Abstract
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; subordination1 Introduction
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 closetoconvex 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., [17]). 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.
Letandbe analytic inUwith. Ifis starlike inUand, then
2 Main results
Our first theorem is given by the following.
Proof
Let
then is analytic in U. By integration from 0 to zntimes, we obtain
Thus, we have
where
It is easily seen from (2.1), (2.2) and (2.5) that
and, in consequence,
Since
we get
and so
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.
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 [[1], Theorem 1], (2.13) gives
i.e.,
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.
(ii) If, thenis closetoconvex in.
Proof
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
and so
Therefore, it follows from (2.22) and (2.24) that
for . This proves that is starlike in .
Then we have
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors have made the same contribution. All authors read and approved the final manuscript.
Acknowledgements
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.
References

Dziok, J, Srivastava, HM: Certain subclasses of analytic functions associated with the generalized hypergeometric function. Integral Transforms Spec. Funct.. 14, 7–18 (2003). Publisher Full Text

Nunokawa, M, Obradovič, M, Owa, S: One criterion for univalency. Proc. Am. Math. Soc.. 106, 1035–1037 (1989). Publisher Full Text

Obradovič, M, Pascu, NN, Radomir, I: A class of univalent functions. Math. Jpn.. 44, 565–568 (1996)

Owa, S: Some sufficient conditions for univalency. Chin. J. Math.. 20, 23–29 (1992)

Samaris, S: Two criteria for univalency. Int. J. Math. Math. Sci.. 19, 409–410 (1996). Publisher Full Text

Silverman, H: Univalence for convolutions. Int. J. Math. Math. Sci.. 19, 201–204 (1996). Publisher Full Text

Yang, DG, Liu, JL: On a class of univalent functions. Int. J. Math. Math. Sci.. 22, 605–610 (1999). Publisher Full Text

Hallenbeck, DJ, Ruscheweyh, S: Subordination by convex functions. Proc. Am. Math. Soc.. 51, 191–195 (1975). Publisher Full Text

Suffridge, TJ: Some remarks on convex maps of the unit disk. Duke Math. J.. 37, 775–777 (1970). Publisher Full Text