Entire majorants via Euler–Maclaurin summation

Abstract

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