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I'm trying to show that a particular function of two variables is concave for all positive values of x and y. I'm pretty sure it is, but I haven't been able to prove it.

The function is (1-e

^{t(y-x)})xy)/(x+e

^{t(y-x)}y)

t is a positive parameter, and as mentioned, x and y are always positive.

I've tried looking at second derivatives and the Hessian, but haven't really got anywhere. I wonder if this is even possible to prove generally, or just for specific values of x and y?

Actually, for my pyrposes it would be sufficient to show that for any given fixed value of y, the function is a concave function of x, and vice versa. I'm not sure if this is any easier...

Any input and suggestions would be much appreciated!