| HU KE.[J].数学年刊A辑,1981,2(1):21~24 |
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| ON MILIN-LEBDEV INEQUALITIES |
| Received:November 03, 1979 Revised:February 12, 1980 |
| DOI： |
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| Let \[\varphi (x) = \sum\limits_{k = 1}^\infty {{A_k}} {x^k},\Phi (x) = {e^{\varphi (x)}} = \sum\limits_{k = 1}^\infty {{D_k}} {x^k}\]
\[\begin{gathered}
\frac{1}{{{{(1 - x)}^\lambda }}} = \sum\limits_{k = 1}^\infty {{d_k}} (\lambda ){x^k} \hfill \ {\overline \Delta _n}(\lambda ) = {\lambda ^{2 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}} \mathop {|{A_k}|}\nolimits_{}^p - \sum\limits_{k = 1}^\infty {\frac{1}{k}} \hfill \\
\end{gathered} \]
Milin-Lebedey proved that
\[\sum\limits_{k = 0}^\infty {\frac{{|{D_k}{|^p}}}{{d_k^{p - 1}(\lambda )}}} \leqslant \exp \{ {\lambda ^{1 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}} |{A_k}{|^p}\} \]
where p>l and \[\lambda \]>0.
In this paper, we have proved the following theorems；
Theorem 1. Let \[p \geqslant 1,\lambda > 0\] and
\[F(x) = \sum\limits_{k = 0}^\infty {\frac{{|{D_k}{|^p}}}{{d_k^p(\lambda )}}} {x^p}\exp \{ - {\lambda ^{1 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}|{A_k}{|^p}{x^k}} \} (2)\]
then F(x) is a decreasing function of x on [0, 1].
This theorem is stronger than the result (1).
Theorem 2. Let \[p \geqslant 2,\lambda > 0\] and
\[{{\bar Q}_n}(\lambda ) = \frac{1}{{n + 1}}\sum\limits_{k = 0}^n {\frac{{|{D_k}{|^p}}}{{d_k^p(\lambda )}}\exp } \{ - \frac{1}{{n + 1}}\sum\limits_{v = 1}^n {\overline {{\Delta _p}} } (\lambda )\} \]
then \[{{\bar Q}_n}(\lambda )\] is a decreasing fimctLon of n(n=l, 2，...)In the case p=2 this is contained in the Miiin-Lebedev's result. |
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