A Refinement of Jensen's Inequality for a Class of Increasing and Concave Functions

• Main Author: Xia Ye
• Format: Article
• Language: English
• Published: SpringerOpen 2008-01-01
• Series: Journal of Inequalities and Applications
• Online Access: http://www.journalofinequalitiesandapplications.com/content/2008/717614

<p>Abstract</p> <p>Suppose that <inline-formula> <graphic file="1029-242X-2008-717614-i1.gif"/></inline-formula> is strictly increasing, strictly concave, and twice continuously differentiable on a nonempty interval <inline-formula> <graphic file="1029-242X-2008-717614-i2.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2008-717614-i3.gif"/></inline-formula> is strictly convex on <inline-formula> <graphic file="1029-242X-2008-717614-i4.gif"/></inline-formula>. Suppose that <inline-formula> <graphic file="1029-242X-2008-717614-i5.gif"/></inline-formula>, where <inline-formula> <graphic file="1029-242X-2008-717614-i6.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2008-717614-i7.gif"/></inline-formula> for <inline-formula> <graphic file="1029-242X-2008-717614-i8.gif"/></inline-formula>, and suppose that <inline-formula> <graphic file="1029-242X-2008-717614-i9.gif"/></inline-formula>. Let <inline-formula> <graphic file="1029-242X-2008-717614-i10.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2008-717614-i11.gif"/></inline-formula>. We show <inline-formula> <graphic file="1029-242X-2008-717614-i12.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2008-717614-i13.gif"/></inline-formula>, for suitably chosen <inline-formula> <graphic file="1029-242X-2008-717614-i14.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2008-717614-i15.gif"/></inline-formula>. These results can be viewed as a refinement of the Jensen's inequality for the class of functions specified above. Or they can be viewed as a generalization of a refined arithmetic mean-geometric mean inequality introduced by Cartwright and Field in 1978. The strength of the above result is in bringing the variations of the <inline-formula> <graphic file="1029-242X-2008-717614-i16.gif"/></inline-formula>'s into consideration, through <inline-formula> <graphic file="1029-242X-2008-717614-i17.gif"/></inline-formula>.</p>