On a conjecture for the fifth coefficients for the class ${\mathcal U}(\lambda)$

Abstract

Let $f$ be function that is analytic in the unit disk ${\mathbb D}=\{z:|z|<1\}$, normalized such that $f(0)=f'(0)-1=0$, i.e., of type $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$. If additionally, \[ \left| \left(\frac{z}{f(z)}\right)^2 f'(z) -1\right|<\lambda \quad\quad (z\in{\mathbb D}), \] then $f$ belongs to the class ${\mathcal U}(\lambda)$, $0<\lambda\le1$. In this paper we prove sharp upper bound of the modulus of the fifth coefficient of $f$ from ${\mathcal U}(\lambda)$ satisfying \[ \frac{f(z)}{z}\prec \frac{1}{(1+z)(1+\lambda z)}, \] ("$\prec$" is the usual subordination) in the case when $0.400436\ldots \le\lambda\le1$.