On Some Inequalities for the Gamma Function

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On the Expressive Power of Deep Neural Networks (2016, background): “We also then recover a chi distribution with k terms, each with standard deviation σ w k where the second step follows from (Laforgia and Natalini, 2013), and holds for k > 1 .”
Inequalities and Bounds for the Incomplete Gamma Function (2013, background): “Also, we will need a result which is established in [6].”
Ridge regression and asymptotic minimax estimation over spheres of growing dimension (2016, unknown): “By Theorem 1 of Kečkić and Vasić [36], which gives convenient bounds on the ratio of two gamma functions, E(s−ad ) = (n/2)aΓ(n/2− a) Γ(n/2) ≤ (n/2) a(n/2− a)n/2−a−1 (n/2)n/2−1 ea ≤ ea.”
Weibull-Pareto Distribution and Its Applications (2013, methodology): “We need the following inequalities introduced by Keckic and Vasic (1971) for the proof:”