On the Growth of Analytic Functions in the Class $${\mathcal {U}}(\lambda )$$ U
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On the Growth of Analytic Functions in the Class U (λ) A. Vasudevarao · H. Yanagihara
Received: 10 May 2013 / Revised: 14 September 2013 / Accepted: 30 September 2013 / Published online: 22 November 2013 © Springer-Verlag Berlin Heidelberg 2013
Abstract For 0 < λ ≤ 1, let U(λ) be the class of analytic functions in the unit disk D with f (0) = f (0) − 1 = 0 satisfying | f (z)(z/ f (z))2 − 1| < λ in D. Then, it is known that every f ∈ U(λ) is univalent in D. Let U(λ) = { f ∈ U(λ) : f (0) = 0}. The sharp distortion and growth estimates for the subclass U(λ) were known and many other properties are exclusively studied in Fourier and Ponnusamy (Complex Var. Elliptic Equ. 52(1):1–8, 2007), Obradovi´c and Ponnusamy ( Complex Variables Theory Appl. 44:173–191, 2001) and Obradovi´c and Ponnusamy (J. Math. Anal. Appl. 336:758–767, 2007). In contrast to the subclass U(λ), the full class U(λ) has been less well studied. The sharp distortion and growth estimates for the full class U(λ) are still unknown. In the present article, we shall prove the sharp estimate | f (0)| ≤ 2(1 + λ) for the full class U(λ). Furthermore, we shall determine the region of variability { f (z 0 ) : f ∈ U(λ)} for any fixed z 0 ∈ D\{0}. This leads to the sharp growth theorem, i.e., the sharp lower and upper estimates for | f (z 0 )| with f ∈ U(λ). As an application we shall also give the sharp covering theorems. Keywords Univalent function · Region of variability · Conformal center · Schwarz’s lemma · Coefficient estimate · Starlike function
Communicated by Stephan Ruscheweyh. A. Vasudevarao (B) Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721 302, West Bengal, India e-mail: [email protected] H. Yanagihara Department of Applied Science, Faculty of Engineering, Yamaguchi University, Tokiwadai, Ube, Yamaguchi 755, Japan e-mail: [email protected]
123
614
A. Vasudevarao, H. Yanagihara
Mathematics Subject Classification (1991)
30C45 · 30C50 · 30C80 · 30C25
1 Introduction We denote the complex plane by C and the extended complex plane by C = C ∪ {∞}. For c ∈ C and r > 0 let D(c, r ) = {z ∈ C : |z − c| < r } and D = D(0, 1). Similarly, C\D(0, r ) = {z ∈ C : r < |z| ≤ ∞} and = 1 . let r = Let A denote the space of analytic functions in D and A0 = { f ∈ A : f (0) = f (0) − 1 = 0}. Here, we regard A as a topological vector space endowed with the topology of uniform convergence over compact subsets of D. A function f is said to be univalent in a domain D if it is one-to-one in D. Let S denote the class of univalent functions in A0 . A function f ∈ A is called starlike if f is univalent in D and f (D) is a starlike domain with respect to the origin. The class of starlike functions f ∈ S is denoted by S ∗ . It is known that for f ∈ A0 , f ∈ S ∗ if and only if f (z) Re z >0 f (z)
(1.1)
in D. See [3] and [8] for proofs. For 0 < λ ≤ 1, let U(λ) be the class of functions f ∈ A0 satisfying 2 z − 1 < λ f (z) f (z)
(1.2)
in D. The boundedness of f
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