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

Abstract

For $0<\lambda \le 1$, let $\mathcal{U}(\lambda )$ denote the family of functions $f(z)=z+{\sum }_{n=2}^{\infty }{a}_n{z}^n$ analytic in the unit disk $\ID$ satisfying the condition $\left|{\left(\frac{z}{f(z)}\right)}^2{f}^{\prime }(z)-1\right|<\lambda$ in $\ID$. Although functions in this family are known to be univalent in $\ID$, the coefficient conjecture about $a_n$ for $n\ge 5$ 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 $$\frac{z}{f(z)}=1-(1+\lambda )\varphi (z)+\lambda (\varphi (z){)}^2$$ with $\varphi (z)=e^{i\theta }z$, that solve many extremal problems in $\mathcal{U}(\lambda )$. Secondly, we shall consider the following question: Do there exist functions $\varphi$ analytic in $\ID$ with $|\varphi (z)|<1$ that are not of the form $\varphi (z)=e^{i\theta }z$ 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 $$f(z)=\frac{z}{1-a_{2}z+\lambda z{\int }_0^z\omega (t)dt},$$ where $\omega$ is analytic in $\ID$ such that $|\omega (z)|\le 1$ and $\omega (0)=a$, where $a\in \overline{\ID}$.

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