PTAH inequality


Let σ={(θ1,,θn)n|θi0,i=1nθi=1}.

Let X be a measure spaceMathworldPlanetmath with measure m. Let ai:X be measurable functionsMathworldPlanetmath such that ai(x)0 a.e [m] for i=1,,n.

Note: the notation “a.e. [m]” means that the condition holds almost everywhere with respect to the measure m.

Define p:X×σ by

p(x,λ)=i=1nλiai(x)

where λ=(λ1,,λn).

And define P:σ and Q:σ×σ by

P(λ)=p(x,λ)𝑑m(x)

and

Q(λ,λ)=p(x,λ)logp(x,λ)𝑑m(x).

Define λi¯ by

λi¯=λiP/λijλjP/λj.

TheoremMathworldPlanetmath. Let λσ and λ¯ be defined as above. Then for every λσ we have

Q(λ,λ)Q(λ,λ¯)

with strict inequalityMathworldPlanetmath unless λ=λ¯. Also,

P(λ)P(λ¯)

with strict inequality unless λ=λ¯. The second inequality is known as the PTAH inequality.

The significance of the PTAH inequality is that some of the classical inequalities are all special cases of PTAH.

Consider:

(A) The arithmetic-geometric mean inequality:

xi1nxin

(B) the concavity of logx:

θilogxilogθixi

(C) the Kullback-Leibler inequality:

θiri(rirj)ri

(D) the convexity of xlogx:

(θixi)log(θixi)θixilogxi

(E)

Q(λ,λ)Q(λ,λ¯)

(F)

Q(λ,λ)-Q(λ,λ)P(λ)logP(λ)P(λ)

(G) the maximum-entropy inequality (in logarithmic form)

-i=1npilogpilogn

(H) Hölder’s generalized inequality (http://planetmath.org/GeneralizedHolderInequality)

j=1ni=1mai,jθii=1m(j=1nai,j)θi

(P) The PTAH inequality:

P(λ)P(λ¯)

All the sums and productsPlanetmathPlanetmathPlanetmath range from 1 to n, all the θi,xi,ri are positive and (θi),λ,λ are in σ and the set X is discrete, so that

P(λ)=xp(x,λ)m(x)
Q(λ,λ)=iri(λ)logλi

where m(x)>0 p(x,λ)=iλiai(x) and ai(x)0, ri(λ)=ai(x)p(x,λ)m(x) and

λi¯=λiP/λiλjP/λj,

and λ¯=(λi¯). Then it turns out that (A) to (G) are all special cases of (H), and in fact (A) to (G) are all equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath, in the sense that given any two of them, each is a special case of the other. (H) is a special case of (P), However, it appears that none of the reverse implicationsMathworldPlanetmath holds. According to George Soules:

”The folklore at the Institute for Defense Analyses in Princeton NJ is that the first program to maximize a function P(z) by iterating the growth transformationPlanetmathPlanetmath

zz¯

was written while the programmer was listening to the opera Aida, in which the Egyptian god of creation Ptah is mentioned, and that became the name of the program (and of the inequality). The name is in upper case because the word processor in use in the middle 1960’s had no lower case.”

References

  • 1 George W. Soules, The PTAH inequality and its relationMathworldPlanetmathPlanetmathPlanetmath to certain classical inequalities, Institute for Defense Analyses, Working paper No. 429, November 1974.
Title PTAH inequality
Canonical name PTAHInequality
Date of creation 2013-03-22 16:54:32
Last modified on 2013-03-22 16:54:32
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 27
Author Mathprof (13753)
Entry type Theorem
Classification msc 26D15