Entire majorants via Euler-Maclaurin summation

It is the aim of this article to give extremal majorants of type $2\pi\delta$ for the class of functions $f_n(x)=$sgn$(x)x^n$, where $n\in\mathbb{N}$. As applications we obtain positive definite extensions to $\mathbb{R}$ of $\pm(it)^{-m}$ defined on $\mathbb{R}\backslash[-1,1]$, where $m\in\mathbb{N}$, optimal bounds in Hilbert-type inequalities for the class of functions $(it)^{-m}$, and majorants of type $2\pi$ for functions whose graphs are trapezoids.