Verification of Brannan and Clunie's conjecture for certain subclasses of bi-univalent functions

S. Sivasubramanian¹ ; R. Sivakumar¹ ; S. Kanas² ; Seong-A Kim³

¹ Department of Mathematics University College of Engineering Tindivanam Anna University Tindivanam, India
² Institute of Mathematics University of Rzeszów Rzeszów, Poland
³ Department of Mathematics Education Dongguk University Gyeongju, Korea

Annales Polonici Mathematici, Tome 113 (2015) no. 3, pp. 295-304

Résumé

Let $\sigma $ denote the class of bi-univalent functions $f$, that is, both $f(z)=z+a_2z^2+\cdots $ and its inverse $f^{-1}$ are analytic and univalent on the unit disk. We consider the classes of strongly bi-close-to-convex functions of order $\alpha $ and of bi-close-to-convex functions of order $\beta $, which turn out to be subclasses of $\sigma .$ We obtain upper bounds for $|a_2|$ and $|a_3|$ for those classes. Moreover, we verify Brannan and Clunie's conjecture $|a_2|\leq \sqrt {2}$ for some of our classes. In addition, we obtain the Fekete–Szegö relation for these classes.

Détail

DOI : 10.4064/ap113-3-6

Keywords: sigma denote class bi univalent functions cdots its inverse analytic univalent unit disk consider classes strongly bi close to convex functions order alpha bi close to convex functions order beta which turn out subclasses sigma obtain upper bounds those classes moreover verify brannan clunies conjecture leq sqrt classes addition obtain fekete szeg relation these classes