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Inclusion theorems for convolution product of second order polylogarithms and functions with the derivative in a halfplane. (English) Zbl 0915.30012
Let ${\cal A}$ denote the family of normalized regular functions defined in the unit disc $\Delta=\{z:| z|<1\}$. For $\beta<1$ and real $\eta$, let ${\cal R}_\eta(\beta)$ denote the family of functions $f\in{\cal A}$ such that $\text{Re} [e^{i\eta} (f'(z)-\beta)]>0$ for $a\in\Delta$. Given a generalized second order polylogarithm function $$G(a,b;z)= \sum_{n=1}^\infty \frac{(a+1)(b+1)} {(n+a)(n+b)} z^n,\qquad a,b\ne-1,-2,-3,\dots,$$ 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 $f\in{\cal A}$, 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 $$\align H_f(a,b;z)&:= \frac{(a+1)(b+1)} {b- a}\int_0^1 t^{a-1}(1-t^{b-a})f(tz)dt, \qquad\text{if }b\ne a\\ \intertext{and} H_f(a,a;z)&:= (1+a)^2\int_0^1 (\log 1/t)t^{-1} f(tz)dt, \qquad \text{for Re }a>-1. \endalign$$ 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 {\it R. Ali} and {\it 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 ${\cal R}_0(\beta)$. Several open problems have been raised at the end.
Reviewer: S.Ponnusamy (Helsinki)

30C45Special classes of univalent and multivalent functions
30C80Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)
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