scholarly journals Characterizations of uniform convexity for differentiable functions

2019 ◽  
Vol 13 (3) ◽  
pp. 721-732
Author(s):  
Osama Alabdali ◽  
Allal Guessab ◽  
Gerhard Schmeisser
Keyword(s):  
Convex Functions ◽  
Hessian Matrix ◽  
Uniform Convexity ◽  
Differentiable Functions

We consider convex functions in d real variables. For applications, for example in optimization, various strengthened forms of convexity have been introduced. Among them, uniform convexity is one of the most general, defined by involving a so-called modulus ?. Inspired by three classical characterizations of ordinary convexity, we aim at characterizations of uniform convexity by conditions in terms of the gradient or the Hessian matrix of the considered function for certain classes of moduli ?.

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 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 Approximate Hessian matrices and second-order optimality conditions for nonlinear programming problems with C1-data

1999 ◽  
Vol 40 (3) ◽  
pp. 403-420 ◽  
Author(s):  
V. Jeyakumar ◽  
X. Wang
Keyword(s):  
Nonlinear Programming ◽  
Optimality Conditions ◽  
Jacobian Matrix ◽  
Hessian Matrix ◽  
Second Order ◽  
Second Order Optimality Conditions ◽  
Differentiable Functions ◽  
Continuous Maps ◽  
Mathematical Programming Problems ◽  
Approximate Hessian

AbstractIn this paper, we present generalizations of the Jacobian matrix and the Hessian matrix to continuous maps and continuously differentiable functions respectively. We then establish second-order optimality conditions for mathematical programming problems with continuously differentiable functions. The results also sharpen the corresponding results for problems involving C1.1-functions.

scholarly journals Some New Simpson’s-Formula-Type Inequalities for Twice-Differentiable Convex Functions via Generalized Fractional Operators

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2249
Author(s):  
Muhammad Aamir Ali ◽  
Hasan Kara ◽  
Jessada Tariboon ◽  
Suphawat Asawasamrit ◽  
Hüseyin Budak ◽  
...  
Keyword(s):  
Convex Functions ◽  
Fractional Operators ◽  
The Past ◽  
Differentiable Functions ◽  
Simpson’S Inequality ◽  
Special Cases ◽  
Wide Range ◽  
Generalized Fractional Integral ◽  
The Subject ◽  
Formula Type

From the past to the present, various works have been dedicated to Simpson’s inequality for differentiable convex functions. Simpson-type inequalities for twice-differentiable functions have been the subject of some research. In this paper, we establish a new generalized fractional integral identity involving twice-differentiable functions, then we use this result to prove some new Simpson’s-formula-type inequalities for twice-differentiable convex functions. Furthermore, we examine a few special cases of newly established inequalities and obtain several new and old Simpson’s-formula-type inequalities. These types of analytic inequalities, as well as the methodologies for solving them, have applications in a wide range of fields where symmetry is crucial.

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 General Raina fractional integral inequalities on coordinates of convex functions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Dumitru Baleanu ◽  
Artion Kashuri ◽  
Pshtiwan Othman Mohammed ◽  
Badreddine Meftah
Keyword(s):  
Mathematical Model ◽  
Fractional Calculus ◽  
Convex Function ◽  
Mathematical Analysis ◽  
Integral Inequality ◽  
Convex Functions ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Differentiable Functions ◽  
Special Cases

AbstractIntegral inequality is an interesting mathematical model due to its wide and significant applications in mathematical analysis and fractional calculus. In this study, authors have established some generalized Raina fractional integral inequalities using an $(l_{1},h_{1})$ ( l 1 , h 1 ) -$(l_{2},h_{2})$ ( l 2 , h 2 ) -convex function on coordinates. Also, we obtain an integral identity for partial differentiable functions. As an effect of this result, two interesting integral inequalities for the $(l_{1},h_{1})$ ( l 1 , h 1 ) -$(l_{2},h_{2})$ ( l 2 , h 2 ) -convex function on coordinates are given. Finally, we can say that our findings recapture some recent results as special cases.

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.

scholarly journals A Note on Hermite–Hadamard–Fejer Type Inequalities for Functions Whose n-th Derivatives Are m-Convex or (α,m)-Convex Functions

Axioms ◽  
2021 ◽  
Vol 11 (1) ◽  
pp. 16
Author(s):  
Sanja Kovač
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Remainder Term ◽  
Quadrature Formulas ◽  
Differentiable Functions ◽  
Special Case

In this paper, we develop some Hermite–Hadamard–Fejér type inequalities for n-times differentiable functions whose absolute values of n-th derivatives are (α,m)-convex function. The results obtained in this paper are extensions and generalizations of the existing ones. As a special case, the generalization of the remainder term of the midpoint and trapezoidal quadrature formulas are obtained.

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