Important inequalities: Jensen's comparison of p-norms

Ask the tutor about Important inequalities: Jensen's comparison of p-norms Open the graphing workbench

The Minkowski inequality uses one exponent to measure a sum of two vectors. Keep a single vector still, and only the exponent moves. Which way does the length move?

A larger exponent, a smaller length

For $x \in C^{N}$ and $0 < r < p \leq \infty$ ,

$∥ x ∥_{p} \leq ∥ x ∥_{r} .$

The lengths are the ones named for Hölder. The board freezes $N = 3$ , takes the coordinates real, and starts the smaller exponent at $1$ . The gap keeps the larger exponent strictly above it.

Drag a, b, c, or the gap d. The smaller exponent r stays at least 1, and the larger exponent is r+d, so it stays strictly above r. The upper mark is the larger-exponent length. The lower mark is the smaller-exponent length. The mark under those two is the largest absolute coordinate. Leave only one coordinate nonzero and the three marks share an x-coordinate.

The upper mark stays to the left of the lower one. Leave only one coordinate nonzero and the three marks share an x-coordinate, because every length of $(a, 0, 0)$ is $∣ a ∣$ .

An interval of length 1

On a finite tuple the larger exponent gives the smaller length. On an interval of length $1$ the larger exponent gives the larger integral mean. For $0 < p < r < \infty$ and a function on $[0, 1]$ ,

$(\int_{0}^{1} ∣ f (x) ∣^{p} d x)^{1/ p} \leq (\int_{0}^{1} ∣ f (x) ∣^{r} d x)^{1/ r} .$

The length $1$ is the extra assumption. The board takes $f (t) = (1 - k) + k t^{4}$ with $0 \leq k \leq 1$ .

Drag k or the gap d. The interval is frozen as [0,1], and f(t)=(1-k)+k t^4 with k from 0 to 1. The smaller exponent t stays at least 1/2, and the larger one is t+d. The upper mark is the integral length for the smaller exponent. The lower mark is the integral length for the larger exponent. Set k=0 and the function is the constant 1, so the marks meet.