3 problems

For $0<\varphi_2<\varphi_1$, write $U_i=U(\varphi_i)$ and $Z_{ij}=Z(\varphi_i,\varphi_j)$ for the functions defined in the paper. Positivity conjecture. One has … for every…

Let $\psi(t)$ be the function defined in the preceding series representation, let $\alpha_1=1$, $\alpha_{2j}=0$, and $\alpha_{2j+1}=-\frac{2^{2j}|B_{2j}|}{(2j)!}$ for $j\in N$, and…

For $x\in(0,1)$, consider the inverse trigonometric functions $\arcsin x$ and $\arctan x$. Chen's conjecture. The following two inequalities hold: … and … These inequalities are pr…