INCLUSION THEOREMS FOR CONVOLUTION
PRODUCT OF SECOND ORDER
POLYLOGARITHMS AND FUNCTIONS
WITH THE DERIVATIVE
IN A HALFPLANE

S. PONNUSAMY

Abstract:

For $\beta <1$ and real $\eta$, let ${\cal R}_{\eta}(\beta)$ denote the family of normalized analytic functions f defined in the unit disc $\Delta$ such that ${\rm Re\,}[e^{i\eta}(f^\prime(z)-\beta )]>0$ for $z\in \Delta$. Given a generalized second order polylogarithm function

$$G(a,b;z)=\sum_{n=1}^{\infty}\frac{(a+1)(b+1)}{(n+a)(n+b)}z^n,$$

$$a,b \in{\bf C}\backslash\{-1,-2,-3, \cdots\},$$

we place conditions on the parameters a, b and $\beta$ to guarantee that the Hadamard product of the power series G(a,b;z)*f(z) will be univalent, starlike or convex. We also give conditions on a and b to describe the geometric nature of the function G(a,b;z). By taking f in the class of convex functions, we also find a sufficient condition for G(a,b;z)*f(z) to belong to the class ${\cal R}_{0}(\beta)$. Several open problems have been raised at the end.