Thread: Unique minimum of a simple function

Hi,

I want to prove that the following function has a unique minimum … and I'm surprised at how hard this is!



The scalars are all positive. The function is defined on the open triangle , , .

Can you think of some simple proof?

I've tried a few things:
– take the logarithm
– try to show quasi-convexity (which is a stronger result) by several means
– try to rule out saddle points (see my post http://www.mathhelpforum.com/math-he...ts-194122.html and answer by xxp9)
– try to identify a composition of elementary functions
…to no avail

Below is a contour plot of the inverse (because it's nicer to plot than ). The function tends to zero on the border of the triangle. You clearly see there's a unique maximum of (minimum of ) somewhere inside the triangle.

Hi again,

I figured out that my function is actually convex. One can show in a few steps that is a sum of functions that are all convex on the mentioned triangle.

For some reason (which I won't explain in detail) I had convinced myself that couldn't be convex. I failed to see the wood for the trees. You can consider this issue as closed (sorry for posting!).