Necessary and sufficient conditions for univalent functions

Abstract

Let 𝒜 be the class of analytic functions in the unit disc with the normalization f (0) = f ′(0) − 1 = 0. This article analyses various necessary and sufficient coefficient conditions for functions f ∈ 𝒜 of the form

to be univalent. We present an interesting class of univalent functions associated with the zeta function and also pose an open problem.

Keywords:

• coefficient inequality
• analytic
• Hadamard convolution
• univalent and starlike functions
• zeta function

AMS Subject Classifications::

• Primary: 30C45
• Secondary: 30C20, 30C75, 30C80

Acknowledgements

The work of the second author was supported by MNZZS Grant No. ON174017, Serbia. The research of the first author was supported in part by a Research University Grant from the Universiti Sains Malaysia. The work was completed during the visit of the third author to the Universiti Sains Malaysia.