scholarly journals Levinson-type inequalities via new Green functions and Montgomery identity

2020 ◽  
Vol 18 (1) ◽  
pp. 632-652 ◽  
Author(s):  
Muhammad Adeel ◽  
Khuram Ali Khan ◽  
Ðilda Pečarić ◽  
Josip Pečarić
Keyword(s):  
Convex Functions ◽  
Green Functions ◽  
Montgomery Identity ◽  
Data Points

Abstract In this study, Levinson-type inequalities are generalized by using new Green functions and Montgomery identity for the class of k-convex functions (k ≥ 3). Čebyšev-, Grüss- and Ostrowski-type new bounds are found for the functionals involving data points of two types. Moreover, a new functional is introduced based on {\mathfrak{f}} divergence and then some estimates for new functional are obtained. Some inequalities for Shannon entropies are obtained too.

scholarly journals Conformable Fractional Integrals Versions of Hermite-Hadamard Inequalities and Their Generalizations

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Muhammad Adil Khan ◽  
Yu-Ming Chu ◽  
Artion Kashuri ◽  
Rozana Liko ◽  
Gohar Ali
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Fractional Integrals ◽  
Montgomery Identity

We prove new Hermite-Hadamard inequalities for conformable fractional integrals by using convex function, s-convex, and coordinate convex functions. We prove new Montgomery identity and by using this identity we obtain generalized Hermite-Hadamard type inequalities.

scholarly journals Levinson type inequalities for higher order convex functions via Abel–Gontscharoff interpolation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Muhammad Adeel ◽  
Khuram Ali Khan ◽  
Ðilda Pečarić ◽  
Josip Pečarić
Keyword(s):  
Convex Functions ◽  
Higher Order ◽  
Data Points

Abstract In this paper, Levinson type inequalities are studied for the class of higher order convex functions by using Abel–Gontscharoff interpolation. Cebyšev, Grüss, and Ostrowski-type new bounds are also found for the functionals involving data points of two types.

scholarly journals Majorization, “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law

2018 ◽  
Vol 16 (1) ◽  
pp. 1357-1373 ◽  
Author(s):  
Naveed Latif ◽  
Đilda Pečarić ◽  
Josip Pečarić
Keyword(s):  
Convex Functions ◽  
Green Functions ◽  
The Real ◽  
Utility Distribution ◽  
Definition Of

AbstractIn this paper, we consider the definition of “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law associated with the real utility distribution to give the results for majorizatioQn inequalities by using monotonic sequences. We obtain the equivalent statements between continuous convex functions and Green functions via majorization inequalities, “useful” Csiszár functional and “useful” Zipf-Mandelbrot law. By considering “useful” Csiszár divergence in the integral case, we give the results for integral majorization inequality. Towards the end, some applications are given.

scholarly journals Popoviciu type inequalities for n-convex functions via extension of Montgomery identity

2016 ◽  
Vol 24 (3) ◽  
pp. 161-188
Author(s):  
Asif R. Khan ◽  
Josip Pečarić ◽  
Marjan Praljak
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Montgomery Identity

Abstract Extension of Montgomery's identity is used in derivation of Popoviciu-type inequalities containing sums , where f is an n-convex function. Integral analogues and some related results for n-convex functions at a point are also given, as well as Ostrowski-type bounds for the integral remainders of identities associated with the obtained inequalities.

Generalization of cyclic refinements of Jensen inequality by montgomery identity and Green’s function

2018 ◽  
Vol 11 (04) ◽  
pp. 1850060 ◽  
Author(s):  
Nasir Mehmood ◽  
Saad Ihsan Butt ◽  
Josip Pečarić
Keyword(s):  
Convex Function ◽  
Green’S Function ◽  
Green's Function ◽  
Convex Functions ◽  
Mean Value ◽  
Higher Order ◽  
Jensen Inequality ◽  
Linear Functionals ◽  
Montgomery Identity ◽  
Exponential Convexity

We consider discrete and continuous cyclic refinements of Jensen’s inequality and generalize them from convex function to higher order convex function by means of Lagrange Green’s function and Montgomery identity. We give application of our results by formulating the monotonicity of the linear functionals obtained from generalized identities utilizing the theory of inequalities for [Formula: see text]-convex functions at a point. We compute Grüss and Ostrowski type bounds for generalized identities associated with the obtained inequalities. Finally, we investigate the properties of linear functionals regarding exponential convexity log convexity and mean value theorems.

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 Montgomery identity and Ostrowski-type inequalities via quantum calculus

2021 ◽  
Vol 19 (1) ◽  
pp. 1098-1109
Author(s):  
Thanin Sitthiwirattham ◽  
Muhammad Aamir Ali ◽  
Huseyin Budak ◽  
Mujahid Abbas ◽  
Saowaluck Chasreechai
Keyword(s):  
Convex Functions ◽  
Integral Inequalities ◽  
Montgomery Identity ◽  
Quantum Calculus ◽  
Special Cases ◽  
Quantum Version

Abstract In this paper, we prove a quantum version of Montgomery identity and prove some new Ostrowski-type inequalities for convex functions in the setting of quantum calculus. Moreover, we discuss several special cases of newly established inequalities and obtain different new and existing inequalities in the field of integral inequalities.

scholarly journals Generalized Steffensen’s Inequality by Fink’s Identity

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 329
Author(s):  
Asfand Fahad ◽  
Saad Butt ◽  
Josip Pečarić
Keyword(s):  
Convex Functions ◽  
Type Inequality ◽  
Green Functions ◽  
Linear Functionals ◽  
Hardy Type Inequality ◽  
Steffensen’S Inequality ◽  
Hardy Type ◽  
Exponentially Convex Functions ◽  
Fink’S Identity

By using Fink’s Identity, Green functions, and Montgomery identities we prove some identities related to Steffensen’s inequality. Under the assumptions of n-convexity and n-concavity, we give new generalizations of Steffensen’s inequality and its reverse. Generalizations of some inequalities (and their reverse), which are related to Hardy-type inequality. New bounds of Gr u ¨ ss and Ostrowski-type inequalities have been proved. Moreover, we formulate generalized Steffensen’s-type linear functionals and prove their monotonicity for the generalized class of ( n + 1 ) -convex functions at a point. At the end, we present some applications of our study to the theory of exponentially convex functions. .

scholarly journals Generalizations of Steffensen’s inequality via the extension of Montgomery identity

2018 ◽  
Vol 16 (1) ◽  
pp. 420-428
Author(s):  
Andrea Aglić Aljinović ◽  
Josip Pečarić ◽  
Anamarija Perušić Pribanić
Keyword(s):  
Convex Functions ◽  
Montgomery Identity ◽  
Taylor’S Formula ◽  
Steffensen’S Inequality ◽  
Grüss Type Inequalities ◽  
Nondecreasing Functions ◽  
New Inequalities ◽  
Taylor's Formula

AbstractIn this paper, we obtained new generalizations of Steffensen’s inequality for n-convex functions by using extension of Montgomery identity via Taylor’s formula. Since 1-convex functions are nondecreasing functions, new inequalities generalize Stefensen’s inequality. Related Ostrowski type inequalities are also provided. Bounds for the reminders in new identities are given by using the Chebyshev and Grüss type inequalities.

Sign in / Sign up

Export Citation Format

Share Document