Inequality for generalized Laguerre polynomials

Question

Please. Does anybody know a proof of this inequality $$\Big|\frac{n!\Gamma(\alpha+1)}{\Gamma(n+\alpha+1)} L^{\alpha}_n(x)\Big|\leq e^{\frac{x}{2}}$$ where $\alpha\geq0$ and $x\geq0$ and $L^{\alpha}_n$ is the n-th Laguerre polynomial. Thanks

Answer 1

The inequality and its proof can be found in Chapter 10.18, Eq. (14), p.207 of

A. Erdelyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, "Higher Transcendental Functions", vol. 2, Bateman Manuscript Project (McGraw-Hill, New York, 1953).