Distortion Theorem for Certain Class of Bazilevic Functions (original) (raw)

A Distortion Theorem for a Subclass of Univalent Functions

Journal of Contemporary Mathematical Analysis

We give a distortion theorem for a subclass of functions that are univalent in the unit disk, and defined using the Sˆalˆagean differential operator. The result generalizes and unifies similar well known results for several subclasses of univalent functions defined on the unit disk having form f(z)=z+∑_(n=2)^∞▒a_n z^n normalized such that f(0) = 0 and f ‘ (0) = 1.

Certain subclasses of analytic and bi-univalent functions

Applied Mathematics Letters, 2010

In the present paper, we introduce and investigate two interesting subclasses of normalized analytic and univalent functions in the open unit disk U := {z : z ∈ C and |z| < 1}, whose inverse has univalently analytic continuation to U. Among other results, bounds for the Taylor-Maclaurin coefficients |a 2 | and |a 3 | are found in our investigation.

Some Properties of a Subclass of Univalent Functions

Advances in Mathematics

We give some coefficient bounds and distortion theorems for a new subclass of functions univalent on the unit disk defined using the Sâlâgean differential operator. The results generalizes and unifies some well known results for several subclasses of univalent functions defined on the unit disk having form f (z) = z + ∞ n=2 a n z n , normalized such that f (0) = 0 and f (0) = 1.

A New Sub Class of Univalent Analytic Functions Involving a Linear Operator

This paper deals with a new class T (α, β, a, b; c) that is a subclass of uniformly starlikefunctions involving a linear operator L (a, b; c). Coefficients inequality, Distortion theorem, Extreme points, Radius of starlikeness and radius of convexity for functions belonging to this class are obtained.

Certain Transformations Preserving Families of Univalent Analytic Functions

2015

The article deals with the family mathcalU(lambda){\mathcal U}(\lambda)mathcalU(lambda) of all functions fff normalized and analytic in the unit disk such that big∣big(z/f(z)big)2f′(z)−1big∣<lambda\big |\big (z/f(z)\big )^{2}f'(z)-1\big |<\lambda bigbig(z/f(z)big)2f(z)1big<lambda for some 0<lambdaleq10<\lambda \leq 10<lambdaleq1. The family mathcalU(lambda){\mathcal U}(\lambda)mathcalU(lambda) has been studied extensively in the recent past and functions in this family are known to be univalent in ID\IDID. However, the problem of determining sharp bounds for the second coefficients of functions in this family was solved recently in \cite{VY2013} by Vasudevarao and Yanagihara but the proof was complicated. In this article, we first present a simpler proof. We obtain a number of new subordination results for this family and their consequences. In addition, we show that the family mathcalU(lambda){\mathcal U}(\lambda )mathcalU(lambda) is preserved under a number of elementary transformations such as rotation, conjugation, dilation and omitted value transformations, but surprisingly this family is not preserved under the nnn-th root transformation for any ngeq2n\geq 2ngeq2. This is a basic here which helps to generate a number of new theorems and in particular provides a way for constructions of functions from the family mathcalU(lambda){\mathcal U}(\lambda)mathcalU(lambda). Finally, we deal with a radius problem.

On a generalized class of analytic functions related to Bazilevič functions

Acta et Commentationes Universitatis Tartuensis de Mathematica

Using operator Lp(a, c) introduced by Saitoh (Math. Japon. 44 (1996), 31–38) we define the subclass Hp,nν,μ (a, c; ϕ) of the class A(p, n) and establish containment, subordination and coefficient inequalities of this subclass. We indicate the connections of our results with earlier results obtained by other researchers.