# Are binomial coefficients with fixed “denominator” log-concave?

I'm working on a problem and began suspecting that the following inequality holds.

Let $k\in\mathbb{N}$ be fixed, and define $f(n)={n\choose k}$. Then $f(n)$ is log-concave in $n$, in particular if $N$ is fixed then for any $n\in[N]$

$$f(n)f(N-n)\leqslant f(\left[\frac{N}{2}\right])^2$$

For example, taking $N=24,n=10,k=2$,

$$LHS={10\choose 2}{14\choose 2}=4095\leqslant 4356={12\choose 2}^2=RHS.$$

I tried doing the second derivative test on $\log{f}$, but there were many ugly terms. I was wondering if there is a neater way of showing it, or if the inequality is perhaps false in general.

Thanks!

Well, $$\binom{N}{k}=\frac{\Gamma(N+1)}{\Gamma(N-k+1)\Gamma(k+1)}$$ hence:
$$\frac{d^2}{dN^2}\log\binom{N}{k} = \psi'(N+1)-\psi'(N-k+1) = \sum_{n\geq 0}\left(\frac{1}{(n+N+1)^2}-\frac{1}{(n+N-k+1)^2}\right)$$ and it is not difficult to discuss convexity.
• Thanks, yes I eventually did something similar (assuming $n\geqslant k$, of course). – youngtableaux Aug 28 '15 at 22:07