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For $$0<\lambda \le 1$$ 0 < λ ≤ 1 , let $${\mathcal {U}}( \lambda )$$ U ( λ ) be the class of analytic functions in the unit disk $$\mathbb {D}$$ D with $$f(0)=f'(0)-1=0$$ f ( 0 ) = f ′ ( 0 ) - 1 = 0 satisfying $$| f'(z) (z/f(z))^2 -1 | < \lambda $$ | f ′ ( z ) ( z / f ( z ) ) 2 - 1 | < λ in $$\mathbb {D}$$ D . Then, it is known that every $$f \in {\mathcal {U}}( \lambda )$$ f ∈ U ( λ ) is univalent in $${{\mathbb {D}}}$$ D . Let $$\widetilde{\mathcal {U}}( \lambda ) = \{ f \in {\mathcal {U}}( \lambda ) : f''(0) = 0 \}$$ U ~ ( λ ) = { f ∈ U ( λ ) : f ′ ′ ( 0 ) = 0 } . The sharp distortion and growth estimates for the subclass $$\widetilde{\mathcal {U}}( \lambda )$$ U ~ ( λ ) 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 $$\widetilde{\mathcal {U}}( \lambda )$$ U ~ ( λ ) , the full class $${\mathcal {U}}( \lambda )$$ U ( λ ) has been less well studied. The sharp distortion and growth estimates for the full class $${\mathcal {U}}( \lambda )$$ U ( λ ) are still unknown. In the present article, we shall prove the sharp estimate $$|f''(0)| \le 2(1+ \lambda )$$ | f ′ ′ ( 0 ) | ≤ 2 ( 1 + λ ) for the full class $${\mathcal {U}} ( \lambda )$$ U ( λ ) . Furthermore, we shall determine the region of variability $$\{ f(z_0) : f \in {\mathcal {U}}( \lambda ) \}$$ { f ( z 0 ) : f ∈ U ( λ ) } for any fixed $$z_0 \in {{\mathbb {D}}} \backslash \{ 0 \}$$ z 0 ∈ D \ { 0 } . This leads to the sharp growth theorem, i.e., the sharp lower and upper estimates for $$|f(z_0)|$$ | f ( z 0 ) | with $$f \in {\mathcal {U}} ( \lambda )$$ f ∈ U ( λ ) . As an application we shall also give the sharp covering theorems.
Computational Methods and Function Theory – Springer Journals
Published: Nov 22, 2013
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