The problem of finding the exact radius of univalence and the starlikeness of functions regular in the unit disc defined by integral representations is considered. A generalized interpretations of a whole series of problems that are encountered separately in the literature is given. In this paper within the presented theorem fairly general conditions for estimating the radius of starlikeness are established. The proof of the theorem reduces to finding a lower bound for a functional that depends on the value of the function and its derivative on the class of functions regular in the unit disc that have a positive real part in the disc. When the integral operator is specified and a special choice of functions regular in the disc is obtained, a number of corollaries of the theorem are obtained. The theorem and its consequences generalize similar results obtained earlier by different methods. An explicit expression is given for realizing estimates of the radius of extremal functions.

В настоящей работе методами геометрической теории функций установлены достаточные условия, при которых регулярные в единичном круге $|z|<1$ функции, определяемые интегральными операторами, однолистно отображают круг $|z|<r$, $0<r<1$ на звездообразные области. Изучение геометрических характеристик регулярных функций осуществляется редукцией в класс $P$ регулярных в круге $|z|<1$ функций $p(z)$, $p(0)=1$ имеющих в круге положительную вещественную часть.