Consider the function $f$ defined for $j \in ]0 , 1[$ by

$$f(j) = P_{1-\gamma}\bigg( \frac{1}{j}\bigg) - \frac{1}{j} P_{2-\gamma}\bigg( \frac{1}{j}\bigg) $$

where $P_{\alpha}$ is the Legendre function of order $\alpha$ and the parameter $\gamma$ is inside $[1/2,3]$.

If $\gamma < 3$ then it looks like $f(j) < 0$ for $j \in ]0 , 1[$ when I draw its plot. On the other hand, if $\gamma > 3$ then it looks like $f(j) > 0$ for $j \in ]0 , 1[$ , though I am more interested in the $\gamma<3$ case as I use it in my current (physics) research.

Could I get a formal reason as for why this seems to be the case?

I tried to find an upper bound inequality by using the integral representation of $P_{\alpha}$ :

$$P_{\alpha}(x) = \frac{1}{2\pi} \int_{-\pi}^{\pi} \bigg(x + \sqrt{x^2-1}\cos \theta \bigg)^{\alpha} \rm{d}\theta $$

but I couldn't come up with anything.


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