# Bounds for associated Legendre polynomials

I am trying to analyze the behaviour of the Associated Legendre polynomials $$P_{n}^{m}$$ on $$[0,1]$$. More specifically, I am trying to get upper bounds for $$P_{n}^{m}$$ on $$[0,1]$$. Bernstein's inequality for the Legendre polynomial is a classical result which states the following \begin{align*} |P_{n}(x)| \leq \sqrt{\frac{2}{\pi n}}\frac{1}{(1-x^2)^{1/4}}. \end{align*} I am interested in bounds of the above type for $$P_{n}^{m}$$'s. For a fixed $$n\in \mathbb{N}$$, normalizing $$P_{n}^{m}$$ appropriately as below, for every $$|m|\leq n$$ we have \begin{align*} \frac{(n-m)!}{(n+m)!} \int_{0}^{1} P_{n}^{m}(x)^2 dx = \frac{C}{2n+1}. \end{align*}
So specifically, I am interested in upper bounds on $$[0,1]$$ for the following functions, for a fixed $$n$$, as $$m \in \mathbb{Z}$$ varies in $$[-n,n]$$. \begin{align*} L_{n}^{m}(x) := \sqrt{\frac{(n-m)!}{(n+m)!}} ~P_{n}^{m} \end{align*}

I did a bit of searching and found that bounds for the above collection of normalized Associated Legendre functions are available in this and this and I state them below.

\begin{align} \sqrt{\frac{(n-m)!}{(n+m)!}}~ |P_{n}^{m} (x)| & \leq \frac{1}{2^m m!} \sqrt{\frac{(n+m)!}{(n-m)!}} (1-x^2)^{m/2} := A_{n}^{m}(x) , \tag{1}\label{1}\\ \sqrt{\frac{(n-m)!}{(n+m)!}}~ |P_{n}^{m} (x)| & \leq \frac{1}{n^{1/4}} \frac{1}{(1-x^2)^{1/8}} =: f_{n}(x).\tag{2} \label{2} \end{align}

I tried to see how good these bounds are. It seemed that $$A_{n}^{m}$$ is a good approximate for $$L_{n}^{m}$$ near 1 (which is expected as it captures the vanishing of $$L_{n}^{m}$$ at 1), but elsewhere it is not good.

And the bound $$f_n$$ appears to be just an upper bound which does not necessarily capture any feature of $$L_{n}^{m}$$.

Hence my question is: Are some other better bounds (than the ones in \ref{1} and \ref{2}) known for $$L_{n}^{m}$$'s?

Thanks!

Please have a look at the following paper

https://www.sciencedirect.com/science/article/pii/S0021904598932075