Order of convexity of Integral Transforms and Duality

Recently, Ali et al defined the class $\mathcal{W}_β(α, γ)$ consisting of functions $f$ which satisfy $\Re e^{iϕ}\left((1-α+2γ)\frac{f(z)}{z}+(α-2γ)f'(z)+γzf''(z)-β\right)>0,$ for all $z\in E=\left\{z : |z|<1\right\}$ and for $α, γ\geq0$ and $β<1$, $ϕ\in \mathbb{R}$ (the set of reals). For $f\in{\mathcal{W}_β(α, γ)}$, they discussed the convexity of the integral transform $V_λ(f)(z):=\int_{0}^{1}λ(t)\frac{f(tz)}{t}dt,$ where $λ$ is a non-negative real-valued integrable function satisfying the condition $\displaystyle\int_{0}^{1}λ(t)dt=1$. The aim of present paper is to find conditions on $λ(t)$ such that $V_λ(f)$ is convex of order $δ$ ($0\leqδ\leq1/2$) whenever $f\in{\mathcal{W}}_β(α, γ)$. As applications, we study various choices of $λ(t)$, related to classical integral transforms.

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