Hölder’s and Minkowski’s Inequalities

Introduction

I have seen Hölder’s inequality and Minkowski’s inequality proved in several ways but this seems the most perspicuous (to me at any rate).

Young’s Inequality

If a, b \ge 0 and p,q \ge 1 such that

\displaystyle   \frac{1}{p} + \frac{1}{q} = 1

then

\displaystyle   ab \le \frac{a^p}{p} + \frac{b^q}{q}

A p and q satisfying the premise are known as conjugate indices.

Proof

Since \log is convex we have

\displaystyle   t\log{x} + (1 - t)\log{y} \le \log{(tx + (1 - t)y)}

Substituting in appropriate values gives

\displaystyle   \frac{1}{p}\log{a^p} + \frac{1}{q}\log{b^q} \le \log{\bigg(\frac{a^p}{p} + \frac{b^q}{q}\bigg)}

or

\displaystyle   \log{a} + \log{b} \le \log{\bigg(\frac{a^p}{p} + \frac{b^q}{q}\bigg)}

Now take exponents.

\blacksquare

Hölders’s Inequality

Let p and q be conjugate indices with 1 < p < \infty and let f \in L^p(\Omega) and g \in L^q(\Omega) then fg \in L^1(\Omega) and

\displaystyle   \|fg\|_{L^1} \le \|f\|_{L^p}\|g\|_{L^q}

Proof

By Young’s inequality

\displaystyle   \int_\Omega \frac{|f(x)|}{\|f\|_{L^p}} \frac{|g(x)|}{\|g\|_{L^q}} \le  \int_\Omega \frac{1}{p}\frac{|f(x)|^p}{\|f\|_{L^p}^p} +              \frac{1}{q}\frac{|g(x)|^q}{\|g\|_{L^q}^q} =  \frac{1}{p} + \frac{1}{q} = 1

\blacksquare

By applying a counting measure to \Omega we also obtain

\displaystyle   \sum |x_i y_i| \le \big(\sum |x_i|^p\big)^{1/p} \big(\sum |y_i|^q\big)^{1/q}

Minkowski’s Inequality

\displaystyle   \|f + g\|_{L^p} \le \|f\|_{L^p} + \|g\|_{L^p}

Proof

By Hölder’s inequality

\displaystyle   \int_\Omega |f + g|^p \le \int_\Omega |f||f + g|^{p-1} +                            \int_\Omega |g||f + g|^{p-1}                            \le \|f\|_{L^p}A + \|g\|_{L^p}A

where

\displaystyle   A = \||f + g|^{p-1}\|_{L^q} = \big(\int_\Omega |f(x) + g(x)|^p\big)^{1/q}

and A is finite since L^p is a vector space.

\blacksquare

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