Hergoltz’s Theorem [1, p. 21] we obtain the extreme points of $\mathcal{R}(\alpha,\beta)$ as follows:

[TABLE]

Since the coefficient bounds are maximized at an extreme point, as an application of (2.1), we have

[TABLE]

where $0\leq\beta<1$ and $-\pi<\alpha\leq\pi$. Equality occurs for $f_{x}(z)$ defined by (2.1).

To prove the first result of this section, i.e. Theorem 2.1, also Theorem 2.2 and Theorem 2.3, we employ the same technique as in [5, Theorem 2.1].

Theorem 2.1**.**

Let $\beta\in[0,1)$ and $\alpha\in(-\pi,\pi]$. If $f\in\mathcal{A}$ belongs to the class $\mathcal{R}(\alpha,\beta)$, then

[TABLE]

This means that $\mathcal{R}(\alpha,\beta)\subset\mathcal{C}(\beta)$.

Proof.

Let $f^{\prime}(z)\neq 0$ for $z\neq 0$ and $p(z)$ be defined by

[TABLE]

Then $p(z)$ is analytic in $\Delta$, $p(0)=1$ and

[TABLE]

where $\phi(r,s;z):=(1-\beta)[r+(1+e^{i\alpha})s/2]$. Since $f\in\mathcal{R}(\alpha,\beta)$, we define the set $\Omega_{\beta}$ as follows:

[TABLE]

For all real $\rho$ and $\sigma$, that $\sigma\leq-(1+\rho^{2})/2$, we get

[TABLE]

This shows that ${\rm Re}\{\phi(p(z),zp^{\prime}(z);z)\}\not\in\Omega_{\beta}$. Thus by Lemma 1.3, we get ${\rm Re}\{p(z)\}>0$ or ${\rm Re}\{f^{\prime}(z)\}>\beta$. This means that $f\in\mathcal{C}(\beta)$ and concluding the proof. ∎

Taking $\beta=0$ in the above Theorem 2.1, we get.

Corollary 2.1**.**

If $f\in\mathcal{L}_{\alpha}$, then ${\rm Re}\{f^{\prime}(z)\}>0$ $(z\in\Delta)$ and thus $f$ is univalent.

Remark 2.1*.*

Since ${\rm Re}\{(1+e^{i\alpha})/2\}=(1+\cos\alpha)/2\geq 0$ where $\alpha\in(-\pi,\pi]$, thus the above Theorem 2.1 is a generalization of the results that earlier were obtained by Chichra [2] and Lewandowski et al. [7].

The problem of finding a lower bound for ${\rm Re}\left\{f(z)/z\right\}$ is called Marx-Strohhäcker problem. Marx and Strohhäcker ([8, 17]) proved that if $f\in\mathcal{K}$, then ${\rm Re}\left\{f(z)/z\right\}>1/2$. In the sequel we consider this problem for the class $\mathcal{R}(\alpha,\beta)$.

Theorem 2.2**.**

Let $\beta\in[0,1)$ and $\alpha\in(-\pi,\pi]$. If $f\in\mathcal{A}$ belongs to the class $\mathcal{R}(\alpha,\beta)$, then we have

[TABLE]

Proof.

Let the function $f\in\mathcal{A}$ belongs to the class $\mathcal{R}(\alpha,\beta)$ where $\beta\in[0,1)$ and $\alpha\in(-\pi,\pi]$. Define the function p as

[TABLE]

Since $f\in\mathcal{A}$, easily seen that p is analytic in $\Delta$ and $\textsl{p}(0)=1$. The equation (2.3), with a simple calculation implies that

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and

[TABLE]

Now, from (2.4) and multiplying (2.5) by $\frac{1+e^{i\alpha}}{2}z$, we get

[TABLE]

where

[TABLE]

Since $f\in\mathcal{R}(\alpha,\beta)$ we consider the following inclusion relation

[TABLE]

where $\Omega_{\beta}$ is defined in (2.2). Let $\rho,\sigma,\mu$ and $\nu$ be real numbers such that

[TABLE]

From (2.6) and by [9] (see also [10, Theorem 2.3b]), since

[TABLE]

we get

[TABLE]

where

[TABLE]

and

[TABLE]

It is easy to see that $F(\alpha,\beta,\rho)\leq\beta$. Since $2\mu+\nu\leq 0$, we have $G(\alpha,\beta,\mu)\leq 0$. Thus ${\rm Re}\{\psi(i\rho,\sigma,\mu+i\nu;z)\}\leq\beta$ and this means that

[TABLE]

Therefore we obtain ${\rm Re}\{\textsl{p}(z)\}>0$ where p is given by (2.3), or equivalently

[TABLE]

This completes the proof. ∎

If we put $\beta=0$ in the above Theorem 2.2, we get.

Corollary 2.2**.**

If $f\in\mathcal{L}_{\alpha}$, then ${\rm Re}\{f(z)/z\}>0$ in the open unit disc $\Delta$.

We shall require the following lemma in order to prove of the next result.

Lemma 2.1**.**

Let $\phi_{b}(z)$ be defined by (1.1) for $b>1/2$. Then $\phi_{b}(\Delta)=\Omega_{b}$ where

[TABLE]

Proof.

If $b=1$, then we have $0<{\rm Re}\{\phi_{b}(z)\}={\rm Re}\{1+z\}<2$. For $b>1/2$ and $b\neq 1$, the function $\phi_{b}(z)$ does not have any poles in $\overline{\Delta}$ and is analytic in $\Delta$. Thus looking for the $\min\{{\rm Re}\{\phi_{b}(z)\}:~{}|z|<1\}$ it is sufficient to consider it on the boundary $\partial\phi_{b}(\Delta)=\{\phi_{b}(e^{i\varphi}):\varphi\in[0,2\pi]\}$. A simple calculation gives us

[TABLE]

So we can see that ${\rm Re}\left\{F_{\alpha}(z)\right\}$ is well defined also for $\varphi=0$ and $\varphi=2\pi$. Define

[TABLE]

Thus for $b>1/2$ and $b\neq 1$, we have $h^{\prime}(x)>0$. Therefore, we get

[TABLE]

This completes the proof. ∎

Theorem 2.3**.**

Let $f\in\mathcal{A}$ be a member of the class $\mathcal{L}_{\alpha}(b)$ where $b>1/2$ and $\alpha\in(-\pi,\pi]$. Then

[TABLE]

Proof.

Let us $f\in\mathcal{L}_{\alpha}(b)$. Then by Lemma 1.1, Lemma 2.1 and by definition of the subordination principle we have

[TABLE]

First, we assume that

[TABLE]

Then by Corollary 2.2 we have ${\rm Re}\{f^{\prime}(z)\}>0$. Now we let

[TABLE]

Put $\xi=2b$ and so $\xi>1$. Let $f^{\prime}(z)\neq 0$ for $z\neq 0$. Consider

[TABLE]

Then $q(z)$ is analytic in $\Delta$ and $q(0)=1$. A simple check gives us

[TABLE]

where $\eta(x,y;z)=(1-\xi)[x+(1+e^{i\alpha})y/2]+\xi$. Now we define

[TABLE]

Again with a simple calculation we deduce that

[TABLE]

This shows that ${\rm Re}\{\eta(i\rho,\sigma;z)\}\not\in\Omega_{\xi}$ and therefore ${\rm Re}\{q(z)\}>0$, or equivalently ${\rm Re}\{f^{\prime}(z)\}<\xi$. This is the end of proof. ∎

Theorem 2.4**.**

Assume that $b>1/2$, $\alpha\in(-\pi,\pi]$ and $f\in\mathcal{L}_{\alpha}(b)$. Then for each $|z|=r<1$ we have

[TABLE]

Proof.

Let $f\in\mathcal{L}_{\alpha}(b)$. Then from the definition of subordination and by Lemma 1.1, there exists a $\omega\in\mathcal{B}$ such that

[TABLE]

We define

[TABLE]

which readily yields

[TABLE]

For $|z|=r<1$, using the known fact that (see [1]) $|\omega(z)|\leq|z|$ we find that

[TABLE]

Hence $W(z)$ maps the disk $|z|<r<1$ onto the disc which the center $C=1$ and the radius $\delta$ given by

[TABLE]

Therefore,

[TABLE]

Now, the assertion follows from (2.9) and this fact that ${\rm Re}\{z\}\leq|z|$. ∎

Remark 2.2*.*

We obtained two lower and upper bounds for

[TABLE]

when $f\in\mathcal{L}_{\alpha}(b)$. From (2.7), we have

[TABLE]

while by (2.8)

[TABLE]

It is easy to check that $U(r)<2b$ if $b\geq 1$ (or $b\rightarrow 1^{+}$) while $G(r)\geq 0$ for $1/2<b\leq 1$ (or $b\rightarrow 1^{-}$).

Corollary 2.3**.**

Let $f\in\mathcal{L}_{\alpha}(1)$. Then we have

[TABLE]

Corollary 2.4**.**

By a simple geometric observation and applying (2.9) and (2.10), we have

[TABLE]

3. The radius of univalence of 2-th section sum of $f\in\mathcal{R}(\alpha,\beta)$

In this section, we obtain the radius of univalence of 2-th section sum of $f\in\mathcal{R}(\alpha,\beta)$. We recall that the Taylor polynomial $s_{k}(z)=s_{k}(f)(z)$ of $f$ defined by

[TABLE]

is called the $k-th\,section/partial$ sum of $f$. In [18], proved that every section $s_{k}(z)$ of a $f\in\mathcal{S}$ is univalent in the disk $|z|<1/4$ and the number $1/4$ is best possible as the second partial sum of the Koebe function $k(z)=z/(1-z)^{2}$ shows. Next, we find the radius of univalence of the 2-th section sum of $f\in\mathcal{R}(\alpha,\beta)$.

Theorem 3.1**.**

The 2-th section sum of $f\in\mathcal{R}(\alpha,\beta)$ is univalent in the disc

[TABLE]

The number $\frac{\sqrt{10+6\cos\alpha}}{4(1-\beta)}$ cannot be replaced by a greater one.

Proof.

Let $f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}\in\mathcal{R}(\alpha,\beta)$ and $s_{2}(z)=z+a_{2}z^{2}$ be its second section. By a simple calculation and since $|a_{2}|\leq\frac{2(1-\beta)}{\sqrt{10+6\cos\alpha}}$ we have

[TABLE]

which is positive provided $|z|<\frac{\sqrt{10+6\cos\alpha}}{4(1-\beta)}$. Therefore $s_{2}(z)$ is close-to-convex (univalent) in the disk $|z|<\frac{\sqrt{10+6\cos\alpha}}{4(1-\beta)}$. To show that this bound is sharp, we consider the function $f_{x}$ defined by (2.1). The second partial sum $s_{2}(f_{x})(z)$ of $f_{x}$ is $z+\frac{4(1-\beta)}{2(3+e^{i\alpha})}z^{2}$. Thus we get

[TABLE]

Hence ${\rm Re}\{s_{2}^{\prime}(z)\}=0$ when $z=-\frac{(3+e^{i\alpha})}{4(1-\beta)}$. This completes the proof. ∎

We finish this paper with the following conjecture:

Conjecture. Every section of $f\in\mathcal{R}(\alpha,\beta)$ is univalent in the disc $|z|<\frac{\sqrt{10+6\cos\alpha}}{4(1-\beta)}$.