Coefficient problems on the class $U(\lambda)$

Abstract

For $0<\lambda \leq 1$, let ${\mathcal U}(\lambda)$ denote the family of functions $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ analytic in the unit disk $\ID$ satisfying the condition $\left |\left (\frac{z}{f(z)}\right )^{2}f'(z)-1\right |<\lambda $ in $\ID$. Although functions in this family are known to be univalent in $\ID$, the coefficient conjecture about $a_n$ for $n\geq 5$ remains an open problem. In this article, we shall first present a non-sharp bound for $|a_n|$. Some members of the family ${\mathcal U}(\lambda)$ are given by $$ \frac{z}{f(z)}=1-(1+\lambda)\phi(z) + \lambda (\phi(z))^2 $$ with $\phi(z)=e^{i\theta}z$, that solve many extremal problems in ${\mathcal U}(\lambda)$. Secondly, we shall consider the following question: Do there exist functions $\phi$ analytic in $\ID$ with $|\phi (z)|<1$ that are not of the form $\phi(z)=e^{i\theta}z$ for which the corresponding functions $f$ of the above form are members of the family ${\mathcal U}(\lambda)$? Finally, we shall solve the second coefficient ($a_2$) problem in an explicit form for $f\in {\mathcal U}(\lambda)$ of the form $$f(z) =\frac{z}{1-a_2z+\lambda z\int_0^z\omega(t)\,dt}, $$ where $\omega$ is analytic in $\ID$ such that $|\omega(z)|\leq 1$ and $\omega(0)=a$, where $a\in \overline{\ID}$.