# A relation on Legendre functions with real index

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.