Convexity of integral transforms and function spaces

Abstract

For β<1, let 𝒫_γ(β) denote the class of all normalized analytic functions f in the unit disc Δ such that

for some ϕ∊ℝ. Let ^∗(μ), 0≤μ<1, denote the usual class of starlike functions of order μ. Define (μ)={f: zf′(z)∊^∗(μ)}, the class of all convex functions of order μ. In this paper, we consider integral transforms of the form

The aim of this paper is to find conditions on λ (t) so that each of the transformations carries 𝒫_γ(β) into ^∗(μ) or (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.

Keywords:

• Duality
• Hypergeometric functions
• Univalent
• Starlike and convex functions

Acknowledgements

The work of the first two authors were supported by Department of Science and Technology, India (no. DST/MS/092/98).