scholarly journals Some δ -Tempered Fractional Hermite–Hadamard Inequalities Involving Harmonically Convex Functions and Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Nousheen Akhtar ◽  
Muhammad Uzair Awan
Keyword(s):  
Convex Functions ◽  
Hadamard’S Inequality ◽  
Harmonic Convex ◽  
Harmonically Convex Functions

The main objective of this paper is to obtain some new δ -tempered fractional versions of Hermite–Hadamard’s inequality using the class of harmonic convex functions. In order to show the significance of the main results, we also discuss some interesting applications.

scholarly journals A Study of Uniform Harmonic χ -Convex Functions with respect to Hermite-Hadamard’s Inequality and Its Caputo-Fabrizio Fractional Analogue and Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Miguel Vivas-Cortez ◽  
Muhammad Uzair Awan ◽  
Muhammad Zakria Javed ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Convex Functions ◽  
Integral Identity ◽  
Fractional Integral ◽  
Fractional Integrals ◽  
Auxiliary Result ◽  
Real Numbers ◽  
Positive Real ◽  
Hadamard’S Inequality ◽  
Special Cases ◽  
Harmonic Convex

In this paper, we introduce the notion of uniform harmonic χ -convex functions. We show that this class relates several other unrelated classes of uniform harmonic convex functions. We derive a new version of Hermite-Hadamard’s inequality and its fractional analogue. We also derive a new fractional integral identity using Caputo-Fabrizio fractional integrals. Utilizing this integral identity as an auxiliary result, we obtain new fractional Dragomir-Agarwal type of inequalities involving differentiable uniform harmonic χ -convex functions. We discuss numerous new special cases which show that our results are quite unifying. Finally, in order to show the significance of the main results, we discuss some applications to means of positive real numbers.

scholarly journals Fractional Hermite–Hadamard type inequalities for interval-valued functions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Xuelong Liu ◽  
Gouju Ye ◽  
Dafang Zhao ◽  
Wei Liu
Keyword(s):  
Convex Functions ◽  
Harmonically Convex Functions ◽  
Interval Valued

Abstract We introduce the concept of interval harmonically convex functions. By using two different classes of convexity, we get some further refinements for interval fractional Hermite–Hadamard type inequalities. Also, some examples are presented.

scholarly journals New Modifications of Integral Inequalities via ℘-Convexity Pertaining to Fractional Calculus and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1753
Author(s):  
Saima Rashid ◽  
Aasma Khalid ◽  
Omar Bazighifan ◽  
Georgia Irina Oros
Keyword(s):  
Integral Operator ◽  
Convex Functions ◽  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Type Inequality ◽  
Fractional Integral Operator ◽  
Integral Inequalities ◽  
Convex Mappings ◽  
Special Cases ◽  
Harmonically Convex Functions

Integral inequalities for ℘-convex functions are established by using a generalised fractional integral operator based on Raina’s function. Hermite–Hadamard type inequality is presented for ℘-convex functions via generalised fractional integral operator. A novel parameterized auxiliary identity involving generalised fractional integral is proposed for differentiable mappings. By using auxiliary identity, we derive several Ostrowski type inequalities for functions whose absolute values are ℘-convex mappings. It is presented that the obtained outcomes exhibit classical convex and harmonically convex functions which have been verified using Riemann–Liouville fractional integral. Several generalisations and special cases are carried out to verify the robustness and efficiency of the suggested scheme in matrices and Fox–Wright generalised hypergeometric functions.

scholarly journals Some new Simpson-type inequalities for generalized p-convex function on fractal sets with applications

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Thabet Abdeljawad ◽  
Saima Rashid ◽  
Zakia Hammouch ◽  
İmdat İşcan ◽  
Yu-Ming Chu
Keyword(s):  
Convex Functions ◽  
Fractional Derivatives ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Auxiliary Result ◽  
Fractal Sets ◽  
Different Types ◽  
Novel Applications ◽  
Class Of Functions ◽  
Harmonically Convex Functions

Abstract The present article addresses the concept of p-convex functions on fractal sets. We are able to prove a novel auxiliary result. In the application aspect, the fidelity of the local fractional is used to establish the generalization of Simpson-type inequalities for the class of functions whose local fractional derivatives in absolute values at certain powers are p-convex. The method we present is an alternative in showing the classical variants associated with generalized p-convex functions. Some parts of our results cover the classical convex functions and classical harmonically convex functions. Some novel applications in random variables, cumulative distribution functions and generalized bivariate means are obtained to ensure the correctness of the present results. The present approach is efficient, reliable, and it can be used as an alternative to establishing new solutions for different types of fractals in computer graphics.

scholarly journals New generalized fractional versions of Hadamard and Fejér inequalities for harmonically convex functions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaoli Qiang ◽  
Ghulam Farid ◽  
Muhammad Yussouf ◽  
Khuram Ali Khan ◽  
Atiq Ur Rahman
Keyword(s):  
Convex Functions ◽  
Harmonically Convex Functions

scholarly journals ON QUASI CONVEX FUNCTIONS AND HADAMARD'S INEQUALITY

2008 ◽  
Vol 41 (2) ◽  
Author(s):  
K.-L. Tseng ◽  
G.-S. Yang ◽  
S. S. Dragomir
Keyword(s):  
Convex Functions ◽  
Hadamard’S Inequality

scholarly journals Fractional Ostrowski type inequalities for differentiable harmonically convex functions

2021 ◽  
Vol 7 (3) ◽  
pp. 3939-3958
Author(s):  
Thanin Sitthiwirattham ◽  
◽  
Muhammad Aamir Ali ◽  
Hüseyin Budak ◽  
Sotiris K. Ntouyas ◽  
...  
Keyword(s):  
Convex Functions ◽  
Fractional Integrals ◽  
Real Numbers ◽  
Special Means ◽  
Special Cases ◽  
Harmonically Convex Functions ◽  
New Inequalities

<abstract><p>In this paper, we prove some new Ostrowski type inequalities for differentiable harmonically convex functions using generalized fractional integrals. Since we are using generalized fractional integrals to establish these inequalities, therefore we obtain some new inequalities of Ostrowski type for Riemann-Liouville fractional integrals and $ k $-Riemann-Liouville fractional integrals in special cases. Finally, we give some applications to special means of real numbers for newly established inequalities.</p></abstract>

General Harmonic Convex Functions and Integral Inequalities

2016 ◽  
pp. 443-472 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Muhammad Uzair Awan ◽  
Sabah Iftikhar
Keyword(s):  
Convex Functions ◽  
Integral Inequalities ◽  
Harmonic Convex

scholarly journals New Riemann–Liouville fractional Hermite–Hadamard type inequalities for harmonically convex functions

2019 ◽  
Vol 9 (2) ◽  
pp. 431-441
Author(s):  
Zeynep Şanlı ◽  
Mehmet Kunt ◽  
Tuncay Köroğlu
Keyword(s):  
Convex Functions ◽  
Fractional Integrals ◽  
Differentiable Functions ◽  
Left And Right ◽  
Harmonically Convex Functions

Abstract In this paper, we proved two new Riemann–Liouville fractional Hermite–Hadamard type inequalities for harmonically convex functions using the left and right fractional integrals independently. Also, we have two new Riemann–Liouville fractional trapezoidal type identities for differentiable functions. Using these identities, we obtained some new trapezoidal type inequalities for harmonically convex functions. Our results generalize the results given by İşcan (Hacet J Math Stat 46(6):935–942, 2014).

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