A kind of sharp Wirtinger inequality

Abstract

In this study, we give a kind of sharp Wirtinger inequality \t\t\t∥f∥p≤Cr,p,q∥f(r)∥qfor all 1≤p,q≤∞,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\Vert f \\Vert _{p}\\le C_{r,p,q} \\bigl\\Vert f^{(r)} \\bigr\\Vert _{q} \\quad \\text{for all } 1\\le p,q\\le \\infty , $$\\end{document} where f is defined on [0,1] and satisfies f^{(k_{1})}(0)=f^{(k _{2})}(0)=cdots =f^{(k_{s})}(0)=f^{(m_{s+1})}(1)=cdots =f^{(m_{r})}(1)=0 with 0le k_{1}< k_{2}<cdots <k_{s}le r-1 and 0le m_{s+1}< m_{s+2}< cdots <m_{r}le r-1. First, based on the Birkhoff interpolation, we refer the computation of C_{r,p,q} to the norm of an integral-type operator. Second, we refer the values of C_{r,1,1} and C_{r,infty ,infty } to explicit integral expressions and the value of C_{r,2,2} to the computation of the maximal eigenvalue of a Hilbert–Schmidt operator. Finally, we give three examples to show our method.