Generalized Typically Real Functions

Stanisława Kanas, Anna Tatarczak

Let $f(z)=z+a_2z^2+\cdots$ 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 \[ \Psi_{p,q}(e^{iheta};z)=\frac1{(1-pze^{iheta})(1-qze^{-iheta})}=um_{n=0}^ıfty U_n(p,q,e^{iheta})z^n, \] where $-1\leq p,q|leq1$, $\theta\in\langle0,2\pi\rangle$, $|z|<1$.