scholarly journals Fractional Integral Inequalities for Differentiable Convex Mappings and Applications to Special Means and a Midpoint Formula

2012 ◽  
Vol 8 (2) ◽  
pp. 21-28 ◽  
Author(s):  
Chun Zhu ◽  
Michal Fečkan ◽  
Jinrong Wang
Keyword(s):  
Error Estimates ◽  
Integral Identity ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Real Numbers ◽  
Liouville Type ◽  
Convex Mappings ◽  
Special Means

Abstract In this paper, Riemann-Liouville type fractional integral identity and inequality for differentiable convex mappings are studied. Some applications to special means of real numbers are given. Finally, error estimates for a midpoint formula are also obtained.

scholarly journals Fractional integral inequalities for generalized-$$\mathbf{m }$$-$$((h_{1}^{p},h_{2}^{q});(\eta _{1},\eta _{2}))$$-convex mappings via an extended generalized Mittag–Leffler function

2019 ◽  
Vol 9 (2) ◽  
pp. 231-243
Author(s):  
George Anastassiou ◽  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Auxiliary Result ◽  
Real Numbers ◽  
Positive Real ◽  
Convex Mappings ◽  
Invex Set ◽  
Special Means ◽  
Special Cases

AbstractThe authors discover a new identity concerning differentiable mappings defined on $$\mathbf{m }$$ m -invex set via general fractional integrals. Using the obtained identity as an auxiliary result, some fractional integral inequalities for generalized-$$\mathbf{m }$$ m -$$((h_{1}^{p},h_{2}^{q});(\eta _{1},\eta _{2}))$$ ( ( h 1 p , h 2 q ) ; ( η 1 , η 2 ) ) -convex mappings by involving an extended generalized Mittag–Leffler function are presented. It is pointed out that some new special cases can be deduced from main results. Also these inequalities have some connections with known integral inequalities. At the end, some applications to special means for different positive real numbers are provided as well.

Some new fractional integral inequalities for generalized relative semi-m-(r; h1, h2)-preinvex mappings via generalized Mittag-Leffler function

2019 ◽  
Vol 26 (1/2) ◽  
pp. 41-55 ◽  
Author(s):  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Auxiliary Result ◽  
Real Numbers ◽  
Positive Real ◽  
Invex Set ◽  
Special Means ◽  
Special Cases ◽  
Differentiable Mappings

The authors discover a new identity concerning differentiable mappings defined on m-invex set via fractional integrals. By using the obtained identity as an auxiliary result, some fractional integral inequalities for generalized relative semi- m-(r;h1,h2)-preinvex mappings by involving generalized Mittag-Leffler function are presented. It is pointed out that some new special cases can be deduced from main results of the paper. Also these inequalities have some connections with known integral inequalities. At the end, some applications to special means for different positive real numbers are provided as well.

scholarly journals New Hermite Hadamard type inequalities for twice differentiable convex mappings via Green function and applications

2016 ◽  
Vol 2 (2) ◽  
pp. 107-118 ◽  
Author(s):  
Samet Erden ◽  
Mehmet Zeki Sarikaya
Keyword(s):  
Green Function ◽  
Integral Inequalities ◽  
Real Numbers ◽  
Absolute Value ◽  
Convex Mappings ◽  
Trapezoidal Formula ◽  
Special Means ◽  
Second Derivatives ◽  
Type Integral

Abstract We derive some Hermite Hamamard type integral inequalities for functions whose second derivatives absolute value are convex. Some eror estimates for the trapezoidal formula are obtained. Finally, some natural applications to special means of real numbers are given

scholarly journals Fractional Integral Inequalities for Some Convex Functions

2021 ◽  
Vol 104 (4) ◽  
pp. 14-27
Author(s):  
B.R. Bayraktar ◽  
◽  
A.Kh. Attaev ◽  
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Arithmetic Mean ◽  
Integral Inequalities ◽  
Power Mean ◽  
Logarithmic Mean ◽  
Real Numbers ◽  
Hadamard Inequality ◽  
Special Means

In this paper, we obtained several new integral inequalities using fractional Riemann-Liouville integrals for convex s-Godunova-Levin functions in the second sense and for quasi-convex functions. The results were gained by applying the double Hermite-Hadamard inequality, the classical Holder inequalities, the power mean, and weighted Holder inequalities. In particular, the application of the results for several special computing facilities is given. Some applications to special means for arbitrary real numbers: arithmetic mean, logarithmic mean, and generalized log-mean, are provided.

scholarly journals On Hermite-Hadamard Type Inequalities for Functions Whose First Derivative Absolute Values are Convex and Concave

2017 ◽  
Vol 58 (1) ◽  
pp. 155-166 ◽  
Author(s):  
Shahid Qaisar ◽  
Sabir Hussain
Keyword(s):  
Error Estimates ◽  
Integral Identity ◽  
Real Numbers ◽  
First Derivative ◽  
General Integral ◽  
Special Means

Abstract In this paper, we derive general integral identity by establishing new Hermite-Hadamard type inequalities for functions whose absolute values of derivatives are convex and concave. Cor- responding error estimates for midpoint formula are also included. Moreover, some applications to special means of real numbers are also provided.

scholarly journals New fractional inequalities of midpoint type via s-convexity and their application

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Ohud Almutairi ◽  
Adem Kılıçman
Keyword(s):  
Error Estimates ◽  
Integral Inequalities ◽  
Real Numbers ◽  
Special Means ◽  
Second Derivatives ◽  
The Absolute ◽  
New Inequalities

Abstract In this study, we introduced new integral inequalities of Hermite–Hadamard type via s-convexity and studied their properties. The absolute form of the first and second derivatives for the new inequalities is considered to be s-convex. As an application, the inequalities were applied to the special means of real numbers. We give the error estimates for the midpoint formula.

scholarly journals Hermite-Hadamard Type Inequalities with Applications

2017 ◽  
Vol 59 (1) ◽  
pp. 57-74 ◽  
Author(s):  
M. Adil Khan ◽  
Tahir Ali ◽  
Tahir Ullah Khan
Keyword(s):  
Error Estimates ◽  
Integral Identity ◽  
Real Numbers ◽  
Trapezoidal Formula ◽  
Divergence Measures ◽  
Special Means

AbstractIn this article first, we give an integral identity and prove some Hermite-Hadamard type inequalities for the function f such that |f″|qis convex or concave for q ≥ 1. Second, by using these results, we present applications to f-divergence measures. At the end, we obtain some bounds for special means of real numbers and new error estimates for the trapezoidal formula.

scholarly journals Some new bounds оf Gauss – Jacobi аnd Hermite – Hadamard type integral inequalities

2021 ◽  
Vol 73 (8) ◽  
pp. 1067-1084
Author(s):  
A. Kashuri ◽  
M. Ramosaçaj ◽  
R. Liko
Keyword(s):  
Error Estimates ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Auxiliary Result ◽  
Real Numbers ◽  
Positive Real ◽  
Special Means ◽  
Special Cases ◽  
Type Integral

UDC 517.5 In this paper, authors discover two interesting identities regarding Gauss–Jacobi and Hermite–Hadamard type integral inequalities. By using the first lemma as an auxiliary result, some new bounds with respect to Gauss–Jacobi type integral inequalities are established. Also, using the second lemma, some new estimates with respect to Hermite–Hadamard type integral inequalities via general fractional integrals are obtained. It is pointed out that some new special cases can be deduced from main results. Some applications to special means for different positive real numbers and new error estimates for the trapezoidal are provided as well. These results give us the generalizations, refinement and significant improvements of the new and previous known results. The ideas and techniques of this paper may stimulate further research.

scholarly journals New fractional identities, associated novel fractional inequalities with applications to means and error estimations for quadrature formulas

2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
Muhammad Uzair Awan ◽  
Artion Kashuri ◽  
Kottakkaran Sooppy Nisar ◽  
Muhammad Zakria Javed ◽  
Sabah Iftikhar ◽  
...  
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Quadrature Formulas ◽  
Real Numbers ◽  
Positive Real ◽  
Special Means ◽  
Error Estimations ◽  
Harmonic Convex ◽  
Integral Identities

AbstractIn this paper, the authors derive some new generalizations of fractional trapezium-like inequalities using the class of harmonic convex functions. Moreover, three new fractional integral identities are given, and on using them as auxiliary results some interesting integral inequalities are found. Finally, in order to show the efficiency of our main results, some applications to special means for different positive real numbers and error estimations for quadrature formulas are obtained.

Some new hermite-hadamard type inequalities and their applications

2019 ◽  
Vol 56 (1) ◽  
pp. 103-142 ◽  
Author(s):  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Convex Function ◽  
Error Estimates ◽  
Fractional Integral ◽  
Real Numbers ◽  
Positive Real ◽  
Divergence Measures ◽  
Integral Forms ◽  
Special Means ◽  
Conformable Fractional Integral ◽  
Integral Identities

AbstractIn this paper first, we prove some new generalizations of Hermite-Hadamard type inequalities for the convex functionfand for (s, m)-convex functionfin the second sense in conformable fractional integral forms. Second, by using five new integral identities, we present some new Riemann-Liouville fractional trapezoid and midpoint type inequalities. Third, using these results, we present applications tof-divergence measures. At the end, some new bounds for special means of different positive real numbers and new error estimates for the trapezoidal and midpoint formula are provided as well. These results give us the generalizations of the earlier results.

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