jump to navigation

Hölder’s Inequality November 23, 2008

Posted by putnam120 in Math Related.
trackback

I am only going to do the case where f,g are real functions. The result however, still holds if they are complex.

Statement: Suppose that f,g are integrable functions with respect to \alpha on the interval \left[a,b\right]. Additionally p,q\in\mathbb{R^+} such that \frac 1p+\frac 1q=1. Then we have the following inequality:

\displaystyle\left|\int_a^bfgd\alpha\right|\le\left\{\int_a^b|f|^pd\alpha\right\}^{\frac 1p}\left\{\int_a^b|g|^qd\alpha\right\}^{\frac 1q}.

This can also be stated as ||fg||_1\le ||f||_p||g||_q.

Lemma: If u,v\ge{0} then uv\le\frac{u^p}{p}+\frac{v^q}{q}, where we have the same conditions on p,q as before.

Proof of lemma: Just apply Jensen’s Inequality to e^x and the fact that e^{\ln x}=x. \mathbb{Q.E.D.}

Proof: Without loss of generality we can assume that ||f||_p=||g||_q=1, if not we can just divide f,g by the appropriate constants and make it so. Now from the lemma we have that \forall x\in\left[a,b\right] \displaystyle |f(x)g(x)|\le \frac{|f(x)|^p}{p}+\frac{|g(x)|^q}{q} we then integrate both sides of the inequality and the result follows. \mathbb{Q.E.D.}

Comments»

No comments yet — be the first.