geo_mean takes a vector as its argument. What vector is the argument of geo_mean?
Then are you claiming convexity, or concavity of that expression, and do you have a proof?
Anyhow, presuming you don’t intend to have geo_mean at all, and are interested in the convex expression, (1+x/y)^x, that can be handled as
(1+x/y)^x = exp(x*log((1+x/y))) = exp(rel_entr(x+y,y) + rel_entr(y,x+y))
where the latter makes use of the result by @Michal_Adamaszek
x*log(1+x/y) = rel_entr(x+y,y) + rel_entr(y,x+y) in your previous question Writing x*log(1+x/y) .