Abstract and Applied Analysis

On an Integral Transform of a Class of Analytic Functions

Sarika Verma, Sushma Gupta, and Sukhjit Singh

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Abstract

For α , γ 0 and β < 1 , let 𝒲 β ( α , γ ) denote the class of all normalized analytic functions f in the open unit disc E = { z : | z | < 1 } such that e i ϕ ( ( 1 - α + 2 γ ) (f ( z ) / z) + ( α - 2 γ ) f ( z ) + γ z f ( z ) - β ) > 0 , z E for some ϕ . It is known (Noshiro (1934) and Warschawski (1935)) that functions in 𝒲 β ( 1,0 ) are close-to-convex and hence univalent for 0 β < 1 . For f 𝒲 β ( α , γ ) , we consider the integral transform F ( z ) = V λ ( f ) ( z ) : = 0 1 λ ( t ) ( f ( t z ) / t ) d t , where λ is a nonnegative real-valued integrable function satisfying the condition 0 1 λ ( t ) d t = 1 . The aim of present paper is, for given δ < 1 , to find sharp values of β such that (i) V λ ( f ) 𝒲 δ ( 1,0 ) whenever f 𝒲 β ( α , γ ) and (ii) V λ ( f ) 𝒲 δ ( α , γ ) whenever f 𝒲 β ( α , γ ) .

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 259054, 10 pages.

Dates
First available in Project Euclid: 28 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1364475870

Digital Object Identifier
doi:10.1155/2012/259054

Mathematical Reviews number (MathSciNet)
MR2984036

Zentralblatt MATH identifier
1255.30025

Citation

Verma, Sarika; Gupta, Sushma; Singh, Sukhjit. On an Integral Transform of a Class of Analytic Functions. Abstr. Appl. Anal. 2012 (2012), Article ID 259054, 10 pages. doi:10.1155/2012/259054. https://projecteuclid.org/euclid.aaa/1364475870


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