If $\int_{0+}^{1-} |f(x)|^p dx$ converges for some $p > 1$, then$\int_{0+}^{1-} |f(x)| dx$ converges.

Question

Assume $f$ is continuous on $(0, 1)$. Prove that if $\int_{0+}^{1-} |f(x)|^p dx$ converges for some $p > 1$, then$\int_{0+}^{1-} |f(x)| dx$ converges.

I imagine I would have to use Young's inequality for this, but I'm stuck.

Any help is appreciated!

Answer 1

Hint: use holder-inequality.

$$\int_{0+}^{1-} |f(x)| \cdot 1 dx \le \left(\int_{0+}^{1-} |f(x)|^p dx \right)^{1/p} \left(\int_{0+}^{1-} 1^q dx \right)^{1/q},$$

where $\dfrac1p + \dfrac1q = 1$.

Answer 2

Use that, for every $x$ in $(0,1)$, $$|f(x)|\leqslant1+|f(x)|^p$$ The continuity of $f$ is irrelevant.