- Open Access
On certain univalent functions with missing coefficients
© Cang and Liu; licensee Springer. 2013
- Received: 12 January 2013
- Accepted: 23 March 2013
- Published: 3 April 2013
The main object of the present paper is to show certain sufficient conditions for univalency of analytic functions with missing coefficients.
which are analytic in the unit disk . We write .
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.
Our first theorem is given by the following.
where , then is univalent in U.
for and .
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.
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.
If , then is starlike in ;
If , then is close-to-convex in .
- (i)Let and(2.21)
- (ii)Let and(2.25)
Thus, for . This shows that is close-to-convex in . □
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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