The optimization for the inequalities of power means
<p/> <p>Let <inline-formula><graphic file="1029-242X-2006-46782-i1.gif"/></inline-formula> be the <inline-formula><graphic file="1029-242X-2006-46782-i2.gif"/></inline-formula>th power mean of a sequence <inline-formula><gr...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2006-01-01
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| Series: | Journal of Inequalities and Applications |
| Online Access: | http://www.journalofinequalitiesandapplications.com/content/2006/46782 |
| Summary: | <p/> <p>Let <inline-formula><graphic file="1029-242X-2006-46782-i1.gif"/></inline-formula> be the <inline-formula><graphic file="1029-242X-2006-46782-i2.gif"/></inline-formula>th power mean of a sequence <inline-formula><graphic file="1029-242X-2006-46782-i3.gif"/></inline-formula> of positive real numbers, where <inline-formula><graphic file="1029-242X-2006-46782-i4.gif"/></inline-formula>, and <inline-formula><graphic file="1029-242X-2006-46782-i5.gif"/></inline-formula>. In this paper, we will state the important background and meaning of the inequality <inline-formula><graphic file="1029-242X-2006-46782-i6.gif"/></inline-formula>; a necessary and sufficient condition and another interesting sufficient condition that the foregoing inequality holds are obtained; an open problem posed by Wang et al. in 2004 is solved and generalized; a rulable criterion of the semipositivity of homogeneous symmetrical polynomial is also obtained. Our methods used are the procedure of descending dimension and theory of majorization; and apply techniques of mathematical analysis and permanents in algebra.</p> |
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| ISSN: | 1025-5834 1029-242X |