To obtain some estimates of the coefficients of functions regular and univalent in the circle, let us use the Grunsky condition: for a function univalent in the circle and regular in it, it is necessary and sufficient that inequality (1) be satisfied for any sequence {xₙ} for which the series on the right-hand side of the inequality converges, and the coefficients aₘₙ are determined by the expansion...

Let S(k), k=1, 2, . . . . is the class of functions, regular and univalent in the disk |z| < 1. Sα(k) is a subclass of functions S(k). This class of functions is non-empty for any αk, 0 < αk ≤ 2/k. …