Integral Calculus inequalities problem

[tomelwood]

Homework Statement

Hey, just wondering how I might go about doing this problem, as I really have very little idea...

Prove the following inequality:
[tex]\frac{1}{e}\leq\frac{1}{4\pi^{2}}\int_{R}e^{cos(x-y)}dxdy\leqe[/tex]
(hopefully this reads "one over e is less than or equal to one over four pi squared times the integral over R of e to the power of cos(x-y) dx dy which is less than or equal to e"

Homework Equations

R is the region [0,2pi]x[0,2pi]

The Attempt at a Solution

I think the Mean value, and intermediate value theorem may come into it somewhere, but I really don't know where to begin. Any pointers at all would be greatly appreciated.
Many thanks

 

[losiu99]

Hint: what is the minimal value of [tex]e^{\cos(x-y)}[/tex] on the whole R?