I am working on proving the below inequality, but I am stuck.
Let $g$ be a differentiable function such that $g(0)=0$ and $0
for all $x$. For all $x\geq 0$, prove that $$\int_{0}^{x}(g(t))^{3}dt\leq \left (\int_{0}^{x}g(t)dt \right )^{2}$$
I am working on proving the below inequality, but I am stuck.
Let $g$ be a differentiable function such that $g(0)=0$ and $0
for all $x$. For all $x\geq 0$, prove that $$\int_{0}^{x}(g(t))^{3}dt\leq \left (\int_{0}^{x}g(t)dt \right )^{2}$$