Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 0915.30012
Ponnusamy, S.
Inclusion theorems for convolution product of second order polylogarithms and functions with the derivative in a halfplane. (English)
[J] Rocky Mt. J. Math. 28, No.2, 695-733 (1998). ISSN 0035-7596

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.
[S.Ponnusamy (Helsinki)]

MSC 2000:: *30C45 Special classes of univalent and multivalent functions
30C80 Maximum principle, etc. (one complex variable)

Keywords: starlike; close-to-convex; polylogarithm; subordination; Hadamard product

Citations: Zbl 0851.30005

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]