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.