The main object of the present paper is to show certain sufficient conditions for univalency of analytic functions with missing coefficients.
MSC: 30C45, 30C55.
Keywords:analytic; univalent; subordination
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.
2 Main results
Our first theorem is given by the following.
Thus, we have
It is easily seen from (2.1), (2.2) and (2.5) that
and, in consequence,
Now it follows from (2.4) and (2.7) that
Next we derive the following.
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
Now, from (2.15), we can easily derive the inequalities (2.8), (2.9) and (2.10). □
Finally, we discuss the following theorem.
If we put
It follows from the lemma that
which implies that
Then by (2.20), we have
Therefore, it follows from (2.22) and (2.24) that
Then we have
The authors declare that they have no competing interests.
The authors have made the same contribution. All authors read and approved the final manuscript.
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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