- Research
- Open access
- Published:
Certain sufficient conditions for strongly starlikeness and convexity
Fixed Point Theory and Applications volume 2013, Article number: 88 (2013)
Abstract
The object of the present paper is to derive some sufficient conditions for strongly starlikeness and convexity.
MSC:30C45.
1 Introduction
Let () denote the class of functions of the form
which are analytic in the open unit disc . We write . Let and K be the subclasses of consisting of all starlike functions in U and of all convex functions in U, respectively.
If satisfies
for some γ (), then is said to be strongly starlike of order γ in U, and denoted by . If satisfies
for some γ (), then we say that is strongly convex of order γ in U, and we denote by the class of all such functions. It is obvious that belongs to if and only if . Further, we note that and .
The strongly starlike and convex functions have been extensively studied by several authors (see, e.g., [1–11]). The object of the present paper is to derive some sufficient conditions for strongly starlikeness and strongly convexity. Some previous results are extended.
For our purpose, we have to recall here the following results.
Lemma 1 (see [5])
Let a function be analytic in U and (). If there exists a point such that
and
then
where

and ().
Lemma 2 (see [4])
If satisfies
then .
2 Starlikeness and convexity
Our first result is contained in the following.
Theorem 1 Let . If () satisfies
then , where
Proof
Note that
and
Let
and
Then, by using (2.2), we have that
Since the condition (2.1) implies that
we obtain that
Furthermore, since
we conclude from (2.1) and (2.3) that
which shows that . □
Theorem 2 Let . If () satisfies
then , where is the root of the equation
Proof
Note that
If there exists a point such that
and
then by Lemma 1, we have
Therefore, if , then we have
which contradicts (2.4). If , then applying the same method for the previous case, we also have
which contradicts (2.4). Therefore, there exists no such that . This implies that
Furthermore, since
we conclude that
which shows that . □
Theorem 3 If () satisfies
then .
Proof From (2.5), one can see that

Noting that
By Lemma 2, we have . □
Theorem 4 If () satisfies
then .
Proof
By using the same method as in the proof of Theorem 3, we have
It follows that
Therefore, using Lemma 2, we see that , or . □
References
Gangadharan A, Ravichandran V: Radii of convexity and strong starlikeness for some classes of analytic functions. J. Math. Anal. Appl. 1997, 211: 303–313.
Liu J-L: The Noor integral operator and strongly starlike functions. J. Math. Anal. Appl. 2001, 261: 441–447. 10.1006/jmaa.2001.7489
Liu J-L: Certain sufficient conditions for strongly starlike functions associated with an integral operator. Bull. Malays. Math. Soc. 2011, 34: 21–30.
Mocanu PT: Some starlikeness conditions for analytic functions. Rev. Roum. Math. Pures Appl. 1988, 33: 117–124.
Nunokawa M: On the order of strongly starlikeness of strongly convex functions. Proc. Jpn. Acad., Ser. A, Math. Sci. 1993, 68: 234–237.
Nunokawa M, Owa S, Polatoglu Y, Caglar M, Duman EY: Some sufficient conditions for starlikeness and convexity. Turk. J. Math. 2010, 34: 333–337.
Nunokawa M, Owa S, Saitoh H, Ikeda A, Koike N: Some results for strongly starlike functions. J. Math. Anal. Appl. 1997, 212: 98–106. 10.1006/jmaa.1997.5468
Nunokawa M, Thomas DK: On convex and starlike functions in a sector. J. Aust. Math. Soc. A 1996, 60: 363–368. 10.1017/S1446788700037873
Obradovic M, Owa S: Some sufficient conditions for strongly starlikeness. Int. J. Math. Math. Sci. 2000, 24: 643–647. 10.1155/S0161171200004154
Ponnusamy S, Singh V: Criteria for strongly starlike functions. Complex Var. Theory Appl. 1997, 34: 267–291. 10.1080/17476939708815053
Xu N, Yang D-G, Owa S: On strongly starlike multivalent functions of order β and type α . Math. Nachr. 2010, 283: 1207–1218. 10.1002/mana.200710077
Acknowledgements
Dedicated to Professor Hari M Srivastava.
We would like to express sincere thanks to the referees for careful reading and suggestions which helped us to improve the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors have made the same contribution. 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
Tao, YQ., Liu, JL. Certain sufficient conditions for strongly starlikeness and convexity. Fixed Point Theory Appl 2013, 88 (2013). https://doi.org/10.1186/1687-1812-2013-88
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2013-88