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\noindent\parbox{4.85in}{\hspace{0.1mm}\\[1.5cm]\noindent
Banach J. Math. Anal. 2 (2008), no. 2, 76--84\\
$\frac{\rule{4.55in}{0.05in}}{{}}$\\
{\footnotesize
\textcolor[rgb]{0.65,0.00,0.95}{\textsc{\textbf{\large{B}}anach
\textbf{\large{J}}ournal of \textbf{\large{M}}athematical
\textbf{\large{A}}nalysis}}\\
ISSN: 1735-8787 (electronic)\\
\textcolor[rgb]{0.00,0.00,0.84}{\textbf{http://www.math-analysis.org }}\\
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\title[Hardy inequality with one negative parameter]{The Hardy inequality with one negative parameter}

\author[A. Kufner, K. Kuliev,
Kulieva]{A. Kufner$^1$, K. Kuliev$^{2}$$^*$ and G. Kulieva$^3$}

\address{$^{1}$ Mathematical Institute, Academy of Sciences of the  Czech
Republic, \v{Z}itna 25, 11567 Praha 1, Czech Republic}

\email{\textcolor[rgb]{0.00,0.00,0.84}{kufner@math.cas.cz}}

\address{$^{2}$ Department of Mathematics, University of West Bohemia,
Univerzitn\'{\i} 22, 30614 Pilsen, Czech Republic }
\email{\textcolor[rgb]{0.00,0.00,0.84}{komil@kma.zcu.cz}}


\address{$^{3}$ Department of Mathematics, University of West Bohemia,
Univerzitn\'{\i} 22, 30614 Pilsen, Czech Republic}
\email{\textcolor[rgb]{0.00,0.00,0.84}{kulievag@mail.ru}}

\dedicatory{Dedicated to Professor Josip E. Pe\v{c}ari\'{c}\\
\vspace{.5cm}{\rm Submitted by C. Park}}

\subjclass[2000]{Primary 26D10, 26D15; Secondary 47B38.}

\keywords{Inequalities, Hardy-type inequalities, weights, negative
powers.}

\date{Received: 19 February 2008; Accepted: 3 June 2008.\newline \indent $^{*}$ Corresponding author}

\begin{abstract}
In this paper, necessary and sufficient conditions for the validity
of the Hardy inequality for the case $q<0,\, p>0 $ and for the case
$q>0,\, p<0$ are derived.
\end{abstract} \maketitle

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