A new method of estimation of moduli of initial Taylor coefficients on the class of bounded non-vanishing functions

Abstract

The task of obtaining the sharp estimate of the modulus of the n-th Taylor coefficient on the class B of bounded non-vanishing functions has been reduced to the problem of estimating a functional over the class of normalized bounded functions, which in turn has been reduced to the problem of finding the constrained maximum of a non-negative objective function of 2n-3 real arguments with constraints of the inequality type, that allows us to apply the standard numerical methods of finding constrained extrema. Analytical expressions of the first six objective functions have been obtained and their Lipschitz continuity has been proved. Based on the Lipschitz continuity of the objective function with number n, a method for the sharp estimating of the modulus of the n-th Taylor coefficient on the class B is rigorously proven. An algorithm of finding the global constrained maximum of the objective function is being discussed. The first step of this algorithm involves a brute-force search with a relatively large step. The second step of the algorithm uses a method for finding a local maximum with the initial points obtained at the previous step. The results of the numerical calculations are presented graphically and confirm the Krzyz conjecture for n=1,…,6. Based on these calculations, as well as on so-called asymptotic estimates, a sharp estimate of the moduli of the first six Taylor coefficients on the class B is derived. The obtained results are compared with previously known estimates of the moduli of initial Taylor coefficients on the class B and its subclasses B_t, t≥0. The extremals for B_t subclasses are discussed and the Krzyz hypothesis is updated for B_t subclasses. A brief historical overview of research of the estimations of moduli of initial Taylor coefficients on the class B is provided.