8 problems

Let $p>3$, and let $\sin_p$ denote the generalized $p$-sine function. The numerical estimates concern the Sobolev regularity of $\sin_p(\pi_p\cdot)$ on $(0,1)$, where…

For $y\in(0,1)$, consider the function $p\mapsto\sin_p(y)$, where $\sin_p$ is the generalized sine. Concavity threshold conjecture. There exists $p_0\in(0,1)$ such that…

For $p>0$, define the generalized trigonometric and hyperbolic functions $\sin_p$, $\cos_p$, $\tan_p$, $\sinh_p$, and $\tanh_p$ as the inverse functions of their corresponding inte…

Let $p,q\in(1,\infty)$ and $r,s\in(0,1)$. The generalized functions $\sin_{p,q}$ and $\sinh_{p,q}$ are defined as the inverse functions of $\arcsin_{p,q}$ and…

Let $p,q\in(1,\infty)$ and $r,s\in(0,1)$. Bhayo–Vuorinen's conjecture. If $p,q\in(1,\infty)$ and $r,s\in(0,1)$, then an inequality stated in the source should hold. The supplied ca…

Cusa–Huygens-type conjecture. The displayed inequality holds for every $p\in(2,\infty)$ and every $x\in(0,\pi_p/2)$. This is part of a sequence of Cusa–Huygens-type inequalities fo…

Monotonicity conjecture. The function $f$ is strictly increasing on $(0,\pi_p/2)$. The statement concerns the interaction between generalized trigonometric and hyperbolic functions…

For $p,q>1$, define the generalized sine functions ${\rm sin}_{p,q}$ and ${\rm sinh}_{p,q}$ as the inverses of ${\rm arcsin}_{p,q}$ and ${\rm arsinh}_{p,q}$, respectively, and let…