Certain sufficient conditions for strongly starlikeness and convexity
Fixed Point Theory and Applications volume 2013, Article number: 88 (2013)
The object of the present paper is to derive some sufficient conditions for strongly starlikeness and convexity.
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.
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 )
Let a function be analytic in U and (). If there exists a point such that
Lemma 2 (see )
2 Starlikeness and convexity
Our first result is contained in the following.
Theorem 1 Let . If () satisfies
then , where
Then, by using (2.2), we have that
Since the condition (2.1) implies that
we obtain that
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
If there exists a point such that
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
we conclude that
which shows that . □
Theorem 3 If () satisfies
Proof From (2.5), one can see that
By Lemma 2, we have . □
Theorem 4 If () satisfies
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 . □
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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.
The authors declare that they have no competing interests.
The authors have made the same contribution. All authors read and approved the final manuscript.
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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