On Certain Families of Analytic Functions in the Hornich Space
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On Certain Families of Analytic Functions in the Hornich Space Md Firoz Ali1 · A. Vasudevarao1
Received: 30 January 2018 / Revised: 2 March 2018 / Accepted: 4 March 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract Let (H, ⊕, ) denote the Hornich space consisting of all locally univalent and analytic functions f on the unit disk D := {z ∈ C : |z| < 1} with f (0) = 0 = f (0)−1 for which arg f is bounded in D. For f, g ∈ H and r, s ∈ R, we consider the z integral operator Ir,s (z) := 0 ( f (ξ ))r (g (ξ ))s dξ and determine all values of r and s for which the operator ( f, g) → Ir,s maps a specified subclass of H into another specified subclass of H . We also determine the set of extreme points for different subclasses of H with respect to the Hornich space structure. Using the extreme points, we develop a new approach to obtain the pre-Schwarzian norm estimate for different , whose linear structure is same as subclasses of H . We also consider a larger space H . that of H and study the same problems as stated above for some subclasses of H Keywords Univalent · Starlike · Convex · Close-to-convex · Spiral-like functions · Extreme point · Pre-Schwarzian norm · Banach space · Hornich space Mathematics Subject Classification Primary 30C45 · 30C55
Communicated by Stephan Ruscheweyh.
B
A. Vasudevarao [email protected] Md Firoz Ali [email protected]
1
Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal 721 302, India
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Md. F. Ali, A. Vasudevarao
1 Introduction and Main Results Let A denote the class of all analytic functions in the unit disk D := {z ∈ C : |z| < 1} with the normalization f (0) = 0 = f (0) − 1. Denote by H , the class of locally univalent (one-to-one) functions in A for which arg f is bounded in D. For f, g ∈ H and λ ∈ R, Hornich [7] has provided a real linear structure on H with the following operations: z z f (ξ )g (ξ ) dξ and (λ f )(z) := ( f (ξ ))λ dξ (1.1) ( f ⊕ g)(z) := 0
0
and introduced the norm: f H := sup | arg f (z 1 ) − arg f (z 2 )|, z 1 ,z 2 ∈D
where the branch of ( f )λ = exp(λ log f ) has been chosen so that ( f )λ (0) = 1. From here onwards, we call the operations in (1.1) as Hornich operations. It is known that (H, · H ) is a separable real Banach space (see [7]) with the Hornich operations. Here, the identity map plays the role of null element in H . It has been proved that the sequential convergence with respect to the norm · H implies the locally uniform convergence but the converse need not be true (see [7, p. 39]). The set S consisting of all univalent functions in A has been the central object to study in the theory of univalent functions since the early twentieth century. It is interesting to note that the class H does not contain the whole class S. Let S ∗ , C and K denote the family of starlike, convex, and close-to-convex functions in S, respectively. In the present paper, we mainly focus on some interesting subclasses of H . A locally
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