Distortion Inequalities Research Articles

ON DISTORTION THEOREMS INVOLVING GENERALIZED FRACTIONAL CALCULUS OPERATORS


 
 
 This paper gives new classes of distortion inequalities for various sub- classes of analytic and univalent functions. The results presented involve certain generalized fractional integrals of functions belonging to the general classes $J_{\delta}(n)$ and $L_\delta(n)$. Some useful deductions of our main results are also pointed out. 
 
 

Behavior of domain constants under conformal mappings

Domain constants are numbers attached to regions in the complex plane ℂ. For a region Ω in ℂ, letd(Ω) denote a generic domain constant. If there is an absolute constantM such thatM −1≤d(Ω)/d(Δ)≤M whenever Ω and Δ are conformally equivalent, then the domain constant is called quasiinvariant under conformal mappings. IfM=1, the domain constant is conformally invariant. There are several standard problems to consider for domain constants. One is to obtain relationships among different domain constants. Another is to determine whether a given domain constant is conformally invariant or quasi-invariant. In the latter case one would like to determine the best bound for quasi-invariance. We also consider a third type of result. For certain domain constants we show there is an absolute constantN such that |d(Ω)−d(Δ)|≤N whenever Ω and Δ and conformally equivalent, sometimes determing the best possible constantN. This distortion inequality is often stronger than quasi-invariance. We establish results of this type for six domain constants.

Fractional calculus and certain starlike functions with negative coefficients

A certain subclass T( n, p, λ, α) of starlike functions in the unit disk is introduced. The object of the present paper is to derive several interesting properties of functions belonging to the class T( n, p, λ, α). Various distortion inequalities for fractional calculus of functions in the class T( n, p, λ, α) are also given.

On the area distortion by quasiconformal mappings

We give the sharp constants in the area distortion inequality for quasiconformal mappings in the plane.

On distortion of the real part in a class of analytic functions related to fractional integration

In close relation to fractional integration a one-parameter family of operators acting on a class of analytic functions regular in the unit disk is observed. In considering functionals referring to the real part, the author derives distortion inequalities showing how their bounds change after application of operators. The case of a particular family generated by a special measure is discussed in detail.