estimative of the norm times p

Question

Is it true that for $p\geq 1$ and $f\in L^1(0,t)\cap L^p(0,t)$ $$|\int_{0}^{t}f(x)dx|^{p}\leq p\int_{0}^{t}|f(x)|^pdx$$

for $p=1$ is obvious, but I'm not sure how to show in the case $p>1$.

Answer 1

Suppose $f(x) = 1$ then the inequality reads $t^p \le pt$ or $t^{p-1} \le p$. Choose $p=2, t>2$.

Answer 2

This is not true. For example, let $f(x)=x$ and $p=2$, then $$\frac{t^4}{4}=\bigg|\int_0^txdx\bigg|^2 \text{ and }\frac{2}{3}t^3=2\int_0^tx^2dx$$ hence if $t> 8/3$, then $$\frac{t^4}{4}>\frac{2}{3}t^3.$$