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\markboth{ On the Opial-Olech-Beesack  inequalities }
{ William C. Troy }
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\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)}
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 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 
\begin{equation}
\label{ineq1}  
\int_0^b |y(x)y'(x)|dx \leq {{b}\over {2}}\int_0^b |y'(x)|^2\,dx\,.
\end{equation}
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
 \begin{equation}
\label{ineq2}  
\int_0^c |y(x)y'(x)|dx \leq {{c}\over {4}}\int_0^c |y'(x)|^2dx.
\end{equation}
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 $0 \leq a<b.$ If $y(x)$ is
 continuously differentiable
  on $[a,b]$, and $y(a)=0$,
then 
\begin{equation}
\label{ineq3}  
\int_a^b t^p |y(t)y'(t)|dt \leq  {{1}\over {2}{\sqrt {p+1}}}\int_a^b
(b^{p+1} -at^p)|y'(t)|^2dt.
\end{equation}
$(ii)$ Let b and c be real with $0 \leq b<c.$ If $y(x)$ is continuously differentiable
 on $[b,c]$, and $y(c)=0$, then
\begin{equation}
\label{ineq33}  
\int_b^c t^p |y(t)y'(t)|dt \leq  {{1}\over {2}{\sqrt {p+1}}}\int_b^c
(ct^p-b^{p+1} )|y'(t)|^2dt.
\end{equation}
\end{theorem}

\noindent{\bf Remarks:} \begin{enumerate}

\item[(R1)] If $a=0$ then (3) reduces to  
\begin{equation}
\label{ineq333}  
\int_0^b t^p|y(t)y'(t)|dt \leq  {{b^{p+1}}\over {2}{\sqrt {p+1}}}\int_0^b
 |y'(t)|^2dt.
\end{equation}


\item[(R2)] It remains an open problem to determine the sharpness of
(\ref {ineq3}), (\ref {ineq33}) and (\ref {ineq333}).
\end{enumerate}
We now state our second result which is a generalization of the inequality of
 Olech \cite{olech} and Opial \cite{opial} stated above in
 Theorem{\ref {opialolech}}.
\begin{theorem}
\label{troy2}
Let  $ p>-1$ and $c>0.$ If $y(x)$ is continuously differentiable 
 on $[0,c]$, and $y(0)=y(c)=0$,
then 
\begin{equation}
\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,
\end{equation}
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 
\begin{equation}
\label{ineq4}
I_1 \leq I^{1/2}_2 I^{1/2}_3,
\end{equation}
where
\begin{equation}
\label{ineq5}
I_2 = \int^b_a t^{p}(t-a)|y'(t)|^2 dt
\end{equation}
and
\begin{equation}
\label{ineq6}
I_3 = \int^b_a t^{p}(t-a)^{-1}|y(t)|^2 dt\,. 
\end{equation}
Because $y(a) = 0$ and $y(x)$ is continuously differentiable on $[a,b], y(x)$
satisfies
\begin{equation}
\label{ineq7}
|y(t)|^2 = \bigg|\int^t_a y'(\eta)d\eta\bigg|^2,  \qquad a\leq t\leq b. 
\end{equation}
A further application of Schwarz's inequality to (\ref{ineq7}) gives  
\begin{equation}
\label{ineq8}
|y(t)|^2 \leq (t-a)\int^t_a |y'(\eta)|^2 d\eta, \qquad a\leq t\leq b.
\end{equation}
Combining (\ref{ineq6})and (\ref{ineq8}), we obtain   
 \begin{equation}
\label{ineq9}
I_3 \leq \int^b_a t^p \int^t_a |y'(\eta)|^2 d\eta dt.
\end{equation}
Reversing the order of integration  in (\ref {ineq9}) gives
\begin{equation}
\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.
\end{equation}
Next, we recall a well known result: if $A \ge 0,B \ge 0$ and $ \lambda >0,$ then
\begin{equation}
\label{ineq10a}
(AB)^{1/2} \leq {{\lambda} \over {2}}A + {{1} \over {{2}{\lambda}}}B.
 \end{equation}
We now combine (\ref{ineq4}), (\ref {ineq5}), (\ref {ineq10})
 and (\ref{ineq10a}),   
  to obtain
\begin{equation}
\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, 
\end{equation}
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 
\begin{equation}
\label{ineq44}
I_4 \leq I^{1/2}_5 I^{1/2}_6,
  \end{equation}
where
\begin{equation}
\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.
\end{equation}
Because $y(c) = 0$ and $y(x)$ is continuously differentiable on $[b,c], y(x)$
satisfies
\begin{equation}
\label{ineq46}
|y(t)|^2 = \bigg|\int^c_t y'(\eta)d\eta\bigg|^2,  \qquad b\leq t\leq c. 
\end{equation}
An application of Schwarz's inequality to (\ref{ineq46}) gives 
\begin{equation}
\label{ineq47}
|y(t)|^2 \leq (c-t)\int^c_t |y'(\eta)|^2 d\eta, \qquad b\leq t\leq c.
\end{equation}
Combining (\ref {ineq45}) and (\ref{ineq47}), we obtain   
 \begin{equation}
\label{ineq48}
I_6 \leq \int^c_b t^p \int^c_t |y'(\eta)|^2 d\eta dt.
\end{equation}
Reversing the order of integration  in (\ref {ineq48}) leads to
\begin{equation}
\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.
\end{equation}
Next, we  combine (\ref {ineq44}), (\ref {ineq45}) and (\ref {ineq49}),
and  apply (\ref {ineq10a}) to arrive at
\begin{equation}
\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, 
\end{equation}
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 $0<b<c.$ Then, from $(3)$
and $(4)$ we obtain  
\begin{eqnarray*}
\int_0^c t^p |y(t)y'(t)|dt &=&  \int_0^b t^p |y(t)y'(t)|dt 
+ \int_b^c t^p |y(t)y'(t)|dt \\
&\leq& {{b^{p+1}}\over {2}{\sqrt {p+1}}}\int_0^b 
|y'(t)|^2dt + {{1}\over {2}{\sqrt {p+1}}}\int_b^c(ct^p-b^{p+1})|y'(t)|^2dt
\end{eqnarray*}

\noindent This inequality can now be written in the form
\begin{equation}
\label{ineq51}  
\int_0^c t^p |y(t)y'(t)|dt= {{b^{p+1}}\over 
{2}{\sqrt {p+1}}}\int_0^c|y'(t)|^2dt
+{{1}\over {2}{\sqrt {p+1}}}\int_b^c(ct^p-2b^{p+1})|y'(t)|^2dt
\end{equation}
\noindent Setting $b= c_p=c/2^{1/(p+1)}$ in (\ref {ineq51}),
 we  obtain
$(6)$ and Theorem{\ref{troy2}} is proved. 
\end{section}

\begin{thebibliography}{0}
 
\bibitem{beesack}
P. R. Beesack,
{\it On an integral inequality of Z. Opial},
Trans. Amer. Math. Soc., {\bf 104} (1962), pp. 470-475.


\bibitem{lev}
N. Levinson,
{\it On an inequality of Opial and Beesack}, 
Proc. Amer. Math. Soc.,{\bf 15} (1964), pp. 565-566. 

\bibitem{olech}
C. Olech,
{\it A simple proof of a certain result of Z. Opial}, 
Ann. Polon. Math., {\bf 8}  (1960), pp. 61-63. 


\bibitem{opial}
Z. Opial,
{\it Sur une inegalite}, 
Ann. Polon. Math., {\bf 8}  (1960), pp. 29-32. 

\end{thebibliography}

\noindent{\sc William C. Troy }\\ 
Department of Mathematics \\
University of Pittsburgh \\
Pittsburgh, Pa. 15260 \\
e-mail: troy@math.pitt.edu \\
website: www.math.pitt.edu/$\sim$troy
 
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