Relationship between the integration of the square root and the square root of the integration

Question

Is there any valid equality or inequality relationship between $\int_{t_1}^{t_2}\sqrt{f(t)}dt$ and $\sqrt{\int_{t_1}^{t_2}f(t)dt}$?

$f(t) \geq 0, \forall t\in \left [ t_1,t_2 \right ]$

Answer 1

Yes: by Cauchy-Schwarz we have $$ \int_{t_1}^{t_2}\sqrt{f(t)}\;dt\leq \Big(\int_{t_1}^{t_2}f(t)\;dt\Big)^{\frac{1}{2}}\Big(\int_{t_1}^{t_2}\;dt\Big)^{\frac{1}{2}}=\sqrt{t_2-t_1}\Big(\int_{t_1}^{t_2}f(t)\;dt\Big)^{\frac{1}{2}}$$