Inequalities for Generalized Logarithmic Means

• Main Authors: Xia Wei-Feng, Chu Yu-Ming
• Format: Article
• Language: English
• Published: SpringerOpen 2009-01-01
• Series: Journal of Inequalities and Applications
• Online Access: http://www.journalofinequalitiesandapplications.com/content/2009/763252

<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>