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

  • Mamoru Nunokawa1,
  • Melike Aydog̃an2Email author,
  • Kazuo Kuroki3,
  • Ismet Yildiz4 and
  • Shigeyoshi Owa3
Journal of Inequalities and Applications20122012:245

https://doi.org/10.1186/1029-242X-2012-245

Received: 1 August 2012

Accepted: 8 October 2012

Published: 24 October 2012

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.

Keywords

analyticstarlikeconvexclose-to-convexsubordination

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
1 + Re z f ( z ) f ( z ) > α in  D

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

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

for some real α ( 0 α < 1 ).

This class was also introduced by Robertson [1] 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 [2] and Strohhäcker [3] 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 [4] posed a more general problem: What is the largest number β = β ( α ) so that
K ( α ) S ( β ( α ) ) .

MacGregor [5] 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 [6] asserted MacGregor’s conjecture: If 0 α < 1 and f ( z ) K ( α ) , then f ( z ) S ( β ( α ) ) , where
β ( α ) = { 1 2 α 2 2 ( 1 α ) 2 if  α 1 2 , 1 2 log 2 if  α = 1 2 .
(1)
Ozaki [7] and Kaplan [8] investigated the following functions: If f ( z ) A satisfies
Re f ( z ) g ( z ) > 0 in  D

for some convex function g ( z ) , then f ( z ) is univalent in D . In view of Kaplan [8], 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:
Re z f ( z ) g ( z ) > 0 in  D

for some starlike function g ( z ) A .

Let us define a function f ( z ) A which satisfies
Re z f ( z ) g ( z ) > α in  D

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 < Re p ( z ) < 1 + α 1 β .
This shows that
Re p ( 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
Re p ( z ) > c for | z | < | z 0 |
and
Re p ( z 0 ) = c , p ( z 0 ) c
for some real c ( 0 < c < 1 ). 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
Re q ( z ) > 0 for  | z | < | z 0 |
and
Re q ( z 0 ) = 0 , q ( z 0 ) 0 .
Then, from [[9], Theorem 1], we have
z 0 q ( z 0 ) q ( z 0 ) = i ,
where
1 2 ( a + 1 a ) 1 when  arg q ( z 0 ) = π 2
and
1 2 ( a + 1 a ) 1 when arg q ( z 0 ) = π 2 ,

where q ( z 0 ) = ± i a and a > 0 .

For the case arg q ( z 0 ) = π 2 , we have
z 0 q ( z 0 ) q ( z 0 ) = z 0 p ( z 0 ) p ( z 0 ) c = i
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
1 c 2 < h ( x ) < 1 ( 1 c ) 2 when  1 2 c < 1
and
1 ( 1 c ) 2 < h ( x ) < 1 c 2 when  0 < c < 1 2 .
This shows that
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 .
For the case arg q ( z 0 ) = π 2 , applying the same method as above, we have the same conclusion
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 .

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
Re p ( z ) > c for | z | < | z 0 |
and
Re p ( 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
Re q ( z ) > 0 for  | z | < | z 0 |
and
Re q ( z 0 ) = 0 and q ( z 0 ) 0 ,

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

For the case arg q ( 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 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 arg q ( z 0 ) = π 2 , applying the same method as above, we have the same conclusion as in the case arg q ( 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 < c < 1 2 ,
     
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
    Re p ( z ) > c for  | z | < | z 0 |
     
and
Re p ( 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 < c < 1 2 , applying the same method as above, we also have that
Re z f ( z ) g ( z ) > c in  D .

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 < c ,
     
and
  1. (ii)
    for the case 0 < α < c 1 2 ,
     
Then we have
Re z f ( z ) g ( z ) > β ( c ) > β ( α ) > α in D .
Remark 1 For the case 0 < α < c < 1 , it is trivial that
α < β ( α ) < β ( c ) < 1 .
Example 1 Let f ( z ) A satisfy
1 + Re z f ( z ) f ( z ) > Re 1 A z 1 + A z 1 β ( 1 2 ) 2 β ( 1 2 ) > 1 10 in  D ,
(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 < r < 1 ),
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
Re p ( z ) > c for  | z | < | z 0 |
and
Re p ( z 0 ) = c
for c < 0 , then from the hypothesis of Theorem 2, we have
Re p ( 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
Re z f ( z ) g ( z ) > c in  D .

 □

Remark 2 In view of the definition for close-to-convex functions, if f ( z ) satisfies
Re z f ( z ) g ( z ) > 0 in  D ,

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.

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
University of Gunma, Chuou-Ward, Chiba, Japan
(2)
Department of Mathematics, Işık University, Şile Kampusu, Istanbul, Turkey
(3)
Department of Mathematics, Kinki University, Higashi-Osaka, Japan
(4)
Department of Mathematics, Duzce University, Duzce, Turkey

References

  1. Robertson MS: On the theory of univalent functions. Ann. Math. 1936, 37: 374–408. 10.2307/1968451View ArticleGoogle Scholar
  2. Marx A: Untersuchungen über schlichte Abbildungen. Math. Ann. 1932/33, 107: 40–67.MathSciNetView ArticleGoogle Scholar
  3. Strohhäcker E: Beiträge zur Theorie der schlichten Funktionen. Math. Z. 1933, 37: 356–380. 10.1007/BF01474580MathSciNetView ArticleGoogle Scholar
  4. Jack IS: Functions starlike and convex of order α . J. Lond. Math. Soc. 1971, 3: 469–474. 10.1112/jlms/s2-3.3.469MathSciNetView ArticleGoogle Scholar
  5. MacGregor TH: A subordination for convex functions of order α . J. Lond. Math. Soc. 1975, 9: 530–536. 10.1112/jlms/s2-9.4.530MathSciNetView ArticleGoogle Scholar
  6. Wilken DR, Feng J: A remark on convex and starlike functions. J. Lond. Math. Soc. 1980, 21: 287–290. 10.1112/jlms/s2-21.2.287MathSciNetView ArticleGoogle Scholar
  7. Ozaki S: On the theory of multivalent functions. Sci. Rep. Tokyo Bunrika Daigaku, Sect. A. 1935, 2: 167–188.Google Scholar
  8. Kaplan W: Close-to-convex schlicht functions. Mich. Math. J. 1952, 1: 169–185.View ArticleGoogle Scholar
  9. Nunokawa M: On properties of non-Carathéodory functions. Proc. Jpn. Acad., Ser. A, Math. Sci. 1992, 68(6):152–153. 10.3792/pjaa.68.152MathSciNetView ArticleGoogle Scholar

Copyright

© Nunokawa et al.; licensee Springer 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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