\documentclass[twoside]{article} \pagestyle{myheadings} \markboth{ On the Opial-Olech-Beesack inequalities } { William C. Troy } \begin{document} \setcounter{page}{297} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent USA-Chile Workshop on Nonlinear Analysis, \newline Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 297--301.\newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % On the Opial-Olech-Beesack inequalities % \thanks{ {\em Mathematics Subject Classifications:} 41A44, 58E35. \hfil\break\indent {\em Key words:} Schwarz inequality, integral inequalities. \hfil\break\indent \copyright 2001 Southwest Texas State University. \hfil\break\indent Published January 8, 2001. } } \date{} \author{ William C. Troy } \maketitle \begin{abstract} We investigate two integral inequalities. The first of these generalizes a result proved by Beesack \cite{beesack} in 1962. We then use our inequality to generalize earlier results of Olech \cite{olech} and Opial \cite{opial} on a related problem. \end{abstract} \newtheorem{theorem}{Theorem}[section] \begin{section}{Introduction.} In 1962 Beesack \cite{beesack} proved the integral inequality \begin{theorem} \label{mainbeesack} Let $b > 0.$ If $y(x)$ is real, continuously differentiable on $[0,b]$, and $y(0)=0$,then $$\label{ineq1} \int_0^b |y(x)y'(x)|dx \leq {{b}\over {2}}\int_0^b |y'(x)|^2\,dx\,.$$ Equality holds only for $y=mx$ where $m$ is a constant. \end{theorem} Beesack used this result to obtain a simplification of proofs given earlier by Olech \cite{olech} and Opial\cite{opial} of the inequality \begin{theorem} \label{opialolech} Let $c>0,$ and let $y(x)$ be real, continuously differentiable on $[0,c]$, with $y(0)=y(c)=0.$ Then $$\label{ineq2} \int_0^c |y(x)y'(x)|dx \leq {{c}\over {4}}\int_0^c |y'(x)|^2dx.$$ Equality holds for the function satisfying $y=x$ on $[0,{{c} \over {2}}]$, and $y=c-x$ on $[{{c} \over {2}},c].$ \end{theorem} In $1964$ Levinson \cite{lev} gave a simpler proof of Theorem $1.1$ His proof generalizes to the class of functions which are complex valued. In this paper we have two goals. The first of these is to extend Levinson's arguments and generalize Beesack's inequality. This is done below in Theorem{\rm \ref{maintroy}}. Our second goal is to use the results of Theorem{\rm \ref{maintroy}} and obtain a generalization of Theorem{\rm \ref{opialolech}}. This is done in Theorem{\rm \ref{troy2}}. \begin{theorem} \label{maintroy} Let $p>-1.$ $(i)$ Let a and b be real with $0 \leq a-1$ and $c>0.$ If $y(x)$ is continuously differentiable on $[0,c]$, and $y(0)=y(c)=0$, then $$\label{ineq35} \int_0^c t^p |y(t)y'(t)|dt \leq {{c^{p+1}}\over {4}{\sqrt {p+1}}}\int_0^c |y'(t)|^2dt +{{c}\over {2}{\sqrt {p+1}}}\int_{c_p}^c(t^p-c^p)|y'(t)|^2dt,$$ where $c_p={{c}\over {2^{1/(p+1)}}}$. \end{theorem} \medskip \noindent {\bf Remarks:} \begin{enumerate} \item[(R3)] If $p=0$ then (\ref {ineq35}) reduces to (\ref {ineq2}). \item[(R4)] It remains an open problem to determine the sharpness of (\ref {ineq35}). \end{enumerate} \end{section} \begin{section}{Proof of Theorem{\bf \ref {maintroy}}} $(i)$ We begin by defining the integral $$I_1 = \int^b_a t^p|y(t)y'(t)|dt.$$ Then $I_1$ can be written in the form $$I_1 = \int^b_a (t^{\frac{p}{2}}(t-a)^{{1}\over {2}} |y'(t)|)(t^{\frac{p}{2}}(t-a)^{{-1}\over {2}}|y(t)|) dt,$$ and an application of the Schwarz inequality leads to $$\label{ineq4} I_1 \leq I^{1/2}_2 I^{1/2}_3,$$ where $$\label{ineq5} I_2 = \int^b_a t^{p}(t-a)|y'(t)|^2 dt$$ and $$\label{ineq6} I_3 = \int^b_a t^{p}(t-a)^{-1}|y(t)|^2 dt\,.$$ Because $y(a) = 0$ and $y(x)$ is continuously differentiable on $[a,b], y(x)$ satisfies $$\label{ineq7} |y(t)|^2 = \bigg|\int^t_a y'(\eta)d\eta\bigg|^2, \qquad a\leq t\leq b.$$ A further application of Schwarz's inequality to (\ref{ineq7}) gives $$\label{ineq8} |y(t)|^2 \leq (t-a)\int^t_a |y'(\eta)|^2 d\eta, \qquad a\leq t\leq b.$$ Combining (\ref{ineq6})and (\ref{ineq8}), we obtain $$\label{ineq9} I_3 \leq \int^b_a t^p \int^t_a |y'(\eta)|^2 d\eta dt.$$ Reversing the order of integration in (\ref {ineq9}) gives $$\label{ineq10} I_3 \leq \frac{b^{p+1}}{p+1}\int^b_a |y'(t)|^2 dt - \frac{1}{p+1} \int^b_a t^{p+1}|y'(t)|^2 dt.$$ Next, we recall a well known result: if $A \ge 0,B \ge 0$ and $\lambda >0,$ then $$\label{ineq10a} (AB)^{1/2} \leq {{\lambda} \over {2}}A + {{1} \over {{2}{\lambda}}}B.$$ We now combine (\ref{ineq4}), (\ref {ineq5}), (\ref {ineq10}) and (\ref{ineq10a}), to obtain $$\label{ineq11} I_1 \leq \frac{1}{2}\biggl(\lambda - \frac{1}{\lambda(p+1)}\biggr)\int^b_a t^{p+1}|y'(t)|^2 dt+ \int^b_a\biggl(\frac{b^{p+1}}{2\lambda(p+1)}-{{a \lambda t^p}\over {2}}\biggr) |y'(t)|^2dt,$$ where $\lambda$ is any positive number. Setting $\lambda = {{1}\over {\sqrt{p+1}}}$ in (\ref{ineq11}), we obtain (\ref {ineq3}). This completes the proof of part $(i)$. \medskip \noindent $(ii)$ The proof of part $(ii)$ follows the method used above. We give the details for the sake of completeness. Thus, we define $$I_4 = \int^c_b t^p|y(t)y'(t)|dt.$$ Then $I_4$ can be written in the form $$I_4 = \int^c_b (t^{\frac{p}{2}}(c-t)^{{1}\over {2}} |y'(t)|)(t^{\frac{p}{2}}(c-t)^{{-1}\over {2}}|y(t)|) dt,$$ and once again an application of the Schwarz inequality leads to $$\label{ineq44} I_4 \leq I^{1/2}_5 I^{1/2}_6,$$ where $$\label{ineq45} I_5 = \int^c_b t^{p}(c-t)|y'(t)|^2 dt \qquad and \qquad I_6 = \int^c_b t^{p}(c-t)^{-1}|y(t)|^2 dt.$$ Because $y(c) = 0$ and $y(x)$ is continuously differentiable on $[b,c], y(x)$ satisfies $$\label{ineq46} |y(t)|^2 = \bigg|\int^c_t y'(\eta)d\eta\bigg|^2, \qquad b\leq t\leq c.$$ An application of Schwarz's inequality to (\ref{ineq46}) gives $$\label{ineq47} |y(t)|^2 \leq (c-t)\int^c_t |y'(\eta)|^2 d\eta, \qquad b\leq t\leq c.$$ Combining (\ref {ineq45}) and (\ref{ineq47}), we obtain $$\label{ineq48} I_6 \leq \int^c_b t^p \int^c_t |y'(\eta)|^2 d\eta dt.$$ Reversing the order of integration in (\ref {ineq48}) leads to $$\label{ineq49} %I_6 \leq -\frac{b^{p+1}}{p+1}\int^c_b |y'(t)|^2 dt + \frac{1}{p+1} \int^c_b I_6 \leq \frac{1}{p+1} \int^c_b (t^{p+1}-b^{p+1})|y'(t)|^2 dt.$$ Next, we combine (\ref {ineq44}), (\ref {ineq45}) and (\ref {ineq49}), and apply (\ref {ineq10a}) to arrive at $$\label{ineq50} I_4 \leq \frac{1}{2}\biggl( \frac{1}{\lambda(p+1)}-\lambda\biggr)\int^c_b t^{p+1}|y'(t)|^2 dt+ \int^c_b\biggl( {{c \lambda t^p}\over {2}}-\frac{b^{p+1}}{2\lambda(p+1)}\biggr) |y'(t)|^2dt,$$ where $\lambda$ is any positive number. Setting $\lambda = {{1}\over {\sqrt{p+1}}}$ in (\ref{ineq50}), we obtain (\ref {ineq33}). This completes the proof of part $(ii)$. \end{section} \begin{section}{Proof of Theorem{\bf \ref {troy2}}} Let $b$ be any positive number satisfying \$0