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Let $f\in L^1(0,1)$. I want to show that $$ \left(\int_0^1 f(t) ~\text{d}t\right) ^2\leqslant \int_0^1f^2(t)~\text{d}t.$$

This is my attempt: I want to apply Jensen's inequality: $\varphi\left(\int_0^1 f\right) \leqslant \int_0^1 \varphi(f).$

Let $\varphi(x)=x^2$. Then $\varphi$ is convex. Thus applying Jensen's inequality, gives the result.

Is what I've done right?

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Yes, you may want to give more details of why you know $\varphi$ is convex. – tomcuchta Nov 17 '11 at 4:38
up vote 4 down vote accepted

That seems fine to me [assuming you are okay showing the fact that $\varphi$ is convex :)]. Another way to get at this is to let $I = \int_0 ^ 1 f \ dt$ and then show $$\int_0 ^ 1 (f - I)^2 \ dt = \int_0 ^ 1 f^2 \ dt - I^2$$ and after rearrangement one gets $\int_0 ^ 1 f^2 \ dt \ge I^2$. This is effectively the same as showing that $\mbox{Var}(X) = E(X^2) - [E(X)]^2$ for a random variable $X$. I suppose this isn't quite right if $f \notin L^2$, but in that case the inequality is trivial since $\int f^2 \ dt = \infty$.

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Very nice ${}{}{}{}$ – Colin Nov 17 '11 at 5:50

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