# On some differential inequalities in the unit disk with applications

## Abstract

In this paper we obtain a number of interesting relations associated with some differential inequalities in the open unit disk, $\mathbb{U}=\left\{z:|z|<1\right\}$. Some applications of the main results are also obtained.

MSC:30C45, 30C80.

## 1 Introduction

Let A denote the class of functions of the form

$f\left(z\right)=z+\sum _{n=2}^{\mathrm{\infty }}{a}_{n}{z}^{n}$
(1.1)

which are analytic in the unit disc $\mathbb{U}=\left\{z:|z|<1\right\}$. Also, we denote by K the class of functions $f\left(z\right)\in A$ that are convex in .

A function $f\left(z\right)$ in the class A is said to be in the class ${S}^{\ast }\left(\alpha \right)$ of starlike functions of order α ($0\le \alpha <1$) if it satisfies

$Re\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)>\alpha \phantom{\rule{1em}{0ex}}\left(z\in \mathbb{U}\right)$
(1.2)

for some α ($0\le \alpha <1$). Also, we write $S\left(0\right)={S}^{\ast }$, the class of starlike functions in .

A function $f\left(z\right)\in A$ is in ${S}^{\lambda }$ ($|\lambda |<\frac{\pi }{2}$), the class of λ-spiral-like functions, if it satisfies

$Re\left({e}^{i\lambda }\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)>0\phantom{\rule{1em}{0ex}}\left(z\in \mathbb{U}\right).$
(1.3)

Definition 1.1 Let $f\left(z\right)$ and $F\left(z\right)$ be analytic functions. The function $f\left(z\right)$ is said to be subordinate to $F\left(z\right)$, written $f\left(z\right)\prec F\left(z\right)$, if there exists a function $w\left(z\right)$ analytic in , with $w\left(0\right)=0$ and $|w\left(z\right)|\le 1$, and such that $f\left(z\right)=F\left(w\left(z\right)\right)$. If $F\left(z\right)$ is univalent, then $f\left(z\right)\prec F\left(z\right)$ if and only if $f\left(0\right)=F\left(0\right)$ and $f\left(\mathbb{U}\right)\subset F\left(\mathbb{U}\right)$.

Let be the set of analytic functions $q\left(z\right)$ injective on $\overline{\mathbb{U}}\mathrm{\setminus }E\left(q\right)$, where

$E\left(q\right)=\left\{\zeta \in \partial \mathbb{U}:\underset{z\to \zeta }{lim}q\left(z\right)=\mathrm{\infty }\right\}$

and ${q}^{\prime }\left(\zeta \right)\ne 0$ for $\zeta \in \partial \mathbb{U}\mathrm{\setminus }E\left(q\right)$. Further, let ${\mathbb{D}}_{a}=\left\{q\left(z\right)\in \mathbb{D}:q\left(0\right)=a\right\}$.

In this paper we obtain some interesting relations associated with some differential inequalities in . These relations extend and generalize the Carathéodory functions in which have been studied by many authors e.g., see .

## 2 Main results

To prove our results, we need the following lemma due to Miller and Mocanu [, p.24].

Lemma 2.1 Let $q\left(z\right)\in {\mathbb{D}}_{a}$ and let

$p\left(z\right)=b+{b}_{n}{z}^{n}+\cdots$

be analytic in with $p\left(z\right)\ne b$. If $p\left(z\right)\nprec q\left(z\right)$, then there exist points ${z}_{0}\in \mathbb{U}$ and ${\zeta }_{0}\in \partial \mathbb{U}\mathrm{\setminus }E\left(q\right)$ and on $m\ge n\ge 1$ for which

1. (i)

$p\left({z}_{0}\right)=q\left({\zeta }_{0}\right)$,

2. (ii)

${z}_{0}{p}^{\prime }\left({z}_{0}\right)=m{\zeta }_{0}{q}^{\prime }\left({\zeta }_{0}\right)$.

Theorem 2.1 Let

$P:\mathbb{U}\to \mathbb{C}$

with

$Re\left(\overline{a}P\left(z\right)\right)>0\phantom{\rule{1em}{0ex}}\left(a\in \mathbb{C}\right).$

If p is a function analytic in with $p\left(0\right)=1$ and

$Re\left(p\left(z\right)+P\left(z\right)z{p}^{\prime }\left(z\right)\right)>\frac{E}{2|a{|}^{2}Re\left(\overline{a}P\left(z\right)\right)},$
(2.1)

then

$Re\left(ap\left(z\right)\right)>\alpha ,$

where

$\begin{array}{rcl}E& =& -\left(Re\left(a\right)-\alpha \right){\left(Re\left(\overline{a}P\left(z\right)\right)\right)}^{2}\\ +2Re\left(\overline{a}P\left(z\right)\right)\left[{\left(Im\left(a\right)\right)}^{2}+2\alpha Re\left(a\right)\right]\\ +\left(Re\left(a\right)-\alpha \right){\left(Im\left(a\right)\right)}^{2},\end{array}$
(2.2)

with $Re\left(a\right)>\alpha$.

Proof Let us define $q\left(z\right)$ and $h\left(z\right)$ as follows:

$q\left(z\right)=ap\left(z\right)$

and

$h\left(z\right)=\frac{a-\left(2\alpha -\overline{a}\right)z}{1-z}\phantom{\rule{1em}{0ex}}\left(Re\left(a\right)>\alpha \right).$

The functions q and h are analytic in with $q\left(0\right)=h\left(0\right)=a\in \mathbb{C}$ with

$h\left(\mathbb{U}\right)=\left\{w:Re\left(w\right)>\alpha \right\}.$

Now, we suppose that $q\left(z\right)\nprec h\left(z\right)$. Therefore, by using Lemma 2.1, there exist points

${z}_{0}\in \mathbb{U}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\zeta }_{0}\in \partial \mathbb{U}\mathrm{\setminus }\left\{1\right\}$

such that $q\left({z}_{0}\right)=h\left({\zeta }_{0}\right)$ and ${z}_{0}{q}^{\prime }\left({z}_{0}\right)=m{\zeta }_{0}{h}^{\prime }\left({\zeta }_{0}\right)$, $m\ge n\ge 1$.

We note that

${\zeta }_{0}={h}^{-1}\left(q\left({z}_{0}\right)\right)=\frac{q\left({z}_{0}\right)-a}{q\left({z}_{0}\right)-\left(2\alpha -\overline{a}\right)}$
(2.3)

and

${\zeta }_{0}{h}^{\prime }\left({\zeta }_{0}\right)=\frac{-|q\left({z}_{0}\right)-a{|}^{2}}{2Re\left(a-q\left({z}_{0}\right)\right)}.$
(2.4)

We have $h\left({\zeta }_{0}\right)=\alpha +\rho i$ ($\alpha ,\rho \in \mathbb{R}$), therefore

$\begin{array}{c}Re\left(p\left({z}_{0}\right)+P\left({z}_{0}\right){z}_{0}{p}^{\prime }\left({z}_{0}\right)\right)\hfill \\ \phantom{\rule{1em}{0ex}}=Re\left(\frac{1}{a}h\left({\zeta }_{0}\right)+\frac{1}{a}P\left({z}_{0}\right)m{\zeta }_{0}{h}^{\prime }\left({\zeta }_{0}\right)\right)\hfill \\ \phantom{\rule{1em}{0ex}}=Re\left(\frac{\alpha +\rho i}{a}\right)-m\frac{|\alpha +\rho i-a{|}^{2}}{2Re\left(a-\alpha \right)}Re\left(\frac{P\left({z}_{0}\right)}{a}\right)\hfill \\ \phantom{\rule{1em}{0ex}}\le Re\left(\frac{\alpha +\rho i}{a}\right)-\frac{|\alpha +\rho i-a{|}^{2}}{2Re\left(a-\alpha \right)}Re\left(\frac{P\left({z}_{0}\right)}{a}\right)\hfill \\ \phantom{\rule{1em}{0ex}}=A{\rho }^{2}+B\rho +C\hfill \\ \phantom{\rule{1em}{0ex}}=g\left(\rho \right),\hfill \end{array}$
(2.5)

where

$\begin{array}{c}A=-\frac{Re\left(\overline{a}P\left({z}_{0}\right)\right)}{2|a{|}^{2}Re\left(a-\alpha \right)},\hfill \\ B=\frac{Im\left(a\right)}{|a{|}^{2}}\left(1+\frac{Re\left(\overline{a}P\left({z}_{0}\right)\right)}{Re\left(a\right)-\alpha }\right)\hfill \end{array}$

and

$C=\frac{1}{|a{|}^{2}}\left(\alpha Re\left(a\right)-\frac{{\alpha }^{2}+|a{|}^{2}-2\alpha Re\left(a\right)Re\left(\overline{a}P\left({z}_{0}\right)\right)}{2\left(Re\left(a\right)-\alpha \right)}\right).$

We can see that the function $g\left(\rho \right)$ in (2.5) takes the maximum value at ${\rho }_{1}$ given by

${\rho }_{1}=Im\left(a\right)\left(1+\frac{Re\left(a\right)-\alpha }{Re\left(\overline{a}P\left({z}_{0}\right)\right)}\right).$

Hence, we have

$\begin{array}{c}Re\left(p\left({z}_{0}\right)+P\left({z}_{0}\right)z{p}^{\prime }\left({z}_{0}\right)\right)\hfill \\ \phantom{\rule{1em}{0ex}}\le g\left({\rho }_{1}\right)\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{E}{2|a{|}^{2}Re\left(\overline{a}P\left(z\right)\right)},\hfill \end{array}$

where E is defined by (2.2). This is in contradiction to (2.1). Then we obtain $Re\left(ap\left(z\right)\right)>\alpha$. □

Theorem 2.2 Let $p\left(z\right)$ a nonzero analytic function in with $p\left(0\right)=1$. If

$|p\left(z\right)+\frac{z{p}^{\prime }\left(z\right)}{p\left(z\right)}-1|<\frac{3Re\left(a-\alpha \right)}{2|a|}|p\left(z\right)|,$
(2.6)

then

$Re\left(\frac{a}{p\left(z\right)}\right)>\alpha ,$

where $Re\left(a\right)>\alpha$.

Proof Let us define both $q\left(z\right)$ and $h\left(z\right)$ as follows:

$q\left(z\right)=a/p\left(z\right)$

and

$h\left(z\right)=\frac{a-\left(2\alpha -\overline{a}\right)z}{1-z}\phantom{\rule{1em}{0ex}}\left(Re\left(a\right)>\alpha \right).$

The functions q and h are analytic in with $q\left(0\right)=h\left(0\right)=a\in \mathbb{C}$ with

$h\left(\mathbb{U}\right)=\left\{w:Re\left(w\right)>\alpha \right\}.$

Now, we suppose that $q\left(z\right)\nprec h\left(z\right)$. Therefore, by using Lemma 2.1, there exist points

${z}_{0}\in \mathbb{U}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\zeta }_{0}\in \partial \mathbb{U}\mathrm{\setminus }\left\{1\right\}$

such that $q\left({z}_{0}\right)=h\left({\zeta }_{0}\right)$ and ${z}_{0}{q}^{\prime }\left({z}_{0}\right)=m{\zeta }_{0}{h}^{\prime }\left({\zeta }_{0}\right)$, $m\ge n\ge 1$.

We note that

${\zeta }_{0}{h}^{\prime }\left({\zeta }_{0}\right)=\frac{-|q\left({z}_{0}\right)-a{|}^{2}}{2Re\left(a-q\left({z}_{0}\right)\right)}.$
(2.7)

We have $h\left({\zeta }_{0}\right)=\alpha +\rho i$ ($\rho \in \mathbb{R}$); therefore,

$\begin{array}{rcl}\frac{|p\left({z}_{0}\right)+\frac{z{p}^{\prime }\left({z}_{0}\right)}{p\left({z}_{0}\right)}-1|}{|p\left({z}_{0}\right)|}& =& |\frac{\alpha +\rho i}{a}-\frac{m}{a}\frac{|a-\alpha -i\rho {|}^{2}}{2Re\left(a-\alpha \right)}-1|\\ \ge & \frac{1}{|a|}|\frac{m|a-\alpha -i\rho {|}^{2}}{2Re\left(a-\alpha \right)}+Re\left(a-\alpha \right)|\\ \ge & \frac{1}{|a|}\left(\frac{|a-\alpha -i\rho {|}^{2}}{2Re\left(a-\alpha \right)}+Re\left(a-\alpha \right)\right)\\ \ge & \frac{1}{2|a|Re\left(a-\alpha \right)}\left(3{\left(Re\left(a-\alpha \right)\right)}^{2}+{\left(Im\left(a\right)-\rho \right)}^{2}\right)\\ \ge & \frac{3Re\left(a-\alpha \right)}{2|a|}.\end{array}$

This is in contradiction to (2.6). Then we obtain $Re\left(\frac{a}{p\left(z\right)}\right)>\alpha$. □

## 3 Applications and examples

Putting $P\left(z\right)=\beta$ ($\beta >0$; real) in Theorem 2.1 we have the following corollary.

Corollary 3.1 If p is a function analytic in with $p\left(0\right)=1$ and

$Re\left(p\left(z\right)+\beta z{p}^{\prime }\left(z\right)\right)>\frac{E}{2\beta |a{|}^{2}Re\left(a\right)},$

then

$Re\left(ap\left(z\right)\right)>\alpha ,$

where

$E=-\left(Re\left(a\right)-\alpha \right){\beta }^{2}{\left(Re\left(a\right)\right)}^{2}+2\beta Re\left(a\right)\left[{\left(Im\left(a\right)\right)}^{2}+2\alpha Re\left(a\right)\right]+\left(Re\left(a\right)-\alpha \right){\left(Im\left(a\right)\right)}^{2},$

with $Re\left(a\right)>\alpha$ ($\alpha \ge 0$).

Putting $\beta =1$ in Corollary 3.1, we obtain the following corollary.

Corollary 3.2 If p is a function analytic in with $p\left(0\right)=1$ and

$Re\left(p\left(z\right)+z{p}^{\prime }\left(z\right)\right)>\frac{1}{2Re\left(a\right)}\left(3Re\left(a\right)-\alpha \right)-\frac{Re\left(a\right)}{|a{|}^{2}}\left(2Re\left(a\right)-3\alpha \right),$

then

$Re\left(ap\left(z\right)\right)>\alpha ,$

with $Re\left(a\right)>\alpha$ ($\alpha \ge 0$).

Corollary 3.3 Let $f\left(z\right)\in A$, ${\left(g\left(z\right)\right)}^{a}\in {S}^{\ast }$ and

$Re\left(\frac{{f}^{\prime }\left(z\right)}{{g}^{\prime }\left(z\right)}\right)>\frac{E}{2|a{|}^{2}Re\left(\overline{a}\frac{g\left(z\right)}{z{g}^{\prime }\left(z\right)}\right)},$

then

$Re\left(a\frac{f\left(z\right)}{g\left(z\right)}\right)>\alpha ,$

where $Re\left(a\right)>\alpha$ ($\alpha \ge 0$) and E is defined by (2.2) with $P\left(z\right)=\frac{g\left(z\right)}{z{g}^{\prime }\left(z\right)}$.

Proof Putting $p\left(z\right)=\frac{f\left(z\right)}{g\left(z\right)}$ and $P\left(z\right)=\frac{g\left(z\right)}{z{g}^{\prime }\left(z\right)}$ in Theorem 2.1, we have

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

Since ${\left(g\left(z\right)\right)}^{a}\in {S}^{\ast }$, which gives $Re\left(a\frac{z{g}^{\prime }\left(z\right)}{g\left(z\right)}\right)>0$, therefore, $Re\left(\overline{a}P\left(z\right)\right)>0$. This completes the proof of the corollary. □

Example 3.1 Let $f\left(z\right)\in A$ and

$Re\left({f}^{\prime }\left(z\right)\right)>\frac{1}{2Re\left(a\right)}\left(3Re\left(a\right)-\alpha \right)-\frac{Re\left(a\right)}{|a{|}^{2}}\left(2Re\left(a\right)-3\alpha \right),$

then

$Re\left(a\frac{f\left(z\right)}{z}\right)>\alpha ,$

where $Re\left(a\right)>\alpha$.

Example 3.2 Let $f\left(z\right)\in A$ and

$Re\left(\left(2+\frac{z{f}^{″}\left(z\right)}{{f}^{\prime }\left(z\right)}-\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)>\frac{1}{2Re\left(a\right)}\left(3Re\left(a\right)-\alpha \right)-\frac{Re\left(a\right)}{|a{|}^{2}}\left(2Re\left(a\right)-3\alpha \right),$

then

$Re\left(a\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)>\alpha ,$

where $Re\left(a\right)>\alpha$.

1. (1)

Putting $a={e}^{i\lambda }$ ($|\lambda |<\frac{\pi }{2}$) and $\alpha =0$ in Theorem 2.1, we have Theorem 1 due to Kim and Cho .

2. (2)

Putting $a={e}^{i\lambda }$ ($|\lambda |<\frac{\pi }{2}$), $P\left(z\right)=\beta$ ($\beta >0$; real) and $\alpha =0$ in Theorem 2.1, we have Corollary 1 due to Kim and Cho .

3. (3)

Putting $a=\alpha =0$ and $P\left(z\right)=1$ in Theorem 2.1, we have the result due to Nunokawa et al. .

4. (4)

Putting $a={e}^{i\lambda }$ ($|\lambda |<\frac{\pi }{2}$), $P\left(z\right)=1$ and $\alpha =0$ in Theorem 2.1, we have Corollary 2 due to Kim and Cho .

Putting $p\left(z\right)=\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}$ in Theorem 2.2, we have the following corollary.

Corollary 3.4 Let $p\left(z\right)$ a nonzero analytic function in U with $p\left(0\right)=1$. If

$|\frac{z{f}^{″}\left(z\right)}{{f}^{\prime }\left(z\right)}|<\frac{3Re\left(a-\alpha \right)}{2|a|}|\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}|,$

then

$Re\left(\frac{1}{a}\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)>\alpha ,$

where $Re\left(a\right)>\alpha$.

Remark

1. (1)

Putting $a=1$ and $\alpha =0$ in Corollary 3.4, we have the result due to Attiya and Nasr .

2. (2)

Putting $a=1$ and $\alpha =0$ in Corollary 3.4, we have the result due to Kim and Cho .

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## Acknowledgement

The author would like to express his gratitude to the referee(s) for the valuable advices to improve this paper.

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Attiya, A.A. On some differential inequalities in the unit disk with applications. J Inequal Appl 2014, 32 (2014). https://doi.org/10.1186/1029-242X-2014-32

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• DOI: https://doi.org/10.1186/1029-242X-2014-32

### Keywords

• analytic functions
• starlike functions
• convex functions
• spiral-like functions
• Carathéodory functions 