Certain subclasses of uniformly starlike and convex functions defined by convolution. (English) Zbl 1224.30030
Summary: The aim of this paper is to obtain coefficient estimates, distortion theorems, convex linear combinations and radii of close-to-convexity, starlikeness and convexity for functions belonging to the subclass \(\text{TS}_\gamma(f,g;\alpha,\beta)\) of uniformly starlike and convex functions. We consider integral operators associated with functions in this class. Furthermore, partial sums \(f_n(z)\) of functions \(f(z)\) in the class \(\text{TS}_\gamma(f,g;\alpha,\beta)\) are considered and sharp lower bounds for the ratios of the real parts of \(f(z)\) to \(f_n(z)\) and \(f'(z)\) to \(f'_n(z)\) are determined.
MSC:
30C45 | Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) |