For $\alpha \in \mathbb R$, let the Laguerre polynomials $L_n^{(\alpha)}(x)$ be defined as the Taylor coefficients of the function $G(w):=(1-w)^{-\alpha-1} \exp\left(- \frac{xw}{1-w}\right)$.
It follows from Fejér's formula (see Szegő's book, Orthogonal Polynomials, AMS 1939, formula 8.22.1) that, for $x>0$,
$$ |L_n^\alpha(x)|\leq C(\alpha,x) n^{\alpha/2-1/4}. $$
Is there a direct and simple proof of this inequality? Is there a readable proof (in some textbook) of Fejér's theorem?
(Szegő's book is a bit difficult to follow).