%% Created by Maple V Release 4 (APPLE_PPC_MAC)
%% Source Worksheet: CBS.mws
%% Generated: Sat Nov  2 10:24:04 1996
%\documentclass{article}
%\usepackage{maple2e}
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%\DefineParaStyle{Heading 2}
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%\DefineParaStyle{Maple Output}
%\DefineParaStyle{Title}
%\DefineCharStyle{2D Comment}
%\DefineCharStyle{2D Math}
%\DefineCharStyle{2D Output}
%\DefineCharStyle{Hyperlink}
%\begin{document}

\begin{Title}
\QTR{Hyperlink}{Real Analysis with Maple}
\end{Title}

\setcounter{section}{4}
\section{The Cauchy-Schwarz Inequality}

\subsection{The Cauchy-Bunyakovski-Schwarz Inequality}

\subsubsection{Theorem}

Let 
\mapleinline{inert}{2d}{a[k]}{%
${a_{k}}$%
} and 
\mapleinline{inert}{2d}{b[k]}{%
${b_{k}}$%
} be complex numbers for 
\mapleinline{inert}{2d}{k=1..n}{%
$k=1 .. n$%
}.
                Then

\begin{center}
\mapleinline{inert}{2d}{Sum(abs(a[k]*b[k]),k = 1 .. n) <=
sqrt(Sum(a[k]^2,k = 1 .. n)*Sum(b[k]^2,k = 1 .. n))
}{%
$\sum _{k=1}^{n}\, \left|  \! \,{a_{k}}\,{b_{k}}\, \!  \right| 
\leq \sqrt{(\sum _{k=1}^{n}\,{a_{k}}^{2})\,(\sum _{k=1}^{n}\,{b_{
k}}^{2})}$%
} .
\end{center}

\begin{Heading 3}
\end{Heading 3}

\subsubsection{Proof}

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{restart: with(student):}{%
}
\end{mapleinput}

\end{maplegroup}
A sum of squares must be nonnegative, hence

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{Sum((abs(a[k])*x+abs(b[k]))^2,k=1..n) >=
0;}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
0\leq {\displaystyle \sum _{k=1}^{n}} \,( \left|  \! \,{a_{k}}\,
 \!  \right| \,x +  \left|  \! \,{b_{k}}\, \!  \right| )^{2}
\]
\end{maplelatex}

\end{maplegroup}
Expand the sums and collect like terms in 
\mapleinline{inert}{2d}{x}{%
$x$%
} to find

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{expand(");}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
0\leq x^{2}\,({\displaystyle \sum _{k=1}^{n}} \, \left|  \! \,{a
_{k}}\, \!  \right| ^{2}) + 2\,x\,({\displaystyle \sum _{k=1}^{n}
} \, \left|  \! \,{a_{k}}\, \!  \right| \, \left|  \! \,{b_{k}}\,
 \!  \right| ) + ({\displaystyle \sum _{k=1}^{n}} \, \left|  \! 
\,{b_{k}}\, \!  \right| ^{2})
\]
\end{maplelatex}

\end{maplegroup}
which is a quadratic equation.

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{subs(
  Sum(abs(a[k])^2,k=1..n) = A,}{%
}
\mapleinline{active}{1d}{
  Sum(abs(a[k])*abs(b[k]),k=1..n) = B}{%
}
\mapleinline{active}{1d}{Sum(abs(b[k])^2,k=1..n) = C, "); }{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
0\leq x^{2}\,A + 2\,x\,B + C
\]
\end{maplelatex}

\end{maplegroup}
The minimum must occur at the vertex 
\mapleinline{inert}{2d}{x=-B/A}{%
$x= - \frac {B}{A}$%
} . Substitute and rearrange the inequality, isolating 
\mapleinline{inert}{2d}{B^2}{%
$B^{2}$%
}.

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{subs(x=-B/A, ");
expand("*A) + (B^2=B^2);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
0\leq  - {\displaystyle \frac {B^{2}}{A}}  + C
\]
\end{maplelatex}

\begin{maplelatex}
\[
B^{2}\leq A\,C
\]
\end{maplelatex}

\end{maplegroup}
Replacing 
\mapleinline{inert}{2d}{A}{%
$A$%
}, 
\mapleinline{inert}{2d}{B}{%
$B$%
}, and 
\mapleinline{inert}{2d}{C}{%
$C$%
} with their respective sums finishes the derivation.

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{subs(A=Sum(abs(a[k])^2,k=1 ..n),
B=Sum(abs(a[k])*abs(b[k]),k=1..n), C=Sum(abs(b[k])^2,k=1..n), ");}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
({\displaystyle \sum _{k=1}^{n}} \, \left|  \! \,{a_{k}}\, \! 
 \right| \, \left|  \! \,{b_{k}}\, \!  \right| )^{2}\leq (
{\displaystyle \sum _{k=1}^{n}} \, \left|  \! \,{a_{k}}\, \! 
 \right| ^{2})\,({\displaystyle \sum _{k=1}^{n}} \, \left|  \! \,
{b_{k}}\, \!  \right| ^{2})
\]
\end{maplelatex}

\end{maplegroup}

\subsection{Additional Activities}

The Useful Inequality

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{1/p + 1/q = 1;
a*b <= a^p*t^p/p + b^q*t^(-q)/q;}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
{\displaystyle \frac {1}{p}}  + {\displaystyle \frac {1}{q}} =1
\]
\end{maplelatex}

\begin{maplelatex}
\[
a\,b\leq {\displaystyle \frac {a^{p}\,t^{p}}{p}}  + 
{\displaystyle \frac {b^{q}\,t^{( - q)}}{q}} 
\]
\end{maplelatex}

\end{maplegroup}
Holder�s Inequality

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{Sum(a[k]*b[k],k=1..n) <=}{%
}
\mapleinline{active}{1d}{(Sum(a[k]^p,k=1..n))^(1/p)}{%
}
\mapleinline{active}{1d}{* (Sum(b[k]^q,k=1..n))^(1/q);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
{\displaystyle \sum _{k=1}^{n}} \,{a_{k}}\,{b_{k}}\leq (
{\displaystyle \sum _{k=1}^{n}} \,{a_{k}}^{p})^{(\frac {1}{p})}\,
({\displaystyle \sum _{k=1}^{n}} \,{b_{k}}^{q})^{(\frac {1}{q})}
\]
\end{maplelatex}

\end{maplegroup}
Minkowski�s Inequality

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{(Sum((a[k]+b[k])^p,k=1..n))^(1/p) <=
(Sum(a[k]^p,k=1..n))^(1/p)
+(Sum(b[k]^p,k=1..n))^(1/p);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
({\displaystyle \sum _{k=1}^{n}} \,({a_{k}} + {b_{k}})^{p})^{(
\frac {1}{p})}\leq ({\displaystyle \sum _{k=1}^{n}} \,{a_{k}}^{p}
)^{(\frac {1}{p})} + ({\displaystyle \sum _{k=1}^{n}} \,{b_{k}}^{
p})^{(\frac {1}{p})}
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\end{mapleinput}

\end{maplegroup}
%\end{document}
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