 Research
 Open Access
 Published:
Some properties concerning closetoconvexity of certain analytic functions
Journal of Inequalities and Applications volume 2012, Article number: 245 (2012)
Abstract
Let f(z) be an analytic function in the open unit disk \mathbb{D} normalized with f(0)=0 and {f}^{\prime}(0)=1. With the help of subordinations, for convex functions f(z) in \mathbb{D}, the order of closetoconvexity for f(z) is discussed with some example.
MSC:30C45.
1 Introduction
Let \mathcal{A} be the class of functions f(z) of the form
which are analytic in the open unit disk \mathbb{D}=\{z\in \mathbb{C}z<1\}. A function f(z)\in \mathcal{A} is said to be convex of order α if it satisfies
for some real α (0\leqq \alpha <1). This family of functions was introduced by Robertson [1] and we denote it by \mathcal{K}(\alpha ).
A function f(z)\in \mathcal{A} is called starlike of order α in \mathbb{D} if it satisfies
for some real α (0\leqq \alpha <1).
This class was also introduced by Robertson [1] and we denote it by {\mathcal{S}}^{\ast}(\alpha ). By the definitions for the classes \mathcal{K}(\alpha ) and {\mathcal{S}}^{\ast}(\alpha ), we know that f(z)\in \mathcal{K}(\alpha ) if and only if z{f}^{\prime}(z)\in {\mathcal{S}}^{\ast}(\alpha ).
Marx [2] and Strohhäcker [3] showed that f(z)\in \mathcal{K}(0) implies f(z)\in {\mathcal{S}}^{\ast}(\frac{1}{2}).
This estimate is sharp for an extremal function
Jack [4] posed a more general problem: What is the largest number \beta =\beta (\alpha ) so that
MacGregor [5] determined the exact value of \beta (\alpha ) for each α (0\leqq \alpha <1) as the infimum over the disc \mathbb{D} of the real part of a specific analytic function. It has been conjectured that this infimum is attained on the boundary of \mathbb{D} at z=1.
Wilken and Feng [6] asserted MacGregor’s conjecture: If 0\leqq \alpha <1 and f(z)\in \mathcal{K}(\alpha ), then f(z)\in {\mathcal{S}}^{\ast}(\beta (\alpha )), where
Ozaki [7] and Kaplan [8] investigated the following functions: If f(z)\in \mathcal{A} satisfies
for some convex function g(z), then f(z) is univalent in \mathbb{D}. In view of Kaplan [8], we say that f(z) satisfying the above inequality is closetoconvex in \mathbb{D}.
It is well known that the above definition concerning closetoconvex functions is equivalent to the following condition:
for some starlike function g(z)\in \mathcal{A}.
Let us define a function f(z)\in \mathcal{A} which satisfies
for some real α (0\leqq \alpha <1) and for some starlike function g(z) in \mathbb{D}.
Then we call f(z) closetoconvex of order α in \mathbb{D} with respect to g(z).
It is the purpose of the present paper to investigate the order of closetoconvexity of the functions which satisfy f(z)\in \mathcal{K}(\alpha ) and 0\leqq \alpha <1.
2 Preliminary
To discuss our problems, we have to give here the following lemmas.
Lemma 1 Let p(z)=1+{\sum}_{n=1}^{\mathrm{\infty}}{c}_{n}{z}^{n} be analytic in \mathbb{D} and suppose that
where ≺ means the subordination, 0<\alpha <1 and 0<\beta <1.
Then we have
This shows that
A proof is very easily obtained.
Lemma 2 Let p(z)=1+{\sum}_{n=1}^{\mathrm{\infty}}{c}_{n}{z}^{n} be analytic in \mathbb{D}, and suppose that there exists a point {z}_{0}\in \mathbb{D} such that
and
for some real c (0<c<1). Then we have
Proof Let us put
Then q(z) is analytic in \mathbb{D} and
and
Then, from [[9], Theorem 1], we have
where
and
where q({z}_{0})=\pm ia and a>0.
For the case argq({z}_{0})=\frac{\pi}{2}, we have
and so
If we put
then it is easy to see that
and
This shows that
For the case argq({z}_{0})=\frac{\pi}{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+{\sum}_{n=1}^{\mathrm{\infty}}{c}_{n}{z}^{n} be analytic in \mathbb{D} and suppose that there exists a point {z}_{0}\in \mathbb{D} such that
and
for some real c (c<0).
Then we have
Proof Let us put
Then q(z) is analytic in \mathbb{D}. If p(z) satisfies the hypothesis of the lemma, then there exists a point {z}_{0}\in \mathbb{D} such that
and
then p(z) satisfies the conditions of the lemma.
For the case argq({z}_{0})=\frac{\pi}{2}, applying the same method as in the proof of Lemma 2, we have
Putting
it follows that
Therefore, from (3) we obtain (2) .
For the case argq({z}_{0})=\frac{\pi}{2}, applying the same method as above, we have the same conclusion as in the case argq({z}_{0})=\frac{\pi}{2}. □
3 The order of closetoconvexity
Now, we discuss the closetoconvexity of f(z) with the help of lemmas.
Theorem 1 Let f(z)\in \mathcal{A}, and suppose that there exists a starlike function g(z) such that

(i)
for the case \frac{1}{2}\leqq c<1,
and

(ii)
for the case 0<c<\frac{1}{2},
Then we have
This means that f(z) is closetoconvex of order c in \mathbb{D}.
Proof Let us put
Then it follows that

(i)
For the case \frac{1}{2}\leqq c<1, if there exists a point {z}_{0}\in \mathbb{D} such that
Rep(z)>c\phantom{\rule{1em}{0ex}}\text{for}z{z}_{0}
and
then, applying Lemma 2 and the hypothesis of Theorem 1, we have
and
Thus, it follows that
which contradicts the hypothesis of Theorem 1. (ii) For the case 0<c<\frac{1}{2}, 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)\in \mathcal{A} be convex of order α (0<\alpha <1), and suppose that there exists a starlike function g(z) such that

(i)
for the case \frac{1}{2}\leqq \alpha <c,
and

(ii)
for the case 0<\alpha <c\leqq \frac{1}{2},
Then we have
Remark 1 For the case 0<\alpha <c<1, it is trivial that
Example 1 Let f(z)\in \mathcal{A} satisfy
where
and \beta (\frac{1}{2})=\frac{1}{2log2}. If we consider the starlike function g(z) given by
then we have
which means that f(z) is closetoconvex of order \beta (\frac{1}{2}) in \mathbb{D}.
Next we show
Theorem 2 Let f(z)\in \mathcal{A} and g(z)\in \mathcal{A} be given by
for some α (0\leqq \alpha <1) and some β (0\leqq \beta <1). Further suppose that for arbitrary r (0<r<1),
and
for c<0. Then we have
Proof Let us define the function p(z) by
for c<0. If there exists a point {z}_{0}\in \mathbb{D} such that
and
for c<0, then from the hypothesis of Theorem 2, we have
Therefore, applying Lemma 1 and Lemma 3, we have
This is a contradiction, and therefore we have
□
Remark 2 In view of the definition for closetoconvex functions, if f(z) satisfies
then we can say that f(z) is closetoconvex in \mathbb{D}. But c should be a negative real number in Theorem 2. Therefore, we cannot say that f(z) is closetoconvex in \mathbb{D} in Theorem 2.
References
 1.
Robertson MS: On the theory of univalent functions. Ann. Math. 1936, 37: 374–408. 10.2307/1968451
 2.
Marx A: Untersuchungen über schlichte Abbildungen. Math. Ann. 1932/33, 107: 40–67.
 3.
Strohhäcker E: Beiträge zur Theorie der schlichten Funktionen. Math. Z. 1933, 37: 356–380. 10.1007/BF01474580
 4.
Jack IS: Functions starlike and convex of order α . J. Lond. Math. Soc. 1971, 3: 469–474. 10.1112/jlms/s23.3.469
 5.
MacGregor TH: A subordination for convex functions of order α . J. Lond. Math. Soc. 1975, 9: 530–536. 10.1112/jlms/s29.4.530
 6.
Wilken DR, Feng J: A remark on convex and starlike functions. J. Lond. Math. Soc. 1980, 21: 287–290. 10.1112/jlms/s221.2.287
 7.
Ozaki S: On the theory of multivalent functions. Sci. Rep. Tokyo Bunrika Daigaku, Sect. A. 1935, 2: 167–188.
 8.
Kaplan W: Closetoconvex schlicht functions. Mich. Math. J. 1952, 1: 169–185.
 9.
Nunokawa M: On properties of nonCarathéodory functions. Proc. Jpn. Acad., Ser. A, Math. Sci. 1992, 68(6):152–153. 10.3792/pjaa.68.152
Acknowledgements
The authors thank the referees for their helpful comments and suggestions to improve our manuscript.
Author information
Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly to writing this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Nunokawa, M., Aydog̃an, M., Kuroki, K. et al. Some properties concerning closetoconvexity of certain analytic functions. J Inequal Appl 2012, 245 (2012). https://doi.org/10.1186/1029242X2012245
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029242X2012245
Keywords
 analytic
 starlike
 convex
 closetoconvex
 subordination