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

Suppose that f(x) is strictly increasing, strictly concave, and twice continuously differentiable on a nonempty interval I__, and f_(x) is strictly convex on I. Suppose that xk_[a,b]_I, where 0<a<b, and pk_0 for k=1,_,n, and suppose that _k=1npk=1. Let x_=_k=1npkxk, and _2=_k=1npk(xk_x_)2. We...

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Main Author:	Ye Xia
Format:	Article
Language:	English
Published:	SpringerOpen 2008-07-01
Series:	Journal of Inequalities and Applications
Online Access:	http://dx.doi.org/10.1155/2008/717614

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spelling	doaj-ed9a35858cbd4d42927f03c2fc2cc84a2020-11-25T00:15:22ZengSpringerOpenJournal of Inequalities and Applications1025-58341029-242X2008-07-01200810.1155/2008/717614A Refinement of Jensen's Inequality for a Class of Increasing and Concave FunctionsYe XiaSuppose that f(x) is strictly increasing, strictly concave, and twice continuously differentiable on a nonempty interval I__, and f_(x) is strictly convex on I. Suppose that xk_[a,b]_I, where 0<a<b, and pk_0 for k=1,_,n, and suppose that _k=1npk=1. Let x_=_k=1npkxk, and _2=_k=1npk(xk_x_)2. We show _k=1npkf(xk)_f(x___1_2), _k=1npkf(xk)_f(x___2_2), for suitably chosen _1 and _2. 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 xk's into consideration, through _2.http://dx.doi.org/10.1155/2008/717614
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Ye Xia
spellingShingle	Ye Xia A Refinement of Jensen's Inequality for a Class of Increasing and Concave Functions Journal of Inequalities and Applications
author_facet	Ye Xia
author_sort	Ye Xia
title	A Refinement of Jensen's Inequality for a Class of Increasing and Concave Functions
title_short	A Refinement of Jensen's Inequality for a Class of Increasing and Concave Functions
title_full	A Refinement of Jensen's Inequality for a Class of Increasing and Concave Functions
title_fullStr	A Refinement of Jensen's Inequality for a Class of Increasing and Concave Functions
title_full_unstemmed	A Refinement of Jensen's Inequality for a Class of Increasing and Concave Functions
title_sort	refinement of jensen's inequality for a class of increasing and concave functions
publisher	SpringerOpen
series	Journal of Inequalities and Applications
issn	1025-5834 1029-242X
publishDate	2008-07-01
description	Suppose that f(x) is strictly increasing, strictly concave, and twice continuously differentiable on a nonempty interval I__, and f_(x) is strictly convex on I. Suppose that xk_[a,b]_I, where 0<a<b, and pk_0 for k=1,_,n, and suppose that _k=1npk=1. Let x_=_k=1npkxk, and _2=_k=1npk(xk_x_)2. We show _k=1npkf(xk)_f(x___1_2), _k=1npkf(xk)_f(x___2_2), for suitably chosen _1 and _2. 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 xk's into consideration, through _2.
url	http://dx.doi.org/10.1155/2008/717614
work_keys_str_mv	AT yexia arefinementofjensensinequalityforaclassofincreasingandconcavefunctions AT yexia refinementofjensensinequalityforaclassofincreasingandconcavefunctions
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A Refinement of Jensen's Inequality for a Class of Increasing and Concave Functions

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