Hölder, Minkowski, Cauchy-Schwarz and triangle inequalities

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 23 July 2014

Hölder, Minkowski, Cauchy-Schwarz and triangle inequalities

Let q0. Let p>1{} be given by 1p+1q=1. The functions h:00 given by h(x)=xq 1 x 1 y y=x y=x2 y=x3 y=x4 are increasing with h(1)=1.

The functions h:00 given by h(x)=x1q 1 x 1 y y=x y=x12 y=x13 y=x14 are increasing with h(1)=1.

The functions h:>00 given by h(x)=x-1q 1 x 1 y y=1x y=x-12 y=x-13 are decreasing with h(1)=1.

Thus the functions g:0 given by g(x)=x-1q-1 are decreasing with g(1)=0. So x-1q-10 forx1. If f:>0 is given by f(x)=x1p -1px then dfdx=1px1p-1-1p=1p(x-1q-1) and so f is decreasing for x>1 and f(1)=1-1p=1q. So x1p-1px 1qforx 1.

Let a,b>0 with ab and let x=ab. Then 1q (ab)1p-1p (ab)=1b (a1pb-1p+1-1pa) =1b(a1pb1q-1pa). So 1pa+1qb a1pb1q fora,b>0.

Let x=(x1,,xn)n and y=(y1,,yn)n. Then |xiyi| xpyq 1p ( |xi| xp ) p +1q ( |yi| yq ) q . So i=1n |xiyi| xpyq i=1n ( 1p ( |xi| xp ) p +1q ( |yi| yq ) q ) = 1p+1q=1. So i=1n |xiyi| xpyq. So |i=1nxiyi| i=1n|xiyi| xpyq.

Using |xi+yi| |xi|+|yi| andp-1=p(1-1p) =p1q=pq and |x1+y1|pq ,, |xn+yn|pq q = (i=1n((xi+yi)pq)q)1q = (i=1n|xi+yi|p)1p·pq = (x+yp)pq, gives |x+y|pp = i=1n |xi+yi|p =i=1n |xi+yi| |xi+yi|p-1 i=1n (|xi|+|yi|) |xi+yi|pq = i=1n|xi| |xi+yi|pq+ i=1n|yi| |xi+yi|pq xp ( |x1+y1|pq,, |xn+yn|pq ) q + yp ( |x1+y1|pq,, |xn+yn|pq ) q = xp x+yppq+ yp x+yppq = ( xp+ yp ) x+ypp-1. Dividing both sides by x+ypp-1, then x+yp xp+ yp.

Let x=(x1,,xn)n. For p1 define xp= ( |x1|p++ |xn|p ) 1p . For x=(x1,,xn) in n define |x|=|x|2 =(|x1|2++|xn|2)12 =x12++xn2 and, for x=(x1,,xn) and y=(y1,,yn) in n define x,y= x1y1++xnyn.

Let x,yn with x=(x1,,xn) and y=(y1,,yn). If p>1 and q is given by 1p+1q=1 then |i=1nxiyi| xpyq and x+yp xp+yp.

Let ,xyn with x=(x1,,xn) and y=(y1,,yn). Then |x,y|= |i=1nxiyi| |x||y| and|x+y| |x|+|y|.

Special case: Let x,y. Then |xy|=|x||y| and|x+y| |x|+|y|.

Notes and References

These are a typed copy of handwritten notes from the pdf 140721Holderinequalitiesscanned140721.pdf.

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