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We provide elementary proofs of some inequalities of Horst Alzer.

Let $\alpha$ be such that $0<\alpha <\frac{1}{2}$. In this note we use the Mittag-Leffler partial fractions expansion for $F_{\alpha }\left(\theta \right)=\Gamma \left(1-\alpha -\frac{\theta }{\pi }\right)\Gamma \left(\alpha \right)/\Gamma \left(\alpha -\frac{\theta }{\pi }\right)\Gamma (1-\alpha )$ to obtain a solution of a Wiener-Hopf integral equation.