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Inclusion properties of Generalized Integral Transform using Duality Techniques

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

Re e((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

Vλδ(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 Vλδ(f)(z) ∈ Wβ1δ111) whenever f ∈ Wβ2δ222). Further, several interesting applications for specific choices of λ(t) are discussed.