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.
Keywordsanalytic univalent subordination
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.
2 Main results
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.
- Dziok J, Srivastava HM: Certain subclasses of analytic functions associated with the generalized hypergeometric function. Integral Transforms Spec. Funct. 2003, 14: 7–18. 10.1080/10652460304543MATHMathSciNetView ArticleGoogle Scholar
- Nunokawa M, Obradovič M, Owa S: One criterion for univalency. Proc. Am. Math. Soc. 1989, 106: 1035–1037. 10.1090/S0002-9939-1989-0975653-5MATHView ArticleGoogle Scholar
- Obradovič M, Pascu NN, Radomir I: A class of univalent functions. Math. Jpn. 1996, 44: 565–568.MATHGoogle Scholar
- Owa S: Some sufficient conditions for univalency. Chin. J. Math. 1992, 20: 23–29.MATHGoogle Scholar
- Samaris S: Two criteria for univalency. Int. J. Math. Math. Sci. 1996, 19: 409–410. 10.1155/S0161171296000579MATHMathSciNetView ArticleGoogle Scholar
- Silverman H: Univalence for convolutions. Int. J. Math. Math. Sci. 1996, 19: 201–204. 10.1155/S0161171296000294MATHView ArticleGoogle Scholar
- Yang D-G, Liu J-L: On a class of univalent functions. Int. J. Math. Math. Sci. 1999, 22: 605–610. 10.1155/S0161171299226051MATHMathSciNetView ArticleGoogle Scholar
- Hallenbeck DJ, Ruscheweyh S: Subordination by convex functions. Proc. Am. Math. Soc. 1975, 51: 191–195. 10.1090/S0002-9939-1975-0402713-XMathSciNetView ArticleGoogle Scholar
- Suffridge TJ: Some remarks on convex maps of the unit disk. Duke Math. J. 1970, 37: 775–777. 10.1215/S0012-7094-70-03792-0MATHMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.