Can anyone solve this inequality?

Question

(A-B-C+D)/LN((A-C)/(B-D)) > (A-B+C-D)/LN((A-D)/(B-C))

A>B>C>D>0

A,B,C,D elements of real

WHOOPS! I meant to write "prove", not "solve". It's not uni homework, it's the expression of a particular concept that is always taken to be true. I can't tell you what just yet, because it might draw you away from a general proof.

I've never seen a proof for this concept - it's probably because it comes from somewhere else in physics.

Hey fred 055, thanks for trying. I posted this question on maths irc a few years ago but there aren't as many people on there as on YA.

bpiguy - thanks for that insight. Actually the way you re-wrote the problem is how it is expressed conventionally. I hadn't noticed the undefined behaviour of the lhs before - thanks for that. The function "blips" at that condition (A-C = B-D), a tiny variation in one of the values restores it to a sensible value. (A-C = B-D) is not the limit of the inequality though, that happens as A approaches B and C approaches D. That's what I'm particularly interested in.

Clinkit - correct, LN is the natural logarithm function. I was playing with values in Excel, hence all the capitals.

Answer 1

Rewrite it as

[(A - C) - (B - D)] / [ln(A - C) - ln(B - D)] >
[(A - D) + (B - C)] / [ln(A - D) - ln(B - C)]

Look at the left side. Both A-C and B-D are positive, but A-C can equal B-D, in which case the left side is an indeterminate zero divided by zero. Also, B-D can be greater than A-C, so the left side can be a negative divided by a negative.

Now look at the right side. A-D and B-C are always positive, and (A-D) - (B-C) is always positive. Therefore, the entire right side is always positive. It's a well-behaved function.

Seeking a counterexample, I'd look at the indeterminate 0/0 case on the right side in the limit as A-C and B-D approach each other. Let one equal some constant k, and the other a variable x, then take the limit as x approaches k. You'll have something like limit (x - k)/(ln x - ln k) = 1/(1/x) = x > 0, regardless of whether x approaches from the left or right.

I'm not sure this leads to your proof, but it might be a step in the right direction.

Answer 2

what is LN, do you mean natural logarithm ln

Answer 3

Why? You stuck on a Uni homework question?

Answer 4

A>B>C>D>0
So
A-B+C-D >0
So A-D > B-C
So A-D/B-C >1
LN ((A-D/(B-C)) >0
the solution is very difficukt to find

Answer 5

there all the same number or are nuetral to each other is the answer they all equal 0?

Answer 6

I'd say, its all because of quantum physics, nothing really exists!

Answer 7

sorry

I m in weekend mood

Answer 8

G+E

Answer 9

um......... ah......... AAAAAAHHHHHHHHHH!!!!!!