Recent Developments of Discrete Inequalities for Convex Functions Defined on Linear Spaces with Applications

2018 ◽  
pp. 117-172
Author(s):  
Silvestru Sever Dragomir
Keyword(s):  
Convex Functions ◽  
Linear Spaces ◽  
Discrete Inequalities ◽  
Recent Developments

scholarly journals Refined estimates and generalization of some recent results with applications

2021 ◽  
Vol 6 (10) ◽  
pp. 10728-10741
Author(s):  
Aqeel Ahmad Mughal ◽  
◽  
Deeba Afzal ◽  
Thabet Abdeljawad ◽  
Aiman Mukheimer ◽  
...  
Keyword(s):  
Convex Functions ◽  
Type Result ◽  
Linear Spaces ◽  
New Class ◽  
Discrete Inequalities ◽  
Harmonic Convex ◽  
Class Of Functions

<abstract><p>In this paper, we firstly give improvement of Hermite-Hadamard type and Fej$ \acute{e} $r type inequalities. Next, we extend Hermite-Hadamard type and Fej$ \acute{e} $r types inequalities to a new class of functions. Further, we give bounds for newly defined class of functions and finally presents refined estimates of some already proved results. Furthermore, we obtain some new discrete inequalities for univariate harmonic convex functions on linear spaces related to a variant most recently presented by Baloch <italic>et al.</italic> of Jensen-type result that was established by S. S. Dragomir.</p></abstract>

scholarly journals The differentiability of convex functions on topological linear spaces

1990 ◽  
Vol 42 (2) ◽  
pp. 201-213 ◽  
Author(s):  
Bernice Sharp
Keyword(s):  
Convex Functions ◽  
Inductive Limit ◽  
Asplund Space ◽  
Fréchet Spaces ◽  
Linear Spaces ◽  
Asplund Spaces ◽  
Continuous Convex Function ◽  
Mazur’S Theorem ◽  
Topological Linear ◽  
Differentiability Properties

In this paper topological linear spaces are categorised according to the differentiability properties of their continuous convex functions. Mazur's Theorem for Banach spaces is generalised: all separable Baire topological linear spaces are weak Asplund. A class of spaces is given for which Gateaux and Fréchet differentiability of a continuous convex function coincide, which with Mazur's theorem, implies that all Montel Fréchet spaces are Asplund spaces. The effect of weakening the topology of a given space is studied in terms of the space's classification. Any topological linear space with its weak topology is an Asplund space; at the opposite end of the topological spectrum, an example is given of the inductive limit of Asplund spaces which is not even a Gateaux differentiability space.

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 Bounds for the normalised Jensen functional

2006 ◽  
Vol 74 (3) ◽  
pp. 471-478 ◽  
Author(s):  
Sever S. Dragomir
Keyword(s):  
Convex Functions ◽  
Jensen’S Inequality ◽  
Normed Spaces ◽  
Jensen's Inequality ◽  
Linear Spaces ◽  
Leibler Divergence ◽  
Jensen Functional ◽  
New Inequalities

New inequalities for the general case of convex functions defined on linear spaces which improve the famous Jensen's inequality are established. Particular instances in the case of normed spaces and for complex and real n-tuples are given. Refinements of Shannon's inequality and the positivity of Kullback-Leibler divergence are obtained.

The FM and BCQ Qualifications for Inequality Systems of Convex Functions in Normed Linear Spaces

2021 ◽  
Vol 31 (2) ◽  
pp. 1410-1432
Author(s):  
Chong Li ◽  
Kung Fu Ng ◽  
Jen-Chih Yao ◽  
Xiaopeng Zhao
Keyword(s):  
Convex Functions ◽  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Inequality Systems

A Note on Best Simultaneous Approximation in Normed Linear Spaces

1976 ◽  
Vol 19 (3) ◽  
pp. 359-360 ◽  
Author(s):  
Arne Brøndsted
Keyword(s):  
Linear Space ◽  
Convex Subset ◽  
Convex Functions ◽  
Existence And Uniqueness ◽  
Simultaneous Approximation ◽  
Normed Linear Space ◽  
Present Note ◽  
Linear Spaces ◽  
Best Simultaneous Approximation ◽  
Image Position

The purpose of the present note is to point out that the results of D. S. Goel, A. S. B. Holland, C. Nasim and B. N. Sahney [1] on best simultaneous approximation are easy consequences of simple facts about convex functions. Given a normed linear space X, a convex subset K of X, and points x1, x2 in X, [1] discusses existence and uniqueness of K* ∈ K such that

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