All Questions

Suppose $\Omega$ is a convex bounded open set of $\mathbb{R}^n$ (I would be happy with just $\Omega$ as the $n$-dimensional cube). Let $\mu$ be the uniform measure on $\Omega$ and consider the ...

Let $a$ be a strictly positive $C^\infty$ smooth function on the unit interval. Does there exist a strictly positive $C^\infty$ smooth function $f$ on $I$ such that $$ f’’(x) \leq 0\quad \text{and} \...

Let $f : \mathbb{R}^d \rightarrow \mathbb{R}$ be infinitely often continuously differentiable and convex. For $d = 1$, we know that for any interval $[a, b]$, it holds for $x, y \in [a, b]$ that $$ (f'...

Let $Q=[0,1]\times[0,1]$ and let $a$ be a positive smooth function on $Q$. Does there exist a smooth positive function $u$ On $Q$ such that there holds $$ \frac{\partial^2}{\partial x_1^2}u <0 $$ ...

I have noticed experimentally that the following question has a positive answer. Is it true that for all even and convex functions $f$, $g$: $$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^...

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