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.

<cite authors="Lohöfer, G.">_Lohöfer, G._, [**Inequalities for the associated Legendre functions**](https://doi.org/10.1006/jath.1998.3207), J. Approximation Theory 95, No. 2, 178-193 (1998). [ZBL0923.33002](https://zbmath.org/?q=an:0923.33002).</cite>