Re: Inequality with max I want to understand

On Sat, 24 May 2008 neillclift@xxxxxxx wrote:

Hi,

I was reading a paper and hit an inequality I have never seen before:

(a + b) / (c + d) <= max (a/c, b/d)

In the paper I have the additional constraints that c > 0, d > 0, a >=
0, b >= 0.

Oh let's see. Some old fashion proportions come to mind.
What were they?

.. . a/c = b/d ==> a/c = (a + b)/(c + d) = b/d

Let's see.
.. . a(c + d) = c(a + b)
.. . b(c + d) = d(a + b)

Ok, it's correct. Hm.
.. . a/c <= b/d ==> a/c <= (a + b)/(c + d) <= b/d

Gee whiz, I'm beginning to understand what the inequality is about.
Interpolation of fractions. Interesting.

I am interested in where this comes from and other examples. If there
are books or other resources that contain stuff similar to this I would
like to know.

How's what I've conjured up?

Execise. For all a,b,c,d > 0,
.. . min( a/c, b/d) <= (a + b) / (c + d) <= max (a/c, b/d)

For example,
.. . min( n,m) <= (n + m)/2 <= max( n,m )

I have no idea how I might search for something like this online for
example.

Interpolation of fractions?

I haven't sat down yet for an extended period to try and prove this.
I don't really have an idea of how to try and tackle it either.

Use the basic inequalities for addition, multiplication and division.
.. . r <= s ==> a + r <= a + s
.. . 0 <= a, r <= s ==> ar <= as
.. . 0 < a ==> 0 < 1/a

Mathematica can find counter examples with negative variables but not
with the additional constraints.

Using computers for math is the faster way to dupe yourself into thinking
you're learning some math.

Thanks.
Neill.

.

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