Inclusion properties of Generalized Integral Transform using Duality Techniques

Let W_β^δ(α,γ) be the class of normalized analytic functions f defined in the region |z| < 1 and satisfying

Re e^iθ((1-α+2γ)(f/z)^δ + (α-3γ+γ[(1-1/δ)(zf′/f) + 1/δ(1+zf″/f′)])(f/z)^δ(zf′/f)-β)>0,

with the conditions α ≤ 0, β < 1, γ ≤ 0, δ > 0 φ ∈ ℝ. For a non-negative and real- valued integrable function λ(t) with ∫₀¹ λ(t)dt = 1, the generalized non-linear integral transform is defined as

V_λ^δ(f)(z) = (∫₀¹λ(t)(f(tz)/t)^δdt)^1/δ.

The main aim of the present work is to find conditions on the related parameters such that V_λ^δ(f)(z) ∈ W_β1^δ1(α₁,γ₁) whenever f ∈ W_β2^δ2(α₂,γ₂). Further, several interesting applications for specific choices of λ(t) are discussed.