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.

scholarly journals Sandwich theorem for reciprocally strongly convex functions

2018 ◽  
Vol 52 (2) ◽  
pp. 171-184
Author(s):  
Mireya Bracamonte ◽  
José Giménez ◽  
Jesús Medina
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Stability Result ◽  
Real Interval ◽  
Strongly Convex Functions ◽  
Strongly Convex Function ◽  
Strongly Convex ◽  
Sandwich Theorem ◽  
Approximate Convexity ◽  
Class Of Functions

We introduce the notion of reciprocally strongly convex functions and we present some examples and properties of them. We also prove that two real functions f and g, defined on a real interval [a, b], satisfy for all x, y ∈ [a, b] and t ∈ [0, 1] iff there exists a reciprocally strongly convex function h : [a, b] → R such that f (x) ≤ h(x) ≤ g(x) for all x ∈ [a, b]. Finally, we obtain an approximate convexity result for reciprocally strongly convex functions; namely we prove a stability result of Hyers-Ulam type for this class of functions.

Counterexamples to Smoothing Convex Functions

1986 ◽  
Vol 29 (3) ◽  
pp. 308-313 ◽  
Author(s):  
Patrick Adrian Neale Smith
Keyword(s):  
Riemannian Manifold ◽  
Convex Function ◽  
Convex Functions ◽  
Strongly Convex Functions ◽  
Strongly Convex Function ◽  
Strongly Convex

AbstractGreene and Wu have shown that any continuous strongly convex function on a Riemannian manifold can be uniformly approximated by infinitely differentiable strongly convex functions. This result is not true if the word “strongly” is omitted; in this paper, we give examples of manifolds on which convex functions cannot be approximated by convex functions (k = 0, 1,2,...).

scholarly journals Around Jensen’s inequality for strongly convex functions

2017 ◽  
Vol 92 (1) ◽  
pp. 25-37 ◽  
Author(s):  
Hamid Reza Moradi ◽  
Mohsen Erfanian Omidvar ◽  
Muhammad Adil Khan ◽  
Kazimierz Nikodem
Keyword(s):  
Convex Functions ◽  
Jensen’S Inequality ◽  
Jensen's Inequality ◽  
Strongly Convex Functions ◽  
Strongly Convex

scholarly journals Ostrowski Type Inequalities Pertaining Strongly Convex Functions Via Conformable Fractional Integrals and Their Applications

2019 ◽  
Author(s):  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Fractional Integrals ◽  
Strongly Convex Functions ◽  
Strongly Convex Function ◽  
Strongly Convex ◽  
Error Estimations

In the article, by applied the concept of strongly convex function and one known identity, we establish several Ostrowski type inequalities involving conformable fractional integrals. As applications, some new error estimations for the midpoint formula are provided as well.

scholarly journals On Generalized Strongly Convex Functions and Unified Integral Operators

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Timing Yu ◽  
Ghulam Farid ◽  
Kahkashan Mahreen ◽  
Chahn Yong Jung ◽  
Soo Hak Shim
Keyword(s):  
Integral Operator ◽  
Convex Function ◽  
Convex Functions ◽  
Fractional Integral ◽  
Integral Operators ◽  
Integral Inequalities ◽  
Operator Inequalities ◽  
Strongly Convex Functions ◽  
Strongly Convex ◽  
Left And Right

In this paper, we define a strongly exponentially α , h − m -convex function that generates several kinds of strongly convex and convex functions. The left and right unified integral operators of these functions satisfy some integral inequalities which are directly related to many unified and fractional integral inequalities. From the results of this paper, one can obtain various fractional integral operator inequalities that already exist in the literature.

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 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 Inequalities for generalized Riemann–Liouville fractional integrals of generalized strongly convex functions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ghulam Farid ◽  
Young Chel Kwun ◽  
Hafsa Yasmeen ◽  
Abdullah Akkurt ◽  
Shin Min Kang
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Strongly Convex Functions ◽  
Strongly Convex ◽  
Constant Multiplier ◽  
Error Estimations ◽  
Integral Identities

AbstractSome new integral inequalities for strongly $(\alpha ,h-m)$ ( α , h − m ) -convex functions via generalized Riemann–Liouville fractional integrals are established. The outcomes of this paper provide refinements of some fractional integral inequalities for strongly convex, strongly m-convex, strongly $(\alpha ,m)$ ( α , m ) -convex, and strongly $(h-m)$ ( h − m ) -convex functions. Also, the refinements of error estimations of these inequalities are obtained by using two fractional integral identities. Moreover, using a parameter substitution and a constant multiplier, k-fractional versions of established inequalities are also given.

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.

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