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Prove this inequality on gamma function

Source: 2019 Jozsef Wildt International Math Competition-W. 25

May 18, 2020
Gamma functionSummationinequalitiesfunction

Problem Statement

Let xix_i, yiy_i, ziz_i, wi∈R+,i=1,2,⋯nw_i \in \mathbb{R}^+, i = 1, 2,\cdots n, such that∑i=1nxi=nx, ∑i=1nyi=ny, ∑i=1nwi=nw\sum \limits_{i=1}^nx_i=nx,\ \sum \limits_{i=1}^ny_i=ny,\ \sum \limits_{i=1}^nw_i=nw Γ(zi)≥Γ(wi), ∑i=1nΓ(zi)=nΓ∗(z)\Gamma \left(z_i\right)\geq \Gamma \left(w_i\right),\ \sum \limits_{i=1}^n\Gamma \left(z_i\right)=n\Gamma^* (z)Then∑i=1n(Γ(xi)+Γ(yi))2Γ(zi)−Γ(wi)≥n(Γ(x)+Γ(y))2Γ∗(z)−Γ(w)\sum \limits_{i=1}^n \frac{\left(\Gamma \left(x_i\right)+\Gamma \left(y_i\right)\right)^2}{\Gamma \left(z_i\right)-\Gamma \left(w_i\right)}\geq n\frac{\left(\Gamma \left(x\right)+\Gamma \left(y\right)\right)^2}{\Gamma^* \left(z\right)-\Gamma \left(w\right)}
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