Generalized typically real functions

Abstract

Let f(z)=z+a2z2+... be regular in the unit disk and real valued if and only if z is real and |z| < 1. Then f(z) is said to be typically real function. Rogosinski found the necessary and sufficient condition for a regular function to be typically-real. The main purpose of the paper is a consideration of the generalized typically-real functions defined via the generating function of the generalized Chebyshev polynomials of the second kind ?p,q(ei?;z)=1 /(1-pzei?)(1-qze-i?) = ??,n=0 Un(p,q; ei?)zn, where -1 ? p,q ? 1; ?? ?0,2??i, |z|<1.