# 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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