[seqfan] Re: Help with proof strategy needed.

L. Edson Jeffery lejeffery2 at gmail.com
Sat Jul 18 05:52:34 CEST 2015


Hello Brad,

Thank you for helping. Please refer to your response at

http://list.seqfan.eu/pipermail/seqfan/2015-July/015081.html

​You defined

​x = log(B1) log(A2) - log(A1) log(B2).

Your transformation produced the identity

(1)   2 x = - a c + b d,

where

a = log(A2 A1),

b = log(A2/A1),

c = log(B2/B1),

d = log(B2 B1).

Assuming A2 > A1, B2 > B1 and Bi > Ai, you stated that d > a which is
unaltered by exchanging B1 and A2. I think this is, at any rate, true from
the properties of the log function over the reals. However, your equation
(1) is useful.

Now it has to be shown that

(2)   b > c,

from which it follows that

- a c + b d > 0,

since d > a, by the above conditions. But proving that b > c is equivalent
to proving the original assertion, because

log(B1)/log(A1) - log(B2)/log(A2) = (log(B1)/log(A1)) (1 - c/b).

So if b > c, then it remains to prove that if d < a, then

(3)   - a c + b d < 0.

The inequality d < a occurs for the first few coincident rows of arrays A
and B for which Bi < Ai.

Can you prove the inequalities (2) and (3) from the results of your
argument?

Best regards,

Ed Jeffery



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