scholarly journals Refinements of Jensen’s inequality for convex functions on the co-ordinates in a rectangle from the plane

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 803-814 ◽  
Author(s):  
Adil Khan ◽  
T. Ali ◽  
A. Kılıçman ◽  
Q. Din
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Jensen’S Inequality ◽  
Jensen's Inequality

In this paper our aim is to give refinements of Jensen?s type inequalities for the convex function defined on the co-ordinates of the bidimensional interval in the plane.

scholarly journals Generalized Convex Functions on Fractal Sets and Two Related Inequalities

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Huixia Mo ◽  
Xin Sui ◽  
Dongyan Yu
Keyword(s):  
Convex Function ◽  
Real Line ◽  
Convex Functions ◽  
Jensen’S Inequality ◽  
Jensen's Inequality ◽  
Generalized Convex Functions ◽  
Fractal Sets ◽  
Hadamard’S Inequality

We introduce the generalized convex function on fractal setsRα  (0<α≤1)of real line numbers and study the properties of the generalized convex function. Based on these properties, we establish the generalized Jensen’s inequality and generalized Hermite-Hadamard's inequality. Furthermore, some applications are given.

scholarly journals Integral Inequalities Involving Strongly Convex Functions

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Ying-Qing Song ◽  
Muhammad Adil Khan ◽  
Syed Zaheer Ullah ◽  
Yu-Ming Chu
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Jensen’S Inequality ◽  
Integral Inequalities ◽  
Jensen Inequality ◽  
Jensen's Inequality ◽  
Strongly Convex Functions ◽  
Strongly Convex Function ◽  
Strongly Convex

We study the notions of strongly convex function as well as F-strongly convex function. We present here some new integral inequalities of Jensen’s type for these classes of functions. A refinement of companion inequality to Jensen’s inequality established by Matić and Pečarić is shown to be recaptured as a particular instance. Counterpart of the integral Jensen inequality for strongly convex functions is also presented. Furthermore, we present integral Jensen-Steffensen and Slater’s inequality for strongly convex functions.

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.

On measures of non-degeneracy

1992 ◽  
Vol 29 (3) ◽  
pp. 733-739
Author(s):  
K. B. Athreya
Keyword(s):  
Convex Function ◽  
Branching Processes ◽  
Jensen’S Inequality ◽  
Random Variable ◽  
Jensen's Inequality ◽  
Unique Role

If φ is a convex function and X a random variable then (by Jensen's inequality) ψ φ (X) = Eφ (X) – φ (EX) is non-negative and 0 iff either φ is linear in the range of X or X is degenerate. So if φ is not linear then ψ φ (X) is a measure of non-degeneracy of the random variable X. For φ (x) = x2, ψ φ (X) is simply the variance V(X) which is additive in the sense that V(X + Y) = V(X) + V(Y) if X and Y are uncorrelated. In this note it is shown that if φ ″(·) is monotone non-increasing then ψ φ is sub-additive for all (X, Y) such that EX ≧ 0, P(Y ≧ 0) = 1 and E(X | Y) = EX w.p.l, and is additive essentially only if φ is quadratic. Thus, it confirms the unique role of variance as a measure of non-degeneracy. An application to branching processes is also given.

Sign in / Sign up

Export Citation Format

Share Document