EXACT ESTIMATES OF REGULAR FUNCTIONS AND RADII OF CONVEXITY AND STARLIKENESS OF SOME CLASSES OF STARLIKE AND CLOSE-TO-STARLIKE FUNCTIONS

Abstract

It is known that many problems for subclasses of univalent functions can be transformed into problems of minimizing or maximizing some functionals associated with the studied subclasses of univalent functions. Often, the logarithmic derivative of regular functions acts as such a functional. In this paper, we introduce a two-parameter subclass of functions regular in the unit circle with a positive real part, the expansion into a series of which begins with the nth degree. This class generalizes the well-known R.Goel and D.Shaffer class of regular functions whose values are contained in a circle symmetric with respect to the real axis containing the point 0 on the boundary. In this class of functions, exact estimates of various functionals, including the logarithmic derivative, are obtained. As applications of these estimates, the exact radii of convexity (or starlikeness) of various classes of starlike and close-to-starlike functions given using the class are found. All the results obtained are accurate and generalize many of the previously known results. The application of the estimates obtained in the article is promising, as it contributes to the theory of extreme problems associated with various subclasses of univalent functions.