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

### Volume 113 / 2015

Annales Polonici Mathematici 113 (2015), 295-304
MSC: Primary 30C45; Secondary 30C80, 30C50.
DOI: 10.4064/ap113-3-6

#### Abstract

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.