Holder inequality for integrals

Question

Using Holder inequality, is it correct to express $$\parallel \int\limits_0^t f(s)g(s)ds \parallel ^p \leq T^{p-1}\int \limits_0^t\parallel f(s)\parallel^pds\int \limits_0^t\parallel g(s)\parallel^pds $$

Answer 1

No. Take $f(s)=g(s)=1$, $p=1$ and $t=1/2$.