PTAH inequality

PTAH inequality

Let

σ

=

{

(

θ

1

,

…

,

θ

n

)

∈

ℝ

n

|

θ

i

≥

0

,

∑

i

=

1

n

θ

i

=

1

}

.

Let

X

be a

measure space

with

measure

m

.
Let

a

i

:

X

→

ℝ

be

measurable functions

such that

a

i

⁢

(

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

→

ℝ

by

p

⁢

(

x

,

λ

)

=

∏

i

=

1

n

λ

i

a

i

⁢

(

x

)

where

λ

=

(

λ

1

,

…

,

λ

n

)

.

And define

P

:

σ

→

ℝ

and

Q

:

σ

×

σ

→

ℝ

by

P

⁢

(

λ

)

=

∫

p

⁢

(

x

,

λ

)

⁢

𝑑

m

⁢

(

x

)

and

Q

⁢

(

λ

,

λ

′

)

=

∫

p

⁢

(

x

,

λ

)

⁢

log

⁡

p

⁢

(

x

,

λ

′

)

⁢

𝑑

m

⁢

(

x

)

.

Define

λ

i

¯

by

λ

i

¯

=

λ

i

⁢

∂

⁡

P

/

∂

⁡

λ

i

∑

j

λ

j

⁢

∂

⁡

P

/

∂

⁡

λ

j

.

Theorem

.

Let

λ

∈

σ

and

λ

¯

be defined
as above. Then for every

λ

′

∈

σ

we have

Q

⁢

(

λ

,

λ

′

)

≤

Q

⁢

(

λ

,

λ

¯

)

with

strict inequality

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

:

∏

x

i

1

n

≤

∑

x

i

n

(B) the concavity of

log

⁡

x

:

∑

θ

i

⁢

log

⁡

x

i

≤

log

⁢

∑

θ

i

⁢

x

i

(C) the Kullback-Leibler inequality:

∏

θ

i

r

i

≤

∏

(

r

i

∑

r

j

)

r

i

(D) the convexity of

x

⁢

log

⁡

x

:

(

∑

θ

i

⁢

x

i

)

⁢

log

⁡

(

∑

θ

i

⁢

x

i

)

≤

∑

θ

i

⁢

x

i

⁢

log

⁡

x

i

(E)

Q

⁢

(

λ

,

λ

′

)

≤

Q

⁢

(

λ

,

λ

¯

)

(F)

Q

⁢

(

λ

,

λ

′

)

-

Q

⁢

(

λ

,

λ

)

≤

P

⁢

(

λ

)

⁢

log

⁡

P

⁢

(

λ

′

)

P

⁢

(

λ

)

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

-

∑

i

=

1

n

p

i

⁢

log

⁡

p

i

≤

log

⁡

n

(H) Hölder’s generalized inequality (

http://planetmath.org/GeneralizedHolderInequality

)

∑

j

=

1

n

∏

i

=

1

m

a

i

,

j

θ

i

≤

∏

i

=

1

m

(

∑

j

=

1

n

a

i

,

j

)

θ

i

(P) The PTAH inequality:

P

⁢

(

λ

)

≤

P

⁢

(

λ

¯

)

All the sums and

products

range from 1 to

n

, all the

θ

i

,

x

i

,

r

i

are

positive

and

(

θ

i

)

,

λ

,

λ

′

are in

σ

and the set

X

is
discrete, so that

P

⁢

(

λ

)

=

∑

x

p

⁢

(

x

,

λ

)

⁢

m

⁢

(

x

)

Q

⁢

(

λ

,

λ

′

)

=

∑

i

r

i

⁢

(

λ

)

⁢

log

⁡

λ

i

′

where

m

⁢

(

x

)

>

0

p

⁢

(

x

,

λ

)

=

∏

i

λ

i

a

i

⁢

(

x

)

and

a

i

⁢

(

x

)

≥

0

,

r

i

⁢

(

λ

)

=

∑

a

i

⁢

(

x

)

⁢

p

⁢

(

x

,

λ

)

⁢

m

⁢

(

x

)

and

λ

i

¯

=

λ

i

⁢

∂

⁡

P

/

∂

⁡

λ

i

∑

λ

j

⁢

∂

⁡

P

/

∂

⁡

λ

j

,

and

λ

¯

=

(

λ

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

→

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.

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

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