I have already asked my question in the link below:
https://mathoverflow.net/questions/418497/minima-approximation-for-laguerre-polynomials?noredirect=1#comment1074404_418497
I have suggested to anyone to give me the approximations of the minima for the Laguerre polynomial, but someone have suggested the zero Laguerre polynomials. But i want the general formula for the approximation of the local minima for the generalized Laguerre polynomial. Since, we have the formula given in the Book "Orthogonal polynomials-Szego 1939" in page 240:
$$Max_{a > 0}e^{-x/2} x^{\alpha/2+1/4}|L_{n}^{(\alpha)}(x)| \sim n^{\alpha/2-1/12} \mbox{ for n is very large and } x \in R^{+} \mbox{ and } \alpha \mbox{ is a real number } $$
My question is: Can the local minima of the Laguerre polynomial be written as the following
$$Min_{a > 0}e^{-x/2} x^{\alpha/2+1/4}|L_{n}^{(\alpha)}(x)| \sim n^{k} \mbox{ for n is very large and } x \in R^{+} \mbox{ and } \alpha \mbox{ is a real number } $$
where $-1 \leq k \leq \alpha/2-1/12$? If so why? I have seen that because for that case $n^{k} \leq n^{\alpha/2}$. But I want someone to suggest to me a demonstration if that approximation is true? If not can we deduce another approximation?