The coefficient problem for bounded functions and its applications

Abstract

A review of the solution of the classical coefficient problem on the class "Ω" _0 of bounded in the unit disc functions ω with normalization ω(0)=0, going back to I. Schur, is given. Then the first six inequalities, describing respectively the first six coefficient bodies on the class "Ω" _0, are derived. Next, a method of obtaining similar inequalities for classes M_F of functions subordinated to the holomorphic function F, giving the solution of the coefficient problem for these classes, is given. Then the properties of the mentioned inequalities as well as the relations between them are analyzed. In addition, it is shown that only one n-th inequality is sufficient to describe the n-th body of coefficients on the class "Ω" _0, and hence on M_F. The problems of estimating both the modulus of each initial Taylor coefficient individually and estimating modules of all Taylor coefficients at once are discussed. The problem of obtaining the sharp estimates of the modulus of the Taylor coefficient with number n, i.e. the functional |〖{f}〗_n |, on the class M_F is at first reduced to the problem of estimating the functional over the class "Ω" _0, which in turn is reduced to the problem of finding the maximal modulo of conditional extremum of a real-valued function of 2(n-1) real arguments with constraints of inequality type 0≤x_k≤1, 0≤φ_k <2π, which allows us to apply standard methods of differential calculus to study for extrema, since the target function is infinitely smooth in all of its arguments. For this purpose, the results of the solution of the classical coefficient problem on the class "Ω" _0 are used.