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.


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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