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On certain univalent functions with missing coefficients
Advances in Difference Equations volume 2013, Article number: 89 (2013)
The main object of the present paper is to show certain sufficient conditions for univalency of analytic functions with missing coefficients.
Let be the class of functions of the form
which are analytic in the unit disk . We write .
A function is said to be starlike in () if and only if it satisfies
A function is said to be close-to-convex in () if and only if there is a starlike function such that
Let and be analytic in U. Then we say that is subordinate to in U, written , if there exists an analytic function in U, such that and (). If is univalent in U, then the subordination is equivalent to and .
Recently, several authors showed some new criteria for univalency of analytic functions (see, e.g., [1–7]). In this note, we shall derive certain sufficient conditions for univalency of analytic functions with missing coefficients.
For our purpose, we shall need the following lemma.
Let and be analytic in U with . If is starlike in U and , then
2 Main results
Our first theorem is given by the following.
Theorem 1 Let with for . If
where , then is univalent in U.
then is analytic in U. By integration from 0 to z n-times, we obtain
Thus, we have
It is easily seen from (2.1), (2.2) and (2.5) that
and, in consequence,
for and .
Now it follows from (2.4) and (2.7) that
Hence, is univalent in U. The proof of the theorem is complete. □
Let denote the class of functions with for , which satisfy the condition (2.1) given by Theorem 1.
Next we derive the following.
Theorem 2 Let . Then, for ,
Proof In view of (2.1), we have
Applying Lemma to (2.11), we get
By using the lemma repeatedly, we finally have
According to a result of Hallenbeck and Ruscheweyh [, Theorem 1], (2.13) gives
where is analytic in U and ().
Now, from (2.15), we can easily derive the inequalities (2.8), (2.9) and (2.10). □
Finally, we discuss the following theorem.
Theorem 3 Let and have the form
If , then is starlike in ;
If , then is close-to-convex in .
If we put
then by (2.1) and the proof of Theorem 2 with , we have
It follows from the lemma that
which implies that
Then by (2.20), we have
Also, from (2.8) in Theorem 2 with , we obtain
Therefore, it follows from (2.22) and (2.24) that
for . This proves that is starlike in .
Then we have
Thus, for . This shows that is close-to-convex in . □
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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.