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Let $n\in\mathbb{N}$ and $0<x<1$ be a real number. Is the following a convex function of $x$? $$G_n(x)=\log\left(\frac{(1+x^{4n+1})(1+x^{4n-1})(1+x^{2n})(1-x^{2n+1})}{(1+x^{2n+1})(1-x^{2n+2})}\...

A referee of a submitted paper requested details on the statement that $\int_0^a e^{-tx^2}\,dx$ is log-convex in real $t$, for each $a>0$. While there are a number of ways to prove this statement, ...

$f \in C^2([0,1])$ with $f''$ convex and $f(0) = f'(0) = f''(0) = 0$. Is it true that : $f''(1)+6f(1)\geq 4f'(1)$ ? Source: AoPS