- Open Access
Certain sufficient conditions for strongly starlikeness and convexity
© Tao and Liu; licensee Springer 2013
- Received: 16 January 2013
- Accepted: 25 March 2013
- Published: 10 April 2013
The object of the present paper is to derive some sufficient conditions for strongly starlikeness and convexity.
- analytic function
- starlike function
- convex function
- strongly starlike function
- strongly convex function
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.
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 )
Lemma 2 (see )
Our first result is contained in the following.
which shows that . □
which shows that . □
By Lemma 2, we have . □
Therefore, using Lemma 2, we see that , or . □
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.
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