Geometric Studies on the Class ${\mathcal {U}}(\lambda )$

Abstract

The article deals with the family ${\mathcal {U}}(\lambda )$ of all functions f normalized and analytic in the unit disk such that $\big |\big (z/f(z)\big )^{2}f'(z)-1\big |<\lambda$ for some $0<\lambda \le 1$. The family ${\mathcal {U}}(\lambda )$ has been studied extensively in the recent past and functions in this family are known to be univalent in ${\mathbb D}$. However, the problem of determining sharp bounds for the second coefficients of functions in this family was solved recently by Vasudevarao and Yanagihara but the proof was complicated. In this article, we first present a simpler proof of it. We obtain a number of new subordination results for this family and their consequences. Also, we obtain sharp estimate for the classical Fekete–Szegö inequality for functions in ${\mathcal {U}}(\lambda )$. In addition, we show that the family ${\mathcal {U}}(\lambda )$ is preserved under a number of elementary transformations such as rotation, conjugation, dilation, and omitted-value transformations, but surprisingly this family is not preserved under the n-th root transformation for any $n\ge 2$. This is a basic here which helps to generate a number of new theorems and in particular provides a way for constructions of functions from the family ${\mathcal {U}}(\lambda )$. Finally, we deal with a radius problem and the paper ends with a coefficient conjecture.