scholarly journals Estimates of Upper Bound for Differentiable Functions Associated with k-Fractional Integrals and Higher Order Strongly s-Convex Functions

2020 ◽  
Vol 2020 ◽  
pp. 1-9 ◽  
Author(s):  
Shanhe Wu ◽  
Muhammad Uzair Awan ◽  
Zakria Javed
Keyword(s):  
Upper Bound ◽  
Convex Functions ◽  
Higher Order ◽  
Fractional Integrals ◽  
Differentiable Functions ◽  
Second Derivatives ◽  
Integral Identities

In this paper, we establish two integral identities associated with differentiable functions and the k-Riemann-Liouville fractional integrals. The results are then used to derive the estimates of upper bound for functions whose first or second derivatives absolute values are higher order strongly s-convex functions.

scholarly journals New Simpson type inequalities for twice differentiable functions via generalized fractional integrals

2021 ◽  
Vol 7 (3) ◽  
pp. 3959-3971
Author(s):  
Xuexiao You ◽  
◽  
Fatih Hezenci ◽  
Hüseyin Budak ◽  
Hasan Kara ◽  
...  
Keyword(s):  
Convex Functions ◽  
Fractional Integrals ◽  
Absolute Value ◽  
Differentiable Functions ◽  
Second Derivatives

<abstract><p>Fractional versions of Simpson inequalities for differentiable convex functions are extensively researched. However, Simpson type inequalities for twice differentiable functions are also investigated slightly. Hence, we establish a new identity for twice differentiable functions. Furthermore, by utilizing generalized fractional integrals, we prove several Simpson type inequalities for functions whose second derivatives in absolute value are convex.</p></abstract>

scholarly journals Simpson type inequalities and applications

2021 ◽  
Author(s):  
Muhammad Uzair Awan ◽  
Muhammad Zakria Javed ◽  
Michael Th. Rassias ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Quadrature Formula ◽  
Convex Functions ◽  
Integral Identity ◽  
Higher Order ◽  
Auxiliary Result ◽  
First Order ◽  
New Class ◽  
Differentiable Functions

AbstractA new generalized integral identity involving first order differentiable functions is obtained. Using this identity as an auxiliary result, we then obtain some new refinements of Simpson type inequalities using a new class called as strongly (s, m)-convex functions of higher order of $$\sigma >0$$ σ > 0 . We also discuss some interesting applications of the obtained results in the theory of means. In last we present applications of the obtained results in obtaining Simpson-like quadrature formula.

scholarly journals Hermite–Hadamard-Type Inequalities for Convex Functions via the Fractional Integrals with Exponential Kernel

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 845 ◽  
Author(s):  
Xia Wu ◽  
JinRong Wang ◽  
and Jialu Zhang
Keyword(s):  
Real Number ◽  
Convex Functions ◽  
Fractional Integral ◽  
Second Order ◽  
Fractional Integrals ◽  
Exponential Kernel ◽  
Special Means ◽  
Integral Identities

In this paper, we establish three fundamental integral identities by the first- and second-order derivatives for a given function via the fractional integrals with exponential kernel. With the help of these new fractional integral identities, we introduce a few interesting Hermite–Hadamard-type inequalities involving left-sided and right-sided fractional integrals with exponential kernels for convex functions. Finally, some applications to special means of real number are presented.

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).

scholarly journals Some new Ostrowski and Gr"Uss type inequalities

2007 ◽  
Vol 38 (2) ◽  
pp. 111-120 ◽  
Author(s):  
B. G. Pachpatte
Keyword(s):  
Differentiable Functions ◽  
Second Derivatives ◽  
Integral Identities ◽  
New Inequalities

In this paper we establish some new inequalities of Ostrowski and Gr"uss type, involving three functions whose second derivatives are bounded. The analysis used in the proofs is fairly elementary and based on the integral identities for twice differentiable functions.

scholarly journals Fractional Versions of Hadamard-Type Inequalities for Strongly Exponentially α , h − m -Convex Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-23
Author(s):  
Shasha Li ◽  
Ghulam Farid ◽  
Atiq Ur Rehman ◽  
Hafsa Yasmeen
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Fractional Integrals ◽  
Strongly Convex ◽  
Error Estimations ◽  
Integral Identities

In this article, we prove some fractional versions of Hadamard-type inequalities for strongly exponentially α , h − m -convex functions via generalized Riemann–Liouville fractional integrals. The outcomes of this paper provide inequalities of strongly convex, strongly m -convex, strongly s -convex, strongly α , m -convex, strongly s , m -convex, strongly h − m -convex, strongly α , h − m -convex, strongly exponentially convex, strongly exponentially m -convex, strongly exponentially s -convex, strongly exponentially s , m -convex, strongly exponentially h − m -convex, and exponentially α , h − m -convex functions. The error estimations are also studied by applying two fractional integral identities.

scholarly journals Generalizations of Hermite–Hadamard like inequalities involving $\chi _{{\kappa }}$-Hilfer fractional integrals

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yu-Ming Chu ◽  
Muhammad Uzair Awan ◽  
Sadia Talib ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Fractional Integrals ◽  
Hadamard’S Inequality ◽  
Integral Identities

Abstract The main objective of this paper is to obtain a new κ-fractional analogue of Hermite–Hadamard’s inequality using the class of s-convex functions and $\chi _{{\kappa }}$ χ κ -Hilfer fractional integrals. In order to obtain other main results of the paper we derive two new fractional integral identities using the definitions of $\chi _{{\kappa }}$ χ κ -Hilfer fractional integrals. For the validity of these identities we also take some particular examples. Using these identities we then obtain some more new variants of Hermite–Hadamard’s inequality using s-convex functions.

scholarly journals Quantum Hermite–Hadamard-type inequalities for functions with convex absolute values of second $q^{b}$-derivatives

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Muhammad Aamir Ali ◽  
Hüseyin Budak ◽  
Mujahid Abbas ◽  
Yu-Ming Chu
Keyword(s):  
Convex Functions ◽  
Differentiable Functions ◽  
Right Hand ◽  
Second Derivatives ◽  
New Inequalities

AbstractIn this paper, we obtain Hermite–Hadamard-type inequalities of convex functions by applying the notion of $q^{b}$ q b -integral. We prove some new inequalities related with right-hand sides of $q^{b}$ q b -Hermite–Hadamard inequalities for differentiable functions with convex absolute values of second derivatives. The results presented in this paper are a unification and generalization of the comparable results in the literature on Hermite–Hadamard inequalities.

scholarly journals New version of fractional Simpson type inequalities for twice differentiable functions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Fatih Hezenci ◽  
Hüseyin Budak ◽  
Hasan Kara
Keyword(s):  
Present Article ◽  
Convex Functions ◽  
Absolute Value ◽  
Differentiable Functions ◽  
Second Derivatives

AbstractSimpson inequalities for differentiable convex functions and their fractional versions have been studied extensively. Simpson type inequalities for twice differentiable functions are also investigated. More precisely, Budak et al. established the first result on fractional Simpson inequality for twice differentiable functions. In the present article, we prove a new identity for twice differentiable functions. In addition to this, we establish several fractional Simpson type inequalities for functions whose second derivatives in absolute value are convex. This paper is a new version of fractional Simpson type inequalities for twice differentiable functions.

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