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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!

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Please have a look at the following paper

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

In particular, Eq. (6) to get the answer to your question.

Lohöfer, G., Inequalities for the associated Legendre functions, J. Approximation Theory 95, No. 2, 178-193 (1998). ZBL0923.33002.

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