Some properties concerning close-to-convexity of certain analytic functions
© Nunokawa et al.; licensee Springer 2012
Received: 1 August 2012
Accepted: 8 October 2012
Published: 24 October 2012
Let be an analytic function in the open unit disk normalized with and . With the help of subordinations, for convex functions in , the order of close-to-convexity for is discussed with some example.
for some real α (). This family of functions was introduced by Robertson  and we denote it by .
for some real α ().
This class was also introduced by Robertson  and we denote it by . By the definitions for the classes and , we know that if and only if .
MacGregor  determined the exact value of for each α () as the infimum over the disc of the real part of a specific analytic function. It has been conjectured that this infimum is attained on the boundary of at .
for some convex function , then is univalent in . In view of Kaplan , we say that satisfying the above inequality is close-to-convex in .
for some starlike function .
for some real α () and for some starlike function in .
Then we call close-to-convex of order α in with respect to .
It is the purpose of the present paper to investigate the order of close-to-convexity of the functions which satisfy and .
To discuss our problems, we have to give here the following lemmas.
where ≺ means the subordination, and .
A proof is very easily obtained.
where and .
This completes the proof of the lemma. □
Our next lemma is
for some real c ().
then satisfies the conditions of the lemma.
Therefore, from (3) we obtain (2) .
For the case , applying the same method as above, we have the same conclusion as in the case . □
3 The order of close-to-convexity
Now, we discuss the close-to-convexity of with the help of lemmas.
This means that is close-to-convex of order c in .
- (i)For the case , if there exists a point such that
This completes the proof of the theorem. □
Applying Theorem 1, we have the following corollary.
which means that is close-to-convex of order in .
Next we show
then we can say that is close-to-convex in . But c should be a negative real number in Theorem 2. Therefore, we cannot say that is close-to-convex in in Theorem 2.
The authors thank the referees for their helpful comments and suggestions to improve our manuscript.
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