# Some properties concerning close-to-convexity of certain analytic functions

## Abstract

Let $f(z)$ be an analytic function in the open unit disk $D$ normalized with $f(0)=0$ and $f ′ (0)=1$. With the help of subordinations, for convex functions $f(z)$ in $D$, the order of close-to-convexity for $f(z)$ is discussed with some example.

MSC:30C45.

## 1 Introduction

Let $A$ be the class of functions $f(z)$ of the form

$f(z)=z+ ∑ n = 2 ∞ a n z n$

which are analytic in the open unit disk $D={z∈C||z|<1}$. A function $f(z)∈A$ is said to be convex of order α if it satisfies

for some real α ($0≦α<1$). This family of functions was introduced by Robertson  and we denote it by $K(α)$.

A function $f(z)∈A$ is called starlike of order α in $D$ if it satisfies

for some real α ($0≦α<1$).

This class was also introduced by Robertson  and we denote it by $S ∗ (α)$. By the definitions for the classes $K(α)$ and $S ∗ (α)$, we know that $f(z)∈K(α)$ if and only if $z f ′ (z)∈ S ∗ (α)$.

Marx  and Strohhäcker  showed that $f(z)∈K(0)$ implies $f(z)∈ S ∗ ( 1 2 )$.

This estimate is sharp for an extremal function

$f(z)= z 1 − z .$

Jack  posed a more general problem: What is the largest number $β=β(α)$ so that

$K(α)⊂ S ∗ ( β ( α ) ) .$

MacGregor  determined the exact value of $β(α)$ for each α ($0≦α<1$) as the infimum over the disc $D$ of the real part of a specific analytic function. It has been conjectured that this infimum is attained on the boundary of $D$ at $z=−1$.

Wilken and Feng  asserted MacGregor’s conjecture: If $0≦α<1$ and $f(z)∈K(α)$, then $f(z)∈ S ∗ (β(α))$, where

(1)

Ozaki  and Kaplan  investigated the following functions: If $f(z)∈A$ satisfies

for some convex function $g(z)$, then $f(z)$ is univalent in $D$. In view of Kaplan , we say that $f(z)$ satisfying the above inequality is close-to-convex in $D$.

It is well known that the above definition concerning close-to-convex functions is equivalent to the following condition:

for some starlike function $g(z)∈A$.

Let us define a function $f(z)∈A$ which satisfies

for some real α ($0≦α<1$) and for some starlike function $g(z)$ in $D$.

Then we call $f(z)$ close-to-convex of order α in $D$ with respect to $g(z)$.

It is the purpose of the present paper to investigate the order of close-to-convexity of the functions which satisfy $f(z)∈K(α)$ and $0≦α<1$.

## 2 Preliminary

To discuss our problems, we have to give here the following lemmas.

Lemma 1 Let $p(z)=1+ ∑ n = 1 ∞ c n z n$ be analytic in $D$ and suppose that

$p(z)≺ 1 − α z 1 + β z in D,$

where means the subordination, $0<α<1$ and $0<β<1$.

Then we have

$1 − α 1 + β

This shows that

$Rep(z)>0 in D.$

A proof is very easily obtained.

Lemma 2 Let $p(z)=1+ ∑ n = 1 ∞ c n z n$ be analytic in $D$, and suppose that there exists a point $z 0 ∈D$ such that

$Rep(z)>c for |z|<| z 0 |$

and

$Rep( z 0 )=c,p( z 0 )≠c$

for some real c ($0). Then we have

$Re z 0 p ′ ( z 0 ) p ( z 0 ) ≦{ − 1 − c 2 c when 1 2 ≦ c < 1 , − c 2 ( 1 − c ) when 0 < c < 1 2 .$

Proof Let us put

$q(z)= p ( z ) − c 1 − c ,q(0)=1.$

Then $q(z)$ is analytic in $D$ and

and

$Req( z 0 )=0,q( z 0 )≠0.$

Then, from [, Theorem 1], we have

$z 0 q ′ ( z 0 ) q ( z 0 ) =iℓ,$

where

and

$ℓ≦− 1 2 ( a + 1 a ) ≦−1whenargq( z 0 )=− π 2 ,$

where $q( z 0 )=±ia$ and $a>0$.

For the case $argq( z 0 )= π 2$, we have

$z 0 q ′ ( z 0 ) q ( z 0 ) = z 0 p ′ ( z 0 ) p ( z 0 ) − c =iℓ$

and so If we put

$h(x)= 1 + x 2 c 2 + ( 1 − c ) 2 x 2 (x>0),$

then it is easy to see that

and

This shows that

For the case $argq( z 0 )=− π 2$, applying the same method as above, we have the same conclusion

This completes the proof of the lemma. □

Our next lemma is

Lemma 3 Let $p(z)=1+ ∑ n = 1 ∞ c n z n$ be analytic in $D$ and suppose that there exists a point $z 0 ∈D$ such that

$Rep(z)>c for |z|<| z 0 |$

and

$Rep( z 0 )=c,p( z 0 )≠c$

for some real c ($c<0$).

Then we have

$Re z 0 p ′ ( z 0 ) p ( z 0 ) >− c 2 ( 1 − c ) >0.$
(2)

Proof Let us put

$q(z)= p ( z ) − c 1 − c ,q(0)=1.$

Then $q(z)$ is analytic in $D$. If $p(z)$ satisfies the hypothesis of the lemma, then there exists a point $z 0 ∈D$ such that

and

$Req( z 0 )=0andq( z 0 )≠0,$

then $p(z)$ satisfies the conditions of the lemma.

For the case $argq( z 0 )= π 2$, applying the same method as in the proof of Lemma 2, we have

$Re z 0 p ′ ( z 0 ) p ( z 0 ) =− ( 1 − c ) c a ℓ c 2 + ( 1 − c ) 2 a 2 ≧− c ( 1 − c ) 2 ( 1 + a 2 c 2 + ( 1 − c ) 2 a 2 ) .$

Putting

$h(x)= 1 + x 2 c 2 + ( 1 − c ) 2 x 2 (x>0),$

it follows that

$h ′ (x)= ( 2 c − 1 ) x ( c 2 + ( 1 − c ) 2 x 2 ) 2 <0(x>0).$
(3)

Therefore, from (3) we obtain (2) .

For the case $argq( z 0 )=− π 2$, applying the same method as above, we have the same conclusion as in the case $argq( z 0 )= π 2$. □

## 3 The order of close-to-convexity

Now, we discuss the close-to-convexity of $f(z)$ with the help of lemmas.

Theorem 1 Let $f(z)∈A$, and suppose that there exists a starlike function $g(z)$ such that

1. (i)

for the case $1 2 ≦c<1$, and

1. (ii)

for the case $0, Then we have

$Re z f ′ ( z ) g ( z ) >c in D.$

This means that $f(z)$ is close-to-convex of order c in $D$.

Proof Let us put

$p(z)= z f ′ ( z ) g ( z ) ,p(0)=1.$

Then it follows that

$1+ z f ″ ( z ) f ′ ( z ) = z g ′ ( z ) g ( z ) + z p ′ ( z ) p ( z ) .$
1. (i)

For the case $1 2 ≦c<1$, if there exists a point $z 0 ∈D$ such that

and

$Rep( z 0 )=c,$

then, applying Lemma 2 and the hypothesis of Theorem 1, we have

$p( z 0 )≠c$

and

$Re z 0 p ′ ( z 0 ) p ( z 0 ) ≦− 1 − c 2 c .$

Thus, it follows that

$1 + Re z 0 f ″ ( z 0 ) f ′ ( z 0 ) = Re z 0 g ′ ( z 0 ) g ( z 0 ) + Re z 0 p ′ ( z 0 ) p ( z 0 ) ≦ Re z 0 g ′ ( z 0 ) g ( z 0 ) − 1 − c 2 c ,$

which contradicts the hypothesis of Theorem 1. (ii) For the case $0, applying the same method as above, we also have that

This completes the proof of the theorem. □

Applying Theorem 1, we have the following corollary.

Corollary 1 Let $f(z)∈A$ be convex of order α ($0<α<1$), and suppose that there exists a starlike function $g(z)$ such that

1. (i)

for the case $1 2 ≦α, and

1. (ii)

for the case $0<α, Then we have

$Re z f ′ ( z ) g ( z ) >β(c)>β(α)>α in D.$

Remark 1 For the case $0<α, it is trivial that

$α<β(α)<β(c)<1.$

Example 1 Let $f(z)∈A$ satisfy

(4)

where

$A= 32 β ( 1 2 ) − 10 8 β ( 1 2 ) + 10 ≒0.29605$

and $β( 1 2 )= 1 2 log 2$. If we consider the starlike function $g(z)$ given by

$g(z)= z ( 1 + A z ) 2 ,$

then we have

$Re z f ′ ( z ) g ( z ) >β ( 1 2 ) ≒0.7213,$

which means that $f(z)$ is close-to-convex of order $β( 1 2 )$ in $D$.

Next we show

Theorem 2 Let $f(z)∈A$ and $g(z)∈A$ be given by

$g(z)={ z ( 1 + β z ) α + β β ( β ≠ 0 ) , z e − α z ( β = 0 )$

for some α ($0≦α<1$) and some β ($0≦β<1$). Further suppose that for arbitrary r ($0),

$min | z | = r ( Re z f ′ ( z ) g ( z ) ) = ( Re z 0 f ′ ( z 0 ) g ( z 0 ) ) | z 0 | = r ≠ z 0 f ′ ( z 0 ) g ( z 0 )$

and

$1+Re z f ″ ( z ) f ′ ( z ) ≦− c 2 ( 1 − c ) + 1 − α 1 + β$

for $c<0$. Then we have

$Re z f ′ ( z ) g ( z ) >c in D.$

Proof Let us define the function $p(z)$ by

$p(z)= z f ′ ( z ) g ( z ) ,p(0)=1$

for $c<0$. If there exists a point $z 0 ∈D$ such that

and

$Rep( z 0 )=c$

for $c<0$, then from the hypothesis of Theorem 2, we have

$Rep( z 0 )≠p( z 0 ).$

Therefore, applying Lemma 1 and Lemma 3, we have

$1 + Re z 0 f ″ ( z 0 ) f ′ ( z 0 ) = Re z 0 p ′ ( z 0 ) p ( z 0 ) + Re z 0 g ′ ( z 0 ) g ( z 0 ) > − c 2 ( 1 − c ) + 1 − α 1 + β .$

This is a contradiction, and therefore we have

□

Remark 2 In view of the definition for close-to-convex functions, if $f(z)$ satisfies

then we can say that $f(z)$ is close-to-convex in $D$. But c should be a negative real number in Theorem 2. Therefore, we cannot say that $f(z)$ is close-to-convex in $D$ in Theorem 2.

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## Acknowledgements

The authors thank the referees for their helpful comments and suggestions to improve our manuscript.

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Correspondence to Melike Aydog̃an.

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The authors declare that they have no competing interests.

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All authors contributed equally and significantly to writing this paper. All authors read and approved the final manuscript.

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Nunokawa, M., Aydog̃an, M., Kuroki, K. et al. Some properties concerning close-to-convexity of certain analytic functions. J Inequal Appl 2012, 245 (2012). https://doi.org/10.1186/1029-242X-2012-245 