A kind of sharp Wirtinger inequality

In this study, we give a kind of sharp Wirtinger inequality ∥f∥p≤Cr,p,q∥f(r)∥qfor all 1≤p,q≤∞,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \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]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[0,1]$\end{document} and satisfies f(k1)(0)=f(k2)(0)=⋯=f(ks)(0)=f(ms+1)(1)=⋯=f(mr)(1)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f^{(k_{1})}(0)=f^{(k _{2})}(0)=\cdots =f^{(k_{s})}(0)=f^{(m_{s+1})}(1)=\cdots =f^{(m_{r})}(1)=0$\end{document} with 0≤k1<k2<⋯<ks≤r−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0\le k_{1}< k_{2}<\cdots <k_{s}\le r-1$\end{document} and 0≤ms+1<ms+2<⋯<mr≤r−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0\le m_{s+1}< m_{s+2}< \cdots <m_{r}\le r-1$\end{document}. First, based on the Birkhoff interpolation, we refer the computation of Cr,p,q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{r,p,q}$\end{document} to the norm of an integral-type operator. Second, we refer the values of Cr,1,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{r,1,1}$\end{document} and Cr,∞,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{r,\infty ,\infty }$\end{document} to explicit integral expressions and the value of Cr,2,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{r,2,2}$\end{document} to the computation of the maximal eigenvalue of a Hilbert–Schmidt operator. Finally, we give three examples to show our method.