- Open Access
Generalized conditions for starlikeness and convexity of certain analytic functions
© Uyanik and Owa; licensee Springer. 2011
Received: 4 June 2011
Accepted: 17 October 2011
Published: 17 October 2011
For analytic functions f(z) in the open unit disk with f (0) = 0 and f '(0) = 1, Nunokawa et al. (Turk J Math 34, 333-337, 2010)have shown some conditions for starlikeness and convexity of f(z). The object of the present paper is to derive some generalized conditions for starlikeness and convexity of functions f(z) with examples.
2010 Mathematics Subject Classification: Primary 30C45.
which are analytic in the open unit disk . Let be the subclass of consisting of functions f(z) which are univalent in . A function is said to be starlike with respect to the origin in if is the starlike domain. We denote by the class of all starlike functions f(z) with respect to the origin in . Furthermore, if a function satisfies , then f(z) is said to be convex in . We also denote by the class of all convex functions in . Note that .
To discuss the univalency of , Nunokawa  has given
Lemma 1.1 If satisfies , then . Also, Mocanu  has shown that
In view of Lemmas 1.1 and 1.2, Nunokawa et al.  have proved the following results.
The object of the present paper is to consider some generalized conditions for functions f(z) to be in the classes or .
2 Generalized conditions for starlikeness
We begin with the statement and the proof of generalized conditions for starlikeness.
then . Thus, the theorem is holds true for j = 3.
Therefore, the theorem holds true for j = k. Thus, applying the mathematical induction, we complete the proof of the theorem.
is in the class .
3 Generalized conditions for convexity
For the convexity of f(z), we derive
then . This shows that the theorem is true for j = 3.
then . Thus, the result is true for j = k. Using the mathematical induction, we complete the proof the theorem.
belongs to the class .
If we use the same technique as in the proof of Theorem 2.1 applying Lemma 1.4, then we have
for some j (j = 2, 3, 4, ...), then , where M is given by (2.2).
This paper was completed when the first author was visiting Department of Mathematics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan, from Atat ürk University, Turkey, between February 17 and 26, 2011.
- Nunokawa M: On the order of strongly starlikeness of strongly convex functions. Proc Jpn Acad 1993, 68: 234–237.MathSciNetView ArticleGoogle Scholar
- Mocanu PT: Some starlikeness conditions for analytic function. Rev Roum Math Pures Appl 1988, 33: 117–124.MathSciNetGoogle Scholar
- Nunokawa M, Owa S, Polatoglu Y, Caglar M, Duman EY: Some sufficient conditions for starlikeness and convexity. Turk J Math 2010, 34: 333–337.Google Scholar
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