Generalized typically real functions

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 $-1\le p,q \le 1, \ \theta \in \langle 0,2\pi\rangle, \ |z|<1.$