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# PTAH inequality

Let $\sigma=\{(\theta_{1},\ldots,\theta_{n})\in{\mathbb{R}}^{{n}}|\theta_{i}\geq 0,% \sum_{{i=1}}^{{n}}\theta_{i}=1\}$.

Let $X$ be a measure space with measure $m$. Let $a_{i}:X\to\mathbb{R}$ be measurable functions such that $a_{i}(x)\geq 0$ a.e [m] for $i=1,\ldots,n$.

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

Define $p:X\times\sigma\to\mathbb{R}$ by

$p(x,\lambda)=\prod_{{i=1}}^{n}{\lambda_{i}}^{{a_{i}(x)}}$ |

where $\lambda=(\lambda_{1},\ldots,\lambda_{n})$.

And define $P:\sigma\to\mathbb{R}$ and $Q:\sigma\times\sigma\to\mathbb{R}$ by

$P(\lambda)=\int p(x,\lambda)dm(x)$ |

and

$Q(\lambda,\lambda^{{\prime}})=\int p(x,\lambda)\log p(x,\lambda^{{\prime}})dm(% x).$ |

Define $\overline{\lambda_{i}}$ by

$\overline{\lambda_{i}}=\frac{\lambda_{i}\partial P/\partial\lambda_{i}}{\sum_{% j}\lambda_{j}\partial P/\partial\lambda_{j}}.$ |

Theorem. Let $\lambda\in\sigma$ and $\overline{\lambda}$ be defined as above. Then for every $\lambda^{{\prime}}\in\sigma$ we have

$Q(\lambda,\lambda^{{\prime}})\leq Q(\lambda,\overline{\lambda})$ |

with strict inequality unless $\lambda^{{\prime}}=\overline{\lambda}$. Also,

$P(\lambda)\leq P(\overline{\lambda})$ |

with strict inequality unless $\lambda=\overline{\lambda}$.
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:

$\prod{x_{i}}^{{\frac{1}{n}}}\leq\sum\frac{x_{i}}{n}$ |

(B) the concavity of $\log x$:

$\sum\theta_{i}\log x_{i}\leq\log\sum\theta_{i}x_{i}$ |

(C) the Kullback-Leibler inequality:

$\prod{\theta_{i}}^{{r_{i}}}\leq\prod(\frac{r_{i}}{\sum r_{j}})^{{r_{i}}}$ |

(D) the convexity of $x\log x$:

$(\sum\theta_{i}x_{i})\log(\sum\theta_{i}x_{i})\leq\sum\theta_{i}x_{i}\log x_{i}$ |

(E)

$Q(\lambda,\lambda^{{\prime}})\leq Q(\lambda,\overline{\lambda})$ |

(F)

$Q(\lambda,\lambda^{{\prime}})-Q(\lambda,\lambda)\leq P(\lambda)\log\frac{P(% \lambda^{{\prime}})}{P(\lambda)}$ |

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

$-\sum_{{i=1}}^{n}p_{i}\log p_{i}\leq\log n$ |

(H) Hölder’s generalized inequality

$\sum_{{j=1}}^{n}\prod_{{i=1}}^{m}a_{{i,j}}^{{\theta_{i}}}\leq\prod_{{i=1}}^{m}% \left(\sum_{{j=1}}^{n}a_{{i,j}}\right)^{{\theta_{i}}}$ |

(P) The PTAH inequality:

$P(\lambda)\leq P(\overline{\lambda})$ |

All the sums and products range from 1 to $n$, all the $\theta_{i},x_{i},r_{i}$ are positive and $(\theta_{i}),\lambda,\lambda^{{\prime}}$ are in $\sigma$ and the set $X$ is discrete, so that

$P(\lambda)=\sum_{x}p(x,\lambda)m(x)$ |

$Q(\lambda,\lambda^{{\prime}})=\sum_{i}r_{i}(\lambda)\log{\lambda_{i}}^{{\prime}}$ |

where $m(x)>0$ $p(x,\lambda)=\prod_{i}{\lambda_{i}}^{{a_{i}(x)}}$ and $a_{i}(x)\geq 0$, $r_{i}(\lambda)=\sum a_{i}(x)p(x,\lambda)m(x)$ and

$\overline{\lambda_{i}}=\frac{\lambda_{i}\partial P/\partial\lambda_{i}}{\sum% \lambda_{j}\partial P/\partial\lambda_{j}},$ |

and $\overline{\lambda}=(\overline{\lambda_{i}})$. 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:

”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 transformation

$z\to\overline{z}$ |

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 relation to certain classical inequalities*, Institute for Defense Analyses, Working paper No. 429, November 1974.

## Mathematics Subject Classification

26D15*no label found*

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## Attached Articles

## Corrections

minor things by alozano ✓

the notation "a.e. [m]" means that the condition holds by alozano ✓

The generalized Holder's inequality (H) lies between (G) and (A) through (F). by cappymate ✘

The generalized Holder's inequality (H) lies between (G) and (A) through (F). by cappymate ✓

## Comments

## PTAH question

So, what does "PTAH" stand for?

bob1

## Re: PTAH question

i don't know.

## Re: PTAH question

Is it in connection to the ancient Egyptian god Phtah?

(http://en.wikipedia.org/wiki/Ptah)

## Re: PTAH question

Since you have rejected most of my corrections, the entry is still useless to anyone who has not read the paper you refer to. For example, why did you reject my correction about the a.e [m] abbreviation? This notation is quite unclear to me, and instead of rejecting and explaining just to me what it means, it would be much clearer if you explained this notation yourself in the entry or at least add a link to an entry where this misterious notation is explained, or at least add entries to the "see also" list for the background needed to read this entry. And again, the lambda_i's appear out of nowhere, it would be better to define at least \lambda=(\lambda_1,...), and \sigma as a collection of \lambdas.

Alvaro

## Re: PTAH question

Mathprof,

I see that you have (quietly) accepted my suggestions and made the suggested changes in the PTAH entry. So, why were my corrections rejected in the first place?

Alvaro

## Re: PTAH question

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 transformation

z --> zbar

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.