On the Growth of Analytic Functions in the Class \({\mathcal {U}}(\lambda )\)

Abstract

For \(0<\lambda \le 1\), let \({\mathcal {U}}( \lambda )\) be the class of analytic functions in the unit disk \(\mathbb {D}\) with \(f(0)=f'(0)-1=0\) satisfying \(| f'(z) (z/f(z))^2 -1 | < \lambda \) in \(\mathbb {D}\). Then, it is known that every \(f \in {\mathcal {U}}( \lambda )\) is univalent in \({{\mathbb {D}}}\). Let \(\widetilde{\mathcal {U}}( \lambda ) = \{ f \in {\mathcal {U}}( \lambda ) : f''(0) = 0 \}\). The sharp distortion and growth estimates for the subclass \(\widetilde{\mathcal {U}}( \lambda )\) were known and many other properties are exclusively studied in Fourier and Ponnusamy (Complex Var. Elliptic Equ. 52(1):1–8, 2007), Obradović and Ponnusamy ( Complex Variables Theory Appl. 44:173–191, 2001) and Obradović and Ponnusamy (J. Math. Anal. Appl. 336:758–767, 2007). In contrast to the subclass \(\widetilde{\mathcal {U}}( \lambda )\), the full class \({\mathcal {U}}( \lambda )\) has been less well studied. The sharp distortion and growth estimates for the full class \({\mathcal {U}}( \lambda )\) are still unknown. In the present article, we shall prove the sharp estimate \(|f''(0)| \le 2(1+ \lambda )\) for the full class \({\mathcal {U}} ( \lambda )\). Furthermore, we shall determine the region of variability \(\{ f(z_0) : f \in {\mathcal {U}}( \lambda ) \}\) for any fixed \(z_0 \in {{\mathbb {D}}} \backslash \{ 0 \}\). This leads to the sharp growth theorem, i.e., the sharp lower and upper estimates for \(|f(z_0)|\) with \(f \in {\mathcal {U}} ( \lambda )\). As an application we shall also give the sharp covering theorems.