A certain general class 𝓢(a,c,A,B) of analytic functions involving a linear operator is introduced. The objective is to investigate various properties and characteristics of this class. Several applications of the results (obtained here) to a class of fractional calculus operators are also considered. The results contain some of the earlier work in univalent function theory.
Let Mₚ denote the class of functions f of the form , p a positive integer, in the unit disk E = |z| < 1, f being regular in 0 < |z| < 1. Let , α < 1, where . Results on are derived by proving more general results on differential subordination. These results reduce, by putting p =1, to the recent results of Al-Amiri and Mocanu.
We geometrically define subclasses of starlike functions related to the class of uniformly starlike functions introduced by A. W. Goodman in 1991. We give an analytic characterization of these classes, some radius properties, and examples of functions in these classes. Our classes generalize the class of uniformly starlike functions, and many results of Goodman are special cases of our results.
Some inequalities are proved for coefficients of functions in the class P(α), where α ∈ [0,1), of functions with real part greater than α. In particular, new inequalities for coefficients in the Carathéodory class P(0) are given.