Inequalities for Generalized Logarithmic Means
<p/> <p>For <inline-formula> <graphic file="1029-242X-2009-763252-i1.gif"/></inline-formula>, the generalized logarithmic mean <inline-formula> <graphic file="1029-242X-2009-763252-i2.gif"/></inline-formula> of two positive numbers...
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doaj-d7dde0d617a240f48b64f09ae54be45d2020-11-25T00:36:12ZengSpringerOpenJournal of Inequalities and Applications1025-58341029-242X2009-01-0120091763252Inequalities for Generalized Logarithmic MeansXia Wei-FengChu Yu-Ming<p/> <p>For <inline-formula> <graphic file="1029-242X-2009-763252-i1.gif"/></inline-formula>, the generalized logarithmic mean <inline-formula> <graphic file="1029-242X-2009-763252-i2.gif"/></inline-formula> of two positive numbers <inline-formula> <graphic file="1029-242X-2009-763252-i3.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2009-763252-i4.gif"/></inline-formula> is defined as <inline-formula> <graphic file="1029-242X-2009-763252-i5.gif"/></inline-formula>, for <inline-formula> <graphic file="1029-242X-2009-763252-i6.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2009-763252-i7.gif"/></inline-formula><inline-formula> <graphic file="1029-242X-2009-763252-i8.gif"/></inline-formula> , for <inline-formula> <graphic file="1029-242X-2009-763252-i9.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2009-763252-i10.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2009-763252-i11.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2009-763252-i12.gif"/></inline-formula><inline-formula> <graphic file="1029-242X-2009-763252-i13.gif"/></inline-formula>, for <inline-formula> <graphic file="1029-242X-2009-763252-i14.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2009-763252-i15.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2009-763252-i16.gif"/></inline-formula><inline-formula> <graphic file="1029-242X-2009-763252-i17.gif"/></inline-formula> , for <inline-formula> <graphic file="1029-242X-2009-763252-i18.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2009-763252-i19.gif"/></inline-formula>. In this paper, we prove that <inline-formula> <graphic file="1029-242X-2009-763252-i20.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2009-763252-i21.gif"/></inline-formula> for all <inline-formula> <graphic file="1029-242X-2009-763252-i22.gif"/></inline-formula>, and the constants <inline-formula> <graphic file="1029-242X-2009-763252-i23.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2009-763252-i24.gif"/></inline-formula> cannot be improved for the corresponding inequalities. Here <inline-formula> <graphic file="1029-242X-2009-763252-i25.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2009-763252-i26.gif"/></inline-formula> denote the arithmetic, geometric, and harmonic means of <inline-formula> <graphic file="1029-242X-2009-763252-i27.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2009-763252-i28.gif"/></inline-formula>, respectively.</p>http://www.journalofinequalitiesandapplications.com/content/2009/763252 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Xia Wei-Feng Chu Yu-Ming |
spellingShingle |
Xia Wei-Feng Chu Yu-Ming Inequalities for Generalized Logarithmic Means Journal of Inequalities and Applications |
author_facet |
Xia Wei-Feng Chu Yu-Ming |
author_sort |
Xia Wei-Feng |
title |
Inequalities for Generalized Logarithmic Means |
title_short |
Inequalities for Generalized Logarithmic Means |
title_full |
Inequalities for Generalized Logarithmic Means |
title_fullStr |
Inequalities for Generalized Logarithmic Means |
title_full_unstemmed |
Inequalities for Generalized Logarithmic Means |
title_sort |
inequalities for generalized logarithmic means |
publisher |
SpringerOpen |
series |
Journal of Inequalities and Applications |
issn |
1025-5834 1029-242X |
publishDate |
2009-01-01 |
description |
<p/> <p>For <inline-formula> <graphic file="1029-242X-2009-763252-i1.gif"/></inline-formula>, the generalized logarithmic mean <inline-formula> <graphic file="1029-242X-2009-763252-i2.gif"/></inline-formula> of two positive numbers <inline-formula> <graphic file="1029-242X-2009-763252-i3.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2009-763252-i4.gif"/></inline-formula> is defined as <inline-formula> <graphic file="1029-242X-2009-763252-i5.gif"/></inline-formula>, for <inline-formula> <graphic file="1029-242X-2009-763252-i6.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2009-763252-i7.gif"/></inline-formula><inline-formula> <graphic file="1029-242X-2009-763252-i8.gif"/></inline-formula> , for <inline-formula> <graphic file="1029-242X-2009-763252-i9.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2009-763252-i10.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2009-763252-i11.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2009-763252-i12.gif"/></inline-formula><inline-formula> <graphic file="1029-242X-2009-763252-i13.gif"/></inline-formula>, for <inline-formula> <graphic file="1029-242X-2009-763252-i14.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2009-763252-i15.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2009-763252-i16.gif"/></inline-formula><inline-formula> <graphic file="1029-242X-2009-763252-i17.gif"/></inline-formula> , for <inline-formula> <graphic file="1029-242X-2009-763252-i18.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2009-763252-i19.gif"/></inline-formula>. In this paper, we prove that <inline-formula> <graphic file="1029-242X-2009-763252-i20.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2009-763252-i21.gif"/></inline-formula> for all <inline-formula> <graphic file="1029-242X-2009-763252-i22.gif"/></inline-formula>, and the constants <inline-formula> <graphic file="1029-242X-2009-763252-i23.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2009-763252-i24.gif"/></inline-formula> cannot be improved for the corresponding inequalities. Here <inline-formula> <graphic file="1029-242X-2009-763252-i25.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2009-763252-i26.gif"/></inline-formula> denote the arithmetic, geometric, and harmonic means of <inline-formula> <graphic file="1029-242X-2009-763252-i27.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2009-763252-i28.gif"/></inline-formula>, respectively.</p> |
url |
http://www.journalofinequalitiesandapplications.com/content/2009/763252 |
work_keys_str_mv |
AT xiaweifeng inequalitiesforgeneralizedlogarithmicmeans AT chuyuming inequalitiesforgeneralizedlogarithmicmeans |
_version_ |
1725306328788762624 |