Extremal coefficient properties of Pick functions are proved. Even coefficients of analytic univalent functions f with |f(z)| < M, |z| < 1, are bounded by the corresponding coefficients of the Pick functions for large M. This proves a conjecture of Jakubowski. Moreover, it is shown that the Pick functions are not extremal for a similar problem for odd coefficients.
MSC 2010: 30C10The classical notion of apolarity is defined for two algebraic polynomials of equal degree. The main property of two apolar polynomials p and q is the classical Grace theorem: Every circular domain containing all zeros of p contains at least one zero of q and vice versa. In this paper, the definition of apolarity is extended to polynomials of different degree and an extension of the Grace theorem is proved. This leads to simplification of the conditions of several well-known results...
We give a new proof of Beurling’s result related to the equality of the extremal length and the Dirichlet integral of solution of a mixed Dirichlet-Neuman problem. Our approach is influenced by Gehring’s work in space. Also, some generalizations of Gehring’s result are presented.