## Inclusion theorems for convolution product of second order polylogarithms and functions with the derivative in a halfplane.(English)Zbl 0915.30012

Let $${\mathcal A}$$ denote the family of normalized regular functions defined in the unit disc $$\Delta=\{z:| z|<1\}$$. For $$\beta<1$$ and real $$\eta$$, let $${\mathcal R}_\eta(\beta)$$ denote the family of functions $$f\in{\mathcal A}$$ such that $$\text{Re} [e^{i\eta} (f'(z)-\beta)]>0$$ for $$a\in\Delta$$. Given a generalized second order polylogarithm function $G(a,b;z)= \sum_{n=1}^\infty \frac{(a+1)(b+1)} {(n+a)(n+b)} z^n,\qquad a,b\neq-1,-2,-3,\dots,$ we place conditions on the parameters $$a$$, $$b$$ and $$\beta$$ to guarantee that the Hadamard product of the power series $$H_f(a,b;z)\equiv G(a,b;z)* f(z)$$ will be univalent, starlike or convex. We give conditions on $$a$$ and $$b$$ to describe the geometric nature of the function $$G(a,b;z)$$. We note that for $$f\in{\mathcal A}$$, the function $$H_f(a,b;z)$$ satisfies the differential equation $z^2 H_f''(z)+ (a+b+1)z H_f'(z)+abH_f(z)= (a+1)(b+1)f(z),$ and $$H_f$$ has the integral representation \begin{aligned} H_f(a,b;z)&:= \frac{(a+1)(b+1)} {b- a}\int_0^1 t^{a-1}(1-t^{b-a})f(tz)dt, \qquad\text{if }b\neq a\\ \text{and} H_f(a,a;z)&:= (1+a)^2\int_0^1 (\log 1/t)t^{-1} f(tz)dt, \qquad \text{for Re }a>-1. \end{aligned} If $$a>-1$$, and $$b>a$$ with $$b\to\infty$$, we see that $$H_f(a,b;z)$$ reduces to the well-known Bernardi transform. If $$a=-\alpha$$ and $$b=2-\alpha$$, $$H_f(a,b;z)$$ is the operator considered by R. Ali and V. Singh [Complex Variables, Theory Appl. 26, No. 4, 299-309 (1995; Zbl 0851.30005)] with an additional assumption that $$0\leq\alpha<1$$. Thus, the operator $$H_f(a,b;z)$$ is the natural choice to study its behaviour. By making $$f$$ in the class of convex functions, we also find a sufficient condition for $$H_f(a,b;z)$$ to belong to the class $${\mathcal R}_0(\beta)$$. Several open problems have been raised at the end.

### MSC:

 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination

Zbl 0851.30005
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