Univalence criteria for two integral operators. (English) Zbl 1242.30013

Stanciu, Laura Filofteia; Breaz, Daniel

Abstr. Appl. Anal. 2012, Article ID 652858, 11 p. (2012).

Let \(A\) denote the class of functions \(f(z)=z+\sum_{n=2}^{\infty}a_nz^n\), \(|z|<1\). The authors prove two theorems and numerous corollaries giving univalence conditions for functions represented by the integral operators \[ I(f_1,\dots,f_n;g_1,\dots,g_n)(z)= \left(\beta\int_0^zt^{\beta-1}\prod_{k=1}^n\left(\frac{f_k(t)}{t}\right)^{(a_k-1)/M_k}(g_k'(t))^{\gamma_k}dt\right)^{1/\beta} \] and \[ J(f_1,\dots,f_n;g_1,\dots,g_n)(z) = \left(\left(1+\sum_{k=1}^na_k\right)\int_0^z\prod_{k=1}^n(f_k(t))^{a_k}(g_k'(t))^{\gamma_k}dt\right)^{1/\left(1+\sum_{k=1}^na_k\right)}, \] where, for \(k=1,\dots,n\), \(a_k,\gamma_k\in\mathbb C\), \(\beta\in\mathbb C\setminus\{0\}\), \(f_k,g_k\in A\) and \(M_k\geq1\).

Reviewer: Dmitri V. Prokhorov (Saratov)

MSC:

30C45	Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
47G10	Integral operators

Keywords:

univalence condition; integral operator; Schwarz lemma

References:

[1]	H. Silverman, “Convex and starlike criteria,” International Journal of Mathematics and Mathematical Sciences, vol. 22, no. 1, pp. 75-79, 1999. · Zbl 0921.30009 · doi:10.1155/S0161171299220753
[2]	A. Baricz and B. A. Frasin, “Univalence of integral operators involving Bessel functions,” Applied Mathematics Letters, vol. 23, no. 4, pp. 371-376, 2010. · Zbl 1194.33008 · doi:10.1016/j.aml.2009.10.013
[3]	V. Pescar, “On the univalence of some integral operators,” General Mathematics, vol. 14, no. 2, pp. 77-84, 2006. · Zbl 1164.30348
[4]	D. Blezu and R. N. Pascu, “Univalence criteria for integral operators,” Glasnik Matemati\vcki. Serija III, vol. 36, no. 2, pp. 241-245, 2001. · Zbl 1005.30018
[5]	D. Breaz and N. Breaz, “Univalence conditions for certain integral operators,” Universitatis Babe\cs-Bolyai, Studia, Mathematica, vol. 47, no. 2, pp. 9-15, 2002. · Zbl 1027.30046
[6]	V. Pescar, “A new generalization of Ahlfors’s and Becker’s criterion of univalence,” Malaysian Mathematical Society, Bulletin (Second Series), vol. 19, no. 2, pp. 53-54, 1996. · Zbl 0880.30020
[7]	Z. Nehari, Conformal Mapping, McGraw-Hill, New York, NY, USA, 1952. · Zbl 0048.31503