On the Distortion Theorem for Quasiconformal Mappings with Fixed Boundary Values

AbstractFor a planar domain Ω with at least three boundary points and ρΩ the hyperbolic metric of Ω with constant curvature −1, G. J. Martin poses a problem that asks, if f is a K-quasiconformal self-homeomorphism of Ω with boundary values given by the identity mapping, whether ρΩ(z,f(z))≤logK holds for z,f(z)∈Ω. In this note, we give a negative answer to this question

Sufficient Conditions for Starlike Functions Associated with the Lemniscate of Bernoulli

Let -1\leq B<A\leq 1. Condition on \beta, is determined so that 1+\beta zp'(z)/p^k(z)\prec(1+Az)/(1+Bz)\;(-1<k\leq3) implies p(z)\prec \sqrt{1+z}. Similarly, condition on \beta is determined so that 1+\beta zp'(z)/p^n(z) or p(z)+\beta zp'(z)/p^n(z)\prec\sqrt{1+z}\;(n=0, 1, 2) implies p(z)\prec(1+Az)/(1+Bz) or \sqrt{1+z}. In addition to that condition on \beta is derived so that p(z)\prec(1+Az)/(1+Bz) when p(z)+\beta zp'(z)/p(z)\prec\sqrt{1+z}. Few more problems of the similar flavor are also considered

Argument estimates of certain multivalent functions involving a linear operator

The purpose of this paper is to derive some argument properties of certain multivalent functions in the open unit disk involving a linear operator. We also investigate their integral preserving property in a sector

Starlike and Convex Properties for Hypergeometric Functions

The purpose of the present paper is to give some characterizations for a (Gaussian) hypergeometric function to be in various subclasses of starlike and convex functions. We also consider an integral operator related to the hypergeometric function

Sandwich-Type Theorems for a Class of Multiplier Transformations Associated with the Noor Integral Operators

We obtain some subordination- and superordination-preserving properties for a class of multiplier transformations associated with Noor integral operators defined on the space of normalized analytic functions in the open unit disk. The sandwich-type theorems for these transformations are also considered

First-order differential subordinations associated with Carathéodory functions

In the present paper, we investigated some conditions to be in the class of Carathéodory functions by using the concept of the first-order differential subordinations. Moreover, various interesting special cases were considered in the geometric function theory as applications of main results presented here