Optimal Inequalities for Generalized Logarithmic, Arithmetic, and Geometric Means

• Main Authors: Chu Yu-Ming, Long Bo-Yong
• Format: Article
• Language: English
• Published: SpringerOpen 2010-01-01
• Series: Journal of Inequalities and Applications
• Online Access: http://www.journalofinequalitiesandapplications.com/content/2010/806825

<p/> <p>For <inline-formula> <graphic file="1029-242X-2010-806825-i1.gif"/></inline-formula>, the generalized logarithmic mean <inline-formula> <graphic file="1029-242X-2010-806825-i2.gif"/></inline-formula>, arithmetic mean <inline-formula> <graphic file="1029-242X-2010-806825-i3.gif"/></inline-formula>, and geometric mean <inline-formula> <graphic file="1029-242X-2010-806825-i4.gif"/></inline-formula> of two positive numbers <inline-formula> <graphic file="1029-242X-2010-806825-i5.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2010-806825-i6.gif"/></inline-formula> are defined by <inline-formula> <graphic file="1029-242X-2010-806825-i7.gif"/></inline-formula>, for <inline-formula> <graphic file="1029-242X-2010-806825-i8.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2010-806825-i9.gif"/></inline-formula>, for <inline-formula> <graphic file="1029-242X-2010-806825-i10.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2010-806825-i11.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2010-806825-i12.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2010-806825-i13.gif"/></inline-formula>, for <inline-formula> <graphic file="1029-242X-2010-806825-i14.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2010-806825-i15.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2010-806825-i16.gif"/></inline-formula>, for <inline-formula> <graphic file="1029-242X-2010-806825-i17.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2010-806825-i18.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2010-806825-i19.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2010-806825-i20.gif"/></inline-formula>, respectively. In this paper, we find the greatest value <inline-formula> <graphic file="1029-242X-2010-806825-i21.gif"/></inline-formula> (or least value <inline-formula> <graphic file="1029-242X-2010-806825-i22.gif"/></inline-formula>, resp.) such that the inequality <inline-formula> <graphic file="1029-242X-2010-806825-i23.gif"/></inline-formula> (or <inline-formula> <graphic file="1029-242X-2010-806825-i24.gif"/></inline-formula>, resp.) holds for <inline-formula> <graphic file="1029-242X-2010-806825-i25.gif"/></inline-formula>(or <inline-formula> <graphic file="1029-242X-2010-806825-i26.gif"/></inline-formula>, resp.) and all <inline-formula> <graphic file="1029-242X-2010-806825-i27.gif"/></inline-formula> with <inline-formula> <graphic file="1029-242X-2010-806825-i28.gif"/></inline-formula>.</p>