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Journal of the Ramanujan Mathematical Society

Volume 35, Issue 3, September 2020  pp. 217–226.

On a class of univalent functions defined by a differential inequality

Authors:  Md Firoz Ali, Vasudevarao Allu and Hiroshi Yanagihara
Author institution:Department of Mathematics, National Institute of Technology Durgapur, West Bengal 713 209, India

Summary:  For 0 < λ ≤ 1, let U(λ) be the class of analytic functions f(z) = z+Σ n=2 \infty an zn in the unit disk D satisfying |f'(z)(z/f(z))2 - 1| < λ and U:=U(1). In the present article, we prove that the class U is contained in the closed convex hull of the class of starlike functions and using this fact, we solve some extremal problems such as integral mean problem and arc length problem for functions in U. By means of the so called theory of star functions, we also solve the integral mean problem for functions in U(λ). We also obtain the estimate of the Fekete-Szegö functional and the pre-Schwarzian norm of certain nonlinear integral transform of functions in U(λ). Further, for the class of meromorphic functions which are defined in Δ:={ζ ∈ C:|Δ|>1} and associated with the class U(λ), we obtain a sufficient condition for a function g to be an extreme point of this class.

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