doiSerbia

Generalized typically real functions

Kanas Stanisława (University of Rzeszów, Rzeszów, Poland)
Tatarczak Anna (Maria Curie-Sklodowska University, Lublin, Poland)

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 Ψp,q(eiθ;z)=1 /(1-pzeiθ)(1-qze-iθ) = ∑∞,n=0 Un(p,q; eiθ)zn, where -1 ≤ p,q ≤ 1; θ 0,2πi, |z|<1.

Keywords: typically real functions, generalized Koebe function, univalent functions