scholarly journals Extension of Jensen's Inequality for Operators without Operator Convexity

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Jadranka Mićić ◽  
Zlatko Pavić ◽  
Josip Pečarić
Keyword(s):  
Convex Functions ◽  
Jensen’S Inequality ◽  
Jensen's Inequality ◽  
Quasiarithmetic Means ◽  
Adjoint Operators

We give an extension of Jensen's inequality for -tuples of self-adjoint operators, unital -tuples of positive linear mappings, and real-valued continuous convex functions with conditions on the operators' bounds. We also study operator quasiarithmetic means under the same conditions.

On Some Integral Inequalities Related to Hardy's

1977 ◽  
Vol 20 (3) ◽  
pp. 307-312 ◽  
Author(s):  
Christopher Olutunde Imoru
Keyword(s):  
Integral Inequality ◽  
Convex Functions ◽  
Jensen’S Inequality ◽  
Integral Inequalities ◽  
Hardy’S Inequality ◽  
Jensen's Inequality ◽  
Hardy's Inequality ◽  
Special Case

AbstractWe obtain mainly by using Jensen's inequality for convex functions an integral inequality, which contains as a special case Shun's generalization of Hardy's inequality.

scholarly journals INEQUALITIES IN TERMS OF THE GÂTEAUX DERIVATIVES FOR CONVEX FUNCTIONS ON LINEAR SPACES WITH APPLICATIONS

2011 ◽  
Vol 83 (3) ◽  
pp. 500-517 ◽  
Author(s):  
S. S. DRAGOMIR
Keyword(s):  
Convex Functions ◽  
Jensen’S Inequality ◽  
Jensen's Inequality ◽  
Linear Spaces ◽  
Divergence Measures ◽  
Gateaux Derivatives ◽  
Gâteaux Derivatives

AbstractSome inequalities in terms of the Gâteaux derivatives related to Jensen’s inequality for convex functions defined on linear spaces are given. Applications for norms, mean f-deviations and f-divergence measures are provided as well.

scholarly journals Generalized Jensen-Mercer Inequality for Functions with Nondecreasing Increments

2016 ◽  
Vol 2016 ◽  
pp. 1-12
Author(s):  
Asif R. Khan ◽  
Sumayyah Saadi
Keyword(s):  
Convex Functions ◽  
Jensen’S Inequality ◽  
Higher Dimensions ◽  
Jensen's Inequality ◽  
Valuable Addition ◽  
Interesting Variation ◽  
Integral Version ◽  
The Way

In the year 2003, McD Mercer established an interesting variation of Jensen’s inequality and later in 2009 Mercer’s result was generalized to higher dimensions by M. Niezgoda. Recently, Asif et al. has stated an integral version of Niezgoda’s result for convex functions. We further generalize Niezgoda’s integral result for functions with nondecreasing increments and give some refinements with applications. In the way, we generalize an important result, Jensen-Boas inequality, using functions with nondecreasing increments. These results would constitute a valuable addition to Jensen-type inequalities in the literature.

scholarly journals Estimation of the coefficient of variation of the excess of loss by means of Jensen's inequality for convex functions. The connection with the theory of life times

1969 ◽  
Vol 5 (2) ◽  
pp. 239-248
Author(s):  
J. Van Klinken ◽  
C. J. Groenenberg
Keyword(s):  
Coefficient Of Variation ◽  
Convex Functions ◽  
Human Life ◽  
Jensen’S Inequality ◽  
Life Time ◽  
Upper And Lower Bounds ◽  
Jensen's Inequality ◽  
Limit Value ◽  
Total Claim ◽  
The Mean

This note deals with a way to determine upper and lower bounds for the coefficient of variation of the total claim costs in a year in excess of a certain limit value. In many a publication attention has been paid to the question of calculating, in addition to the mean of the excess costs, the variance of these costs. In this connection we only mention here the study of Vajda: Minimum Variance Reinsurance, ASTIN-Bulletin, September 1962. It is evident that in studying such problems of “statistics of large claims” the coefficient of variation, defined as the ratio of the standard deviation and the mean, may be a useful tool. If we exclude special “dangerous” claim distributions and also distributions with a very “irregular” tail, it appears possible to derive bounds for this coefficient of variation and to indicate its asymptotic behaviour. Hereby good use can be made of Jensen's inequality for convex functions. Jensen's inequality has frequently been applied to problems in the field of life insurance mathematics, but as far as the authors of this note know, not to the question mentioned here.Besides, it may have some interest to point out a connection between the question of estimating the excess of total claim costs and the theory of life times. We may translate “the limit value” as “the age already reached” and “the mean of the excess of loss” as “the expectation of life”. The estimation of the coefficient of variation ot the life time now leads to some well-known biometric formulae on life times and at the same time to an interesting observation as regards human life tables.

Cyclic Refinements of the Different Versions of Operator Jensen's Inequality

2016 ◽  
Vol 31 ◽  
pp. 125-133 ◽  
Author(s):  
Laszlo Horvath ◽  
Khuram Khan ◽  
Josip Pecaric
Keyword(s):  
Convex Functions ◽  
Jensen’S Inequality ◽  
Jensen's Inequality ◽  
Norm Inequalities ◽  
Power Means ◽  
Operator Convex ◽  
Operator Convex Functions

Refinements of the operator Jensen's inequality for convex and operator convex functions are given by using cyclic refinements of the discrete Jensen's inequality. Similar refinements are fairly rare in the literature. Some applications of the results to norm inequalities, the Holder McCarthy inequality and generalized weighted power means for operators are presented.

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