Inequality with Bessel Functions of the first kind

Question

How can be proven the following inequality:

$$\int_0^1dx|J_k(x)'J_k(x)|\lt\frac{1}{2}\int_0^1dx|J_k(x)'^2|$$ where obviously: $J_k(x)'=\frac{d}{dx}J(k,x)$? Thanks.

Answer 1

Given that $k>0$ ensures that $J_k(0)=0$ and $J_k(x),J_k'(x)>0$ over $(0,1)$, your inequality follows from the Cauchy-Schwarz inequality, since: $$\int_{0}^{1}|J_k(x)J_k'(x)|dx = \int_{0}^{1}J_k(x)J_k'(x)dx = \left[\frac{1}{2}J_k(x)^2\right]_{0}^{1}=\frac{1}{2}J_k(1)^2 = \frac{1}{2}\left(\int_{0}^{1}J_k'(x)dx\right)^2.$$ The inequality is strict since $J_k'(x)$ is non-constant over $(0,1)$.