Timo Weidl
On the Lieb-Thirring constants L_gamma,1 for gamma geq 1/2.
(31K, AMS-LATEX)

ABSTRACT.  Let $E_i(H)$ denote the negative eigenvalues of the one-dimensional
Schr\"odinger operator $Hu:=-u^{\prime\prime}-Vu,\ V\geq 0,$
on $L_2({\Bbb R}).$ We prove the inequality
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\begin{equation}\label{1}
\sum_i|E_i(H)|^\gamma\leq L_{\gamma,1}\int_{\Bbb R} V^{\gamma+1/2}(x)dx,
\end{equation}
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for the "limit" case $\gamma=1/2.$ This will imply improved estimates
for the best constants $L_{\gamma,1}$ in \eqref{1} as $1/2<\gamma<3/2.