Verification of Brannan and Clunie's conjecture for certain subclasses of bi-univalent functions
Tom 113 / 2015
Streszczenie
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.