Proving $\log\left(\frac{4^n}{\sqrt{2n+1}{2n\choose n+m}}\right)\geq \frac{m^2}{n}$

Here is a partial proof. I have some ideas, but it will be a while before I can return to it. I'll show that the conjecture is true for $m<\sqrt{\log{(\pi/2)}} \ n \sim .672 \ n.$ Maybe someone else can use these ideas for a full proof.

Use the fact that the central binomial $\binom{2n}{n+m}$ has its max at $m=0.$ Do an asymptotic expansion

$$ \binom{2n}{n+m} \big/\binom{2n}{n}=1-\frac{m^2}{n}+\frac{m^2(1+m^2)}{2n^2}-\frac{m^2(1+4m^2+m^4)}{6n^3}... $$ It 'just so happens' that these are the first three terms in $$\exp{\big(-\frac{m^2}{n}(1-\frac{1}{2n}) \big)} =1-\frac{m^2}{n}+\frac{m^2(1+m^2)}{2n^2}-\frac{m^2(3m^2+m^4)}{6n^3}...$$ match. (The gaussian approximation is well known and I added the factor $(1-1/(2n))$ to match the third term.) The exponential makes a convenient bound to go in this problem (note the flip): $$ \binom{2n}{n} \big/\binom{2n}{n+m} \ge \exp{\big(\frac{m^2}{n}(1-\frac{1}{2n}) \big)} $$

Then $$L:=\log\Big(\frac{4^n}{\sqrt{2n+1}\binom{2n}{n+m}}\Big)= \log\Big(\frac{4^n}{\sqrt{2n+1}\binom{2n}{n}} \binom{2n}{n} \big/\binom{2n}{n+m}\Big)$$ $$ \ge\log\Big(\frac{4^n}{\sqrt{2n+1}\binom{2n}{n}} \exp{\big(\frac{m^2}{n}(1-\frac{1}{2n})}\big) \Big) $$ $$ \geq \frac{1}{2} \log{\big( \pi \ n/(2n+1) \big)} + \frac{m^2}{n}(1-\frac{1}{2n}) $$ where the Stirling approximation has been used for the central binomial. For large $n$ proposer's reduces to $$ \frac{1}{2} \log{\big( \pi /2)} > \frac{m^2}{2n^2} .$$ This is indeed true for $m<\sqrt{\log{(\pi/2)}} \ n.$

The problem that I see with this method is that the Gaussian approximation, even with my correction to get the third order term matched, does not do a good job in the 'wings' (large $m.$) A better function is needed, and I believe there are 'entropy' function formulations that can do this. I don't know if an analytic solution will be available using it, but at least the one I've given gets part of the way there.

With @skbmoore's work, we know that this is true for $m<\sqrt{\log(\pi/2)}n$. I'll now show that it's also true for $m>\frac12n$, which will obviously prove the result.

Rearrange the desired inequality as thus: $$\log\left(\frac{4^n}{\sqrt{2n+1}}\right)\ge\frac{m^2}n+\log\binom{2n}{n+m}=f_n(m).$$ We're basically going to try to show that $f_n(m)$ is decreasing after $m=n/2$ (actually, it's a bit before that; I believe it is somehow related to OEIS A143978).

Observe that $$\frac d{dm}f_n(m)=\frac{2m}n+\psi(n-m+1)-\psi(n+m+1),$$ where $\psi$ is the digamma function. (This is from Wolfram Alpha; I've actually never worked with $\psi$ before today, so please let me know if I mess up anywhere here—I'm a bit out of my depth!) Do note that we're extending $f_n(m)$ to be over $[1,n]$, instead of only the integers.

Apparently, for $z\ne-1,-2,\dots$, there is an equation for the digamma function, namely $$\psi(z+1)=-\gamma+\sum_{k=1}^\infty\left(\frac1k-\frac1{k+z}\right),$$ where $\gamma$ is the Euler-Mascheroni constant. It is fortunate for us, then, that $n-m$ and $n+m$ are never nonnegative integers! This means, in particular, that $$-g_n(m)=\psi(n-m+1)-\psi(n+m+1)=\sum_{k=1}^\infty\left(\frac1{k+n+m}-\frac1{k+n-m}\right).$$

Our goal, then, is going to be to show that $g_n(m)>\frac{2m}n$ for all $n>m>\frac n2$. Then we can show that $f_n(m)=\frac{2m}n-g_n(m)<0$.

First, observe that $$\frac d{dm}g_n(m)=\sum_{k=1}^\infty\left(\frac1{(k+n+m)^2}-\frac1{(k+n-m)^2}\right)>0$$ for all $m$. This means, in particular, that if $m$ is not an integer, then $g_n(m)$ is sandwiched between $g_n(\lfloor m\rfloor)$ and $g_n(\lceil m\rceil)$. Obviously, the function $\frac{2m}n$ is increasing with respect to $m$. This all implies that it is sufficient to show that $$\tag{*}g_n(m)\ge\frac{2m-2}n$$ for integers $m\ge\frac n2$.

However, for integers $m$, we know that $g_n(m)$ telescopes as $$g_n(m)=\sum_{k=1}^{2m}\frac1{k+n-m}.$$ Now, if $(*)$ holds for $m$, then it holds for $m+1$. This can be seen by observing that the left hand side increases by $\frac1{n-m}+\frac1{n+m+1}$, while the right side increases by $\frac2n$.

Thus it suffices to prove the statement $(*)$ for $m=\lceil\frac n2\rceil$. But then \begin{align*}g_n(m)&\ge\sum_{k=1}^n\frac1{k+(n-1)/2}\\\frac{2(m-1)}n&\le1.\end{align*} So it suffices to prove that $h(n)=\sum_{k=1}^n\frac1{k+(n-1)/2}\ge1$.

But it is easy to see that if we define $h(n)$ to be the sum above, but removing the floors, then $\frac d{dn}h(n)<0$. Moreover, as $n\to\infty$, this approaches $\log3>1$, according to Wolfram. If someone wants to give me a tip as to how to actually show this limit, I'd love to hear it, but, honestly, I'm a bit pooped! (Check out @skbmoore's explanation for why $h(n)\to\log3$ in the comments!)

This does, however, prove the conjecture! I'm sure there's a much simpler way to do this, since there's no real intuition here; it's just bashing with the one tool I know how to use (Wolfram Alpha! :D)