A note on the paper "Tang et al. [Bull Iran Math Soc (2019) doi:10.1007/s41980-019-00262-y]"

TL;DR

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Contribution

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Abstract

Very recently Tang et al. [Bull Iran Math Soc (2019) doi:10.1007/s41980-019-00262-y] have studied some majorization results for two certain subclasses of the starlike functions associated with the sine and cosine functions defined by $\mathcal{S}^*_s$ and $\mathcal{S}^*_c$, respectively. In this note we pointed out that the definition of the class $\mathcal{S}^*_c$ and it's result are incorrect and give correct definition and result.

Equations

$$f(z)=z+\sum _{n=2}^{\infty }{a}_n{z}^n(z\in \Delta ).$$

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

$$|\psi (z)|\le 1\mathrm{and}f(z)=\psi (z)g(z)$$

$${\mathcal{S}}_c^{\ast }:=\left\{f\in \mathcal{A}:\frac{z{f}^{\prime }(z)}{f(z)}\prec 1+\cos z\right\}.$$

$$|f^{\prime }(z)|\le |g^{\prime }(z)|,$$

$$(1-r^2)(1-\cosh r)-2r=0.$$

$$\frac{zf^{\prime}(z)}{f(z)}\Big{|}_{z=0}=1$$

$$|{\psi }^{\prime }(z)|\le \frac{1-|\psi (z){|}^2}{1-|z{|}^2}.$$

$$\underset{|z|=r}{\max }|f^{\prime }(z)|\le \underset{|z|=r}{\max }|g^{\prime }(z)|$$

$$|f(z)|\le 1andf(0)=0,$$

$$\frac{z{f}^{\prime }(z)}{f(z)}\prec \cos z.$$

$$|f^{\prime }(z)|\le |g^{\prime }(z)|(|z|\le r_1),$$

$$(1-r^2)\cos r-2r=0(0<r<1).$$

$$\frac{z{g}^{\prime }(z)}{g(z)}=\cos (\varphi (z))(z\in \Delta ),$$

$$\frac{g(z)}{g^{\prime }(z)}=\frac{z}{\cos (\varphi (z))}(z\in \Delta ).$$

$$\cos r\le \cos R\le |\cos (\varphi (z))|\le \cosh R\le \cosh r(R\le r<1).$$

$$\left|\frac{g(z)}{g^{\prime }(z)}\right|\le \frac{r}{\cos r}(0<r<1).$$

$$f(z)=\psi (z)g(z)(z\in \Delta ).$$

$$f^{\prime }(z)={\psi }^{\prime }(z)g(z)+\psi (z)g^{\prime }(z)=g^{\prime }(z)\left({\psi }^{\prime }(z)\frac{g(z)}{g^{\prime }(z)}+\psi (z)\right).$$

$$|f^{\prime }(z)|\le \left(|\psi (z)|+\frac{1-|\psi (z){|}^2}{1-r^2}.\frac{r}{\cos r}\right)|g^{\prime }(z)|(|z|=r<1).$$

$$|f^{\prime }(z)|\le \left(\beta +\frac{1-{\beta }^2}{1-r^2}.\frac{r}{\cos r}\right)|g^{\prime }(z)|(|z|=r<1).$$

$$h(r,\beta ):=\beta +\frac{1-{\beta }^2}{1-r^2}.\frac{r}{\cos r}(0\le \beta \le 1,0<r<1).$$

$$r_1=\max \{r\in [0,1):h(r,\beta )\le 1,\forall \beta \in [0,1]\}.$$

$$h(r,\beta )\le 1$$

$$\iff \beta (1-r^2)\cos r+(1-{\beta }^2)r\le (1-r^2)\cos r$$

$$\iff 0\le (1-\beta )(1-r^2)\cos r-(1-{\beta }^2)r$$

$$\iff 0\le (1-r^2)\cos r-(1+\beta )r=:k(r,\beta ).$$

$$\min \{k(r,\beta ):\beta \in [0,1]\}=k(r,1)=:k(r),$$

$$k(r)=(1-r^2)\cos r-2r(0<r<1).$$

$$f(0)=0=f^{\prime }(0)-1and|f(z)|<1,$$

$$|f^{\prime }(z)|\le 1(|z|\le 0.391389).$$

Full text

*Corresponding Author, ORCID iD: https://orcid.org/0000-0003-1029-5386

Mobile:+989141667438

1. Introduction

We denote by $\mathcal{H}$ the family of analytic functions $f$ on the open unit disc $\Delta=\{z\in\mathbb{C}:|z|<1\}$ and by $\mathcal{A}\subset\mathcal{H}$ the class of normalized functions $f$ of the form

[TABLE]

The subclass of $\mathcal{A}$ consisting of all univalent functions $f(z)$ in $\Delta$ will be denoted by $\mathcal{U}$. Also we denote by $\mathcal{B}\subset\mathcal{H}$ the family of functions $f$ on $\Delta$ satisfying the conditions $\phi(0)=0$ and $|\phi(z)|<1$. A function $\phi$ belongs to the class $\mathcal{B}$ is said to be a Schwarz function on $\Delta$.

Let two functions $f$ and $g$ belong to the class $\mathcal{H}$ and there exists a Schwarz function $\phi$ such that $f(z)=g(\phi(z))$. Then we say that $f$ is subordinate to $g$, written as $f(z)\prec g(z)$ or $f\prec g$. It is clear that if $f\prec g$, then we have

[TABLE]

In particular, if $g$ belong to the class $\mathcal{U}$, then $f\prec g$ if and only if the conditions (1.2) hold true. In the sequel, we recall the following definition from [7].

Definition 1.1**.**

Let $f$ and $g$ belong to the class $\mathcal{H}$. A function $f$ is said to be majorized by $g$ written as $f(z)\ll g(z)$ or $f\ll g$, if there exists an analytic function $\psi$ in $\Delta$, satisfying

[TABLE]

for all $z\in\Delta$.

For an analytic univalent function $\varphi$ with ${\rm Re}\{\varphi(z)\}>0$ $(z\in\Delta)$ and normalized by $\varphi(0)=1$ and $\varphi^{\prime}(0)>0$, Ma and Minda [6] introduced the class $\mathcal{S}^{*}(\varphi)$. A function $f\in\mathcal{A}$ belongs to the class $\mathcal{S}^{*}(\varphi)$ if and only if $zf^{\prime}(z)/f(z)$ is subordinate to $\varphi$. By choosing some special function $\varphi$, several authors have defined many new subclasses of the starlike functions in recent years, see for instance [2, 3, 4, 5, 9, 10].

Very recently, Tang et al. [11] introduced the class $\mathcal{S}^{*}_{c}$ as follows:

[TABLE]

Also, they proved the following theorem:

Theorem A. Let the function $f\in\mathcal{A}$ and suppose that $g\in\mathcal{S}^{*}_{c}$. If $f(z)\ll g(z)$, then for $|z|\leq r_{0}$

[TABLE]

where $r_{0}$ is the smallest positive root of the following equation:

[TABLE]

First of all we notice that the definition of the class $\mathcal{S}^{*}_{c}$ is incorrect, because if $f\in\mathcal{A}$, then

[TABLE]

and $(1+\cos z)|_{z=0}=2$ while $(zf^{\prime}(z)/f(z))\prec(1+\cos z)$. Therefore Theorem A is incorrect, too. In this note, we first will define the class $\mathcal{S}^{*}_{c}$ correctly and obtain majorization property (the correct version of Theorem A) for it.

In order to prove the main result, we need the following lemma, see [8].

Lemma 1.1**.**

Let $\psi(z)$ be analytic in $\Delta$ and satisfying $|\psi(z)|\leq 1$ for all $z\in\Delta$. Then

[TABLE]

2. Main Result

First, we recall that if $f(z)$ is majorized by $g(z)$ in $\Delta$ and $g(0)=0$, then

[TABLE]

for each number $r$ in the interval $[0,\sqrt{2}-1]$, see [7]. By choosing $g(z)=z$, the above assertion generalizes the following well-known theorem [1]:

Theorem B. If the function $f\in\mathcal{H}$ satisfies the conditions

[TABLE]

then $|f^{\prime}(z)|\leq 1$ for $|z|\leq\sqrt{2}-1$.

Following we define the correct version of the class $\mathcal{S}^{*}_{c}$ and denote by $\mathbf{S}^{*}_{c}$.

Definition 2.1**.**

We say that a function $f\in\mathcal{A}$ belongs to the class $\mathbf{S}^{*}_{c}$ if and only if

[TABLE]

Since $\cos z$ is univalent in $\Delta$ and $\cos z|_{z=0}=1$, thus the above Definition 2.1 is well-defined. Figure 1 shows the image of $\Delta$ under the function $\cos z$.

Theorem 2.1**.**

Let $f\in\mathcal{A}$ and $g\in\mathbf{S}^{*}_{c}$. If $f(z)$ is majorized by $g(z)$ in $\Delta$, then

[TABLE]

where $r_{1}\approx 0.391389$ is the smallest positive root of the following equation:

[TABLE]

Proof.

Let $f\in\mathcal{A}$ be given by (1.1) and $g\in\mathbf{S}^{*}_{c}$. Thus by Definition 2.1 and by the subordination principle there exists a Schwarz function $\phi$ such that

[TABLE]

or equivalently

[TABLE]

Let $\phi(z)=Re^{it}$ where $R\leq r=|z|<1$ and $-\pi\leq t\leq\pi$. It is a simple exercise that

[TABLE]

From (2.3) and (2.4) we get

[TABLE]

Also, since $f$ is majorized by $g$ in $\Delta$, thus we can find an analytic function $\psi$ in $\Delta$ with $|\psi(z)|\leq 1$ such that

[TABLE]

Taking a simple derivation of relation (2.6) we get

[TABLE]

If we apply (2.5) and Lemma 1.1 in (2.7), then we obtain

[TABLE]

Letting $|\psi(z)|=\beta$ $(0\leq\beta\leq 1)$, the last inequality (2.8) becomes

[TABLE]

Now we define

[TABLE]

To obtain $r_{1}$ it is enough to consider it as follows

[TABLE]

Thus

[TABLE]

Since $\frac{\partial}{\partial\beta}k(r,\beta)=-r<0$, therefore we see that the above function $k(r,\beta)$ gets its minimum value in $\beta=1$, namely

[TABLE]

where

[TABLE]

Because $k(0)=1>0$ and $k(1)=-2<0$, therefore we conclude that there exists a $r_{1}$ such that $k(r)\geq 0$ for all $r\in[0,r_{1}]$ where $r_{1}$ is the smallest positive root of the Eq. (2.2) and concluding the proof. ∎

Since the identity function $g(z)=z$ belongs to the class $\mathbf{S}^{*}_{c}$, thus letting $g(z)=z$ in the above Theorem 2.1 we get the following result. Indeed, we improve the bound $\sqrt{2}-1$ in the Theorem B.

Corollary 2.1**.**

If the function $f\in\mathcal{H}$ satisfies the conditions

[TABLE]

then

[TABLE]