Generalized Typically Real Functions

Abstract

Let $f(z) = z+a_2 z^2+\cdots$ be regular in the
unit disk  and real valued if and only if $z$ is real and $|z| < 1$.
Then $f$ 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 presented paper is a
consideration of the generalized typically-real functions defined
via the generating function of the generalized Chebyshev polynomials
of the second kind
$$
\displaystyle
\Psi_{p,q}(e^{i\theta};z)=\frac{1}{(1-pze^{i\theta})(1-qze^{-i\theta})}=\sum_{n=0}^\infty
U_n(p,q;e^{i\theta})z^n, $$ where $-1\le p,q \le 1, \ \theta \in
\langle 0,2\pi\rangle, \ |z|<1.$