Re: Inequality with max I want to understand


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.
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.
I have no idea how I might search for something like this online for
example.
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.
Mathematica can find counter examples with negative variables but not
with the additional constraints.

Since we can multiply the numerators or denominators by a positive constant
without changing the inequality, we may assume wlog that a+b = c+d = 1.
Thus the inequality says
1 <= max(a/c, (1-a)/(1-c)) for 0 < a < 1, 0 < c < 1.
Let the right side be R. There are three cases:
1) for a = c we have R = 1;
2) for a > c we have R >= a/c > 1;
3) for a < c we have 1-a > 1-c, and R >= (1-a)/(1-c) > 1.
So the inequality is true.
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

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