Constrained coefficients problem for generalized typically real functions

Abstract

For −1 ≤ p, q ≤ 1 let T p,q denote the class of generalized typically real functions i.e. the class of functions of a formwhere |z| < 1, and μ(θ) is the unique probability measure on the interval 0, 2π . For the same range of parameters, let the generalized Chebyshev polynomials of the second kind U n (p, q; e iθ ) be defined as followsWe see thatThe main purpose of a paper is a solution of coefficients problems in T p,q . Problem related to the well-known Zalcman conjecture is presented.

An Extension of the Chebyshev Polynomials

“…In [15] we prove coefficients bounds of functions from the class T p,q . In the case when p = q = 1 these results become the well known estimates in the class of typically real functions T R .…”