New properties for certain integral operators. (English) Zbl 1218.30043

Mohammed, Aabed; Darus, Maslina

Int. J. Math. Anal., Ruse 4, No. 41-44, 2101-2109 (2010).

Summary: The purpose of the present paper is to use the so-called pre-Schwarzian derivative to obtain some properties of a certain integral operator. We first establish the relationship between the two integral operators \(F_n\) and \(F_{\gamma_1,\dots ,\gamma_n}\) given by D. Breaz and N. Breaz [Stud. Univ. Babeş-Bolyai, Math. 47, No. 3, 13–19 (2002; Zbl 1027.30018)] and D. Breaz, S. Owa and N. Breaz [Acta Univ. Apulensis, Math. Inform. 16, 11–16 (2008; Zbl 1212.30031)], respectively, for the familiar classes of starlike functions of order \(\alpha\), \(S^*(\alpha)\), and convex functions of order \(\alpha\), \(K(\alpha )\). Furthermore, using the the norm of the pre-Schwarzian derivative, some other properties of the integral operator \(F_{\gamma_1,\dots ,\gamma_n}\) are obtained.

MSC:

30C45

Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)

Keywords:

univalent function; convex function; starlike function; pre-Schwarzian derivative

Citations:

Zbl 1027.30018; Zbl 1212.30031