Note: the notation “a.e. [m]” means that the condition holds almost everywhere with respect to the measure .
And define and by
The significance of the PTAH inequality is that some of the classical inequalities are all special cases of PTAH.
(A) The arithmetic-geometric mean inequality:
(B) the concavity of :
(C) the Kullback-Leibler inequality:
(D) the convexity of :
(G) the maximum-entropy inequality (in logarithmic form)
(H) Hölder’s generalized inequality (http://planetmath.org/GeneralizedHolderInequality)
(P) The PTAH inequality:
where and , and
and . Then it turns out that (A) to (G) are all special cases of (H), and in fact (A) to (G) are all equivalent, 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 implications holds. According to George Soules:
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.”
- 1 George W. Soules, The PTAH inequality and its relation to certain classical inequalities, Institute for Defense Analyses, Working paper No. 429, November 1974.
|Date of creation||2013-03-22 16:54:32|
|Last modified on||2013-03-22 16:54:32|
|Last modified by||Mathprof (13753)|