Integral Transforms of Certain Subclasses of Analytic Functions

Duality techniques for certain integral transforms to be starlike

“…In this paper, we discuss the above problem for µ ∈ [0, 1/2] and hence, our results extend and improve the work of Fournier and Ruscheweyh [5] (γ = 0 = µ), Ponnusamy and Rønning [12] (γ = 0, µ ∈ [0, 1/2]), Kim and Rønning [7] (γ ∈ [1/2, 1], µ = 0). One of the main differences between these papers and the present one is that it handles an important inequality in a simpler and nicer way (see Lemma 5.1).…”

“…For f ∈ P 1 (β), the starlikeness of the transformation V λ (f ) was studied first by Fournier and Ruscheweyh [5] and was extended to starlikeness of order α by Ponnusamy and Rønning [12]. Further applications, extensions and improvements have been obtained in [1,2,7,14]. In this paper, we are interested in the following problem.…”

“…In a recent paper, Kim and Rønning [7] answered this problem for the case µ = 0 and the same method has been used earlier to discuss similar problems involving P 1 (β) [5,12,14]. In this paper, we discuss the above problem for µ ∈ [0, 1/2] and hence, our results extend and improve the work of Fournier and Ruscheweyh [5] (γ = 0 = µ), Ponnusamy and Rønning [12] (γ = 0, µ ∈ [0, 1/2]), Kim and Rønning [7] (γ ∈ [1/2, 1], µ = 0).…”

“…In fact, this convolution operator was introduced and investigated systematically by Hohlov [6]. Later, this operator has been studied extensively by Ponnusamy [11], Ponnusamy and Rønning [13] and many others [2,3,7]. In fact, the corresponding convolution operators involving a generalized hypergeometric function (in place of the Gauss hypergeometric function) were also investigated in a recent paper by Dziok and Srivastava [4].…”

Geometric Properties of an Integral Operator

“…For α = γ = 1, Fournier and Ruscheweyh [12] and Ali and Singh [3] used the Duality Principle [19,20] to prove starlikeness and convexity of the linear integral transform V λ,α ( f ), when f varies in the class P γ (α, β). For α = 1, Kim and Rønning [13] and Choi et al [9] studied starlikeness and convexity of the linear transform V λ,α ( f ), f ∈ P γ (α, β). In 2008, Aghalary et al [1] discussed the univalence of integral transform V λ,α ( f ) of the functions f in the class P γ (α, β).…”