scholarly journals Application of Functionals in Creating Inequalities

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
Zlatko Pavić ◽  
Shanhe Wu ◽  
Vedran Novoselac
Keyword(s):  
Convex Functions ◽  
Functional Form ◽  
Closed Interval ◽  
Jensen’S Inequality ◽  
Linear Functionals ◽  
Jensen's Inequality ◽  
Hadamard Inequality ◽  
Positive Linear Functionals ◽  
Main Inequality

The paper deals with the fundamental inequalities for convex functions in the bounded closed interval. The main inequality includes convex functions and positive linear functionals extending and refining the functional form of Jensen’s inequality. This inequality implies the Jensen, Fejér, and, thus, Hermite-Hadamard inequality, as well as their refinements.

scholarly journals The Applications of Functional Variants of Jensen's Inequality

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Zlatko Pavić
Keyword(s):  
Convex Functions ◽  
Functional Form ◽  
Jensen’S Inequality ◽  
Linear Functionals ◽  
Jensen's Inequality ◽  
Positive Linear Functionals ◽  
Functional Variants ◽  
Several Variables

The paper is inspired by McShane's results on the functional form of Jensen's inequality for convex functions of several variables. The work is focused on applications and generalizations of this important result. At that, the generalizations of Jensen's inequality are obtained using the positive linear functionals.

scholarly journals Functions Like Convex Functions

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Zlatko Pavić
Keyword(s):  
Convex Functions ◽  
Convex Sets ◽  
Jensen’S Inequality ◽  
Convexity Property ◽  
Linear Functionals ◽  
Jensen's Inequality ◽  
Positive Linear Functionals ◽  
Functional Forms ◽  
Common Center ◽  
The Common

The paper deals with convex sets, functions satisfying the global convexity property, and positive linear functionals. Jensen's type inequalities can be obtained by using convex combinations with the common center. Following the idea of the common center, the functional forms of Jensen's inequality are considered in this paper.

scholarly journals Cauchy-type means for positive linear functionals

2011 ◽  
Vol 42 (4) ◽  
pp. 511-530
Author(s):  
M. Anwar ◽  
J. Pecaric ◽  
M. Rodi´c Lipanovi´c
Keyword(s):  
Jensen’S Inequality ◽  
Integral Form ◽  
Mean Value ◽  
Cauchy Type ◽  
Linear Functionals ◽  
Jensen's Inequality ◽  
Discrete Form ◽  
Mean Value Theorems ◽  
Positive Linear Functionals

Some mean-value theorems of the Cauchy type, which are connected with Jensen's inequality, are given in \cite{Mercer2} in discrete form and in \cite{PPSri} in integral form. Here we give the generalization of that result for positive linear functionals. Using that result, new means of Cauchy type for positive linear functionals are given. Monotonicity of these new means is also discussed.

scholarly journals Uniform Treatment of Jensen’s Inequality by Montgomery Identity

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Tahir Rasheed ◽  
Saad Ihsan Butt ◽  
Đilda Pečarić ◽  
Josip Pečarić ◽  
Ahmet Ocak Akdemir
Keyword(s):  
Information Theory ◽  
Integral Inequality ◽  
Convex Functions ◽  
Jensen’S Inequality ◽  
Fractional Integrals ◽  
Jensen's Inequality ◽  
Hadamard Inequality ◽  
Montgomery Identity ◽  
Discrete Inequality ◽  
Uniform Treatment

We generalize Jensen’s integral inequality for real Stieltjes measure by using Montgomery identity under the effect of n − convex functions; also, we give different versions of Jensen’s discrete inequality along with its converses for real weights. As an application, we give generalized variants of Hermite–Hadamard inequality. Montgomery identity has a great importance as many inequalities can be obtained from Montgomery identity in q − calculus and fractional integrals. Also, we give applications in information theory for our obtained results, especially for Zipf and Hybrid Zipf–Mandelbrot entropies.

scholarly journals The Jensen's inequality and functional form of Jensen's inequality for 3-convex functions at a point

2020 ◽  
Vol 22 (02) ◽  
pp. 131-141
Author(s):  
Yu Ming Chu ◽  
Imran Abbas Baloch ◽  
Absar Ul Haq ◽  
Manuel De La Sen
Keyword(s):  
Convex Functions ◽  
Functional Form ◽  
Jensen’S Inequality ◽  
Jensen's Inequality

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.

INEQUALITIES OF JENSEN’S TYPE FOR POSITIVE LINEAR FUNCTIONALS ON HERMITIAN UNITAL BANACH -ALGEBRAS

2020 ◽  
Vol 102 (2) ◽  
pp. 308-318
Author(s):  
S. S. DRAGOMIR
Keyword(s):  
Convex Functions ◽  
Banach Algebras ◽  
General Setting ◽  
Linear Functionals ◽  
Positive Linear Functionals ◽  
Hermitian Unital

We establish inequalities of Jensen’s and Slater’s type in the general setting of a Hermitian unital Banach $\ast$-algebra, analytic convex functions and positive normalised linear functionals.

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