Timeline for The missing link: an inequality

4 events

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Jan 23, 2017 at 16:59	comment	added	Matt Young		@Iosif Pinelis Yes, that's true, one would need to prove uniform convergence. I imagined that one could show an explicit asymptotic for $n^{-2} F_n''(1-y/n)$ with an error term depending on $n$ and $y$. However, I didn't want to do it myself!
Jan 23, 2017 at 14:40	comment	added	Iosif Pinelis		@MattYoung : For this argument to succeed, even for large $n$, one would also need to show that the convergence of $n^{-2}F_n''(1-\frac{y}{n})$ to the positive expression $\frac{16 e^{4y}}{(1+e^{4y})^2}$ is uniform in all small enough values of $y>0$. This seems likely to be true, since $(1-y/n)^n$ converges uniformly over any bounded set of values of $y$ (as $n\to\infty$). However, Mathematica by itself would probably not be enough to show the uniformity.
Jan 10, 2017 at 18:44	comment	added	T. Amdeberhan		I like this argument. on the other hand, I had some large $n$ analysis already. Please see Section 6 of math.temple.edu/~tewodros/Catalan_Convexity.pdf
Jan 10, 2017 at 18:14	history	answered	Matt Young	CC BY-SA 3.0