Let $\phi\colon[0,1] \to \mathbb R$ be such that $\phi,\phi^\prime,\phi^{\prime\prime}$ are continuous on $[0,1]$, then the following inequality holds:
$\int_0^1\cos x\frac{x\phi^\prime(x)-\phi(x)+\phi(0)}{x^2}\mathrm dx < \frac32\|\phi^{\prime\prime}\|_\infty$
I have no idea how to solve this problem, could you help me please?