TITLE: About an inequality with central binomial coefficients ${2N \choose N }<2^N {N \choose N/2 }<2 {2N \choose N }$.
QUESTION [10 upvotes]: For every even positive number $N$ we have:

$${2N \choose N }<2^N {N \choose N/2 }<2 {2N \choose N }$$

(Furthermore, $\frac{2^N {N \choose N/2 }}{{2N \choose N }} \to \sqrt{2} $ for large $N$; i.e., the middle expression tends to the geometric mean of those lower-upper bounds)
I'm interested in some simple proof (perhaps some visual/combinatorial one, perhaps counting paths?) of that pair of inequalities.

REPLY [4 votes]: It seems that nobody else has proved the upper inequality, so I'll give a couple (nonvisual, noncombinatorial, debatably simple) proofs.  In both
 I'll prove the slightly stronger inequality
$$ 2^n \binom{n}{n/2} < \binom{2n+1}{n} $$
which implies the desired one because $\binom{2n+1}{n} = \frac{2n+1}{n+1}\binom{2n}{n} < 2\binom{2n}{n}$.
First proof: This proof actually gives both inequalities at once.  First note that
$$\begin{aligned}
&(n+1)(n+2)(n+3)(n+4)\dotsm (n+n-1)(n+n) \\
&< (n+2)(n+2)(n+4)(n+4)\dotsm (n+n)(n+n) \\
&< (n+2)(n+3)(n+4)(n+5)\dotsm (n+n)(n+n+1)
\end{aligned}$$
In other words,
$$ \frac{(2n)!}{n!} < 2^n \left(\frac{n!}{\frac n2!}\right)^2 <\frac{(2n+1)!}{(n+1)!} $$
Dividing by $n!$ yields
$$ \binom{2n}{n} < 2^n \binom{n}{n/2} < \binom{2n+1}{n} $$
(That proof is, I think, the "standard" one.  If you take logs in those initial inequalities and try to estimate the resulting sums by integrals, you're on the road to proving Stirling's formula and Wallis's product and all that.  See Gowers' blog, for example.)
Second proof: This one is a little silly.  By the AM/GM inequality, if $0<t<1$ then
$$ \sqrt{t(1-t)} \le \frac12 $$
and so
$$ 2t(1-t) \le t^{1/2} (1-t)^{1/2} $$
Raising both sides to the power $n$ and integrating over $[0,1]$ yields
$$ 2^n B(n+1,n+1) \le B(\tfrac n2+1,\tfrac n2+1) $$
where $B(\cdot,\cdot)$ is the Beta function.  In other words,
$$ \frac{2^n}{(2n+1)\binom{2n}{n}} \le \frac{1}{(n+1)\binom{n}{n/2}} $$
Rearranging yields the desired inequality.