Convexity of integral transforms and function spaces

Balasubramanian, R. ; Ponnusamy, S. ; Prabhakaran, D. J. (2007) Convexity of integral transforms and function spaces Integral Transforms and Special Functions, 18 (1). pp. 1-14. ISSN 1065-2469

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For β<1, let Pγ(β) denote the class of all normalized analytic functions f in the unit disc Δ such that Re{e((1−γ)f(z)/z+γf'(z)-β)}>0, z ∈ Δ, for some φ∈R. Let S(μ), 0≤μ<1, denote the usual class of starlike functions of order μ. Define K(μ)={f: zf'(z)∈S(μ)}, the class of all convex functions of order μ. In this paper, we consider integral transforms of the form Vλ (f)(z) = ∫01λ(t) f(tz)/t dt and νλ(f)(z) = z ∫01λ(t) 1-ρtz/1-tz dt ∗ f(z) . The aim of this paper is to find conditions on λ(t) so that each of the transformations carries Pγ(β) into S(μ) or K(μ). A number of applications for certain special choices of λ(t) are also established. These results extend the previously known results by a number of authors.

Item Type:Article
Source:Copyright of this article belongs to Taylor and Francis Ltd.
Keywords:Duality; Hypergeometric Functions; Univalent; Starlike and Convex Functions
ID Code:1308
Deposited On:04 Oct 2010 07:52
Last Modified:01 Feb 2023 07:20

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