As I got no answers, I reposted this question in MO.
I have been trying to find the supremum of this bivariate function over a specific region. However, the expressions that I get are horrible. I tried Mathematica, but it did not provide a good answer as it gives you something that is clearly not the supremum, as it is smaller than some values I experimented. I hope someone here can help me. I am searching for an upper bound for the function $$\frac{\log\left(\frac{\binom{n}{d}}{2^{d}4}\right)}{d\log(\frac{n}{d})}$$ over the region $\{(n,d)\in\mathbb{N}\mid n>d>0\}.$ I do not need specifically the supremum (although, of course, that would be the best), a good upper bound would also be helpful. I tried feeding Mathematica with a real extension of this function developing the binomial coefficient using Euler's $\Gamma$ function, which is smooth over the region I am interested in, but Mathematica fails to provide a good upper bound.
Could you help me bounding this function over the region of the integers provided? Does it in fact diverge? If so, how can I see that it diverges?