Re: Inequality with max I want to understand

In article <250520080450125841%edgar@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"G. A. Edgar" <edgar@xxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

In article
<2c9eb255-e0ba-41eb-ab8a-dd3e8289915b@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<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.
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.

Thanks.
Neill.

Another way to prove it is to write (a + b) / (c + d) as
a convex combination of a/c and b/d ... Here is where you
need to know certain things are nonnegative.

Thanks. I put a bit of effort into finding a nifty method
and failed. This is the kind of thing I had in mind: convexity.
I will not forget.

--
Michael Press
.

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