Important inequalities: the Minkowski inequality

Hölder's inequality bounds a pairing by two lengths. Those lengths still have to say what happens when the two vectors are added.

Add the arrows

The board freezes $N=2$ and takes both vectors real, so each vector is an arrow in the plane. $\Vert x{\Vert }_p$ and $\Vert x{\Vert }_{\infty }$ are the lengths named on the Hölder page.

For $x,y\in {\mathbb{C}}^N$ and $1\le p\le \infty$,

$\Vert x+y{\Vert }_p\le \Vert x{\Vert }_p+\Vert y{\Vert }_p.$

At $p=1$ the length is the sum of the absolute coordinates. On each coordinate, $|x_k+y_k|\le |x_k|+|y_k|$ is the comparison for two complex numbers on the triangle inequality. Adding those estimates gives the two sides here.

At $p=\infty$ the length is the largest absolute coordinate. The lower axis compares those lengths.

Changing the exponent while the vector stays fixed is a different comparison, on p-norm Jensen inequality.

The integral stops short of infinity

For an integral, the same addition runs over $1\le r<\infty$, so the endpoint $r=\infty$ is not in this statement. Whenever the integrals on the right exist,

$({\int }_G|f(x)+g(x){|}^{r}dx{)}^{1/r}\le ({\int }_G|f(x){|}^{r}dx{)}^{1/r}+({\int }_G|g(x){|}^{r}dx{)}^{1/r}.$

The board freezes the region as $[0,1]$ and the two functions as $f(t)=at$ and $g(t)=1$.