On Coefficients Problems for Typically Real Functions Related to Gegenbauer Polynomials

Abstract

Abstract. We solve problems concerning the coefficients of functions in the class T (λ) of typically real functions associated with Gegenbauer polynomials. The main aim is to determine the estimates of two expressions: |a4 − a2a3| and |a2a4 − a3 2 |. The second one is known as the second Hankel determinant. In order to obtain these bounds, we consider the regions of variability of selected pairs of coefficients for functions in T (λ). Furthermore, we find the upper and the lower bounds of functionals of Fekete-Szegö type. Finally, we present some conclusions for the classes T and T (1/2).Mathematics Subject Classification. 30C50.

A Certain Subclass of Analytic Functions with Negative Coefficients Defined by Gegenbauer Polynomials

“…where c λ n (m) denotes the Gegenbauer polynomial of degree n. Varying the parameter λ in (7), we obtain the class of typically real functions studied by [1,2,6,9,10,12,14] and [17].…”