scholarly journals On Levinson's inequality involving averages of 3-convex functions

2020 ◽  
pp. 45-45
Author(s):  
Ana Vukelic
Keyword(s):  
Convex Functions ◽  
Divided Difference ◽  
Higher Order ◽  
Cauchy Type ◽  
Linear Functionals ◽  
New Family ◽  
Exponentially Convex Functions ◽  
Levinson’S Inequality ◽  
Favard’S Inequality ◽  
New Inequalities

By using the Levinson inequality we give the extension for 3-convex functions of Wulbert's result from Favard's Inequality on Average Values of Convex Functions, Math. Comput. Model. 37 (2003), 1383{1391. Also, we obtain inequalities with divided differences, and as a consequence, the convexity of higher order for function defined by divided difference is proved. Further, we construct a new family of exponentially convex functions and Cauchy-type means by looking at linear functionals associated with these new inequalities.

Refinements of the majorization-type inequalities via green and fink identities and related results

2018 ◽  
Vol 68 (4) ◽  
pp. 773-788 ◽  
Author(s):  
Sadia Khalid ◽  
Josip Pečarić ◽  
Ana Vukelić
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Convex Functions ◽  
Type Inequality ◽  
Cauchy Type ◽  
Linear Functionals ◽  
Ostrowski’S Inequality ◽  
New Family ◽  
Exponentially Convex Functions ◽  
The Difference

Abstract In this work, the Green’s function of order two is used together with Fink’s approach in Ostrowski’s inequality to represent the difference between the sides of the Sherman’s inequality. Čebyšev, Grüss and Ostrowski-type inequalities are used to obtain several bounds of the presented Sherman-type inequality. Further, we construct a new family of exponentially convex functions and Cauchy-type means by looking to the linear functionals associated with the obtained inequalities.

scholarly journals Some Estimates for Generalized Riemann-Liouville Fractional Integrals of Exponentially Convex Functions and Their Applications

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 807 ◽  
Author(s):  
Saima Rashid ◽  
Thabet Abdeljawad ◽  
Fahd Jarad ◽  
Muhammad Aslam Noor
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Fractional Integral ◽  
Fractional Integrals ◽  
Order Derivative ◽  
Fundamental Identity ◽  
First Order ◽  
Special Means ◽  
Exponentially Convex Functions ◽  
New Inequalities

In the present paper, we investigate some Hermite-Hadamard ( HH ) inequalities related to generalized Riemann-Liouville fractional integral ( GRLFI ) via exponentially convex functions. We also show the fundamental identity for GRLFI having the first order derivative of a given exponentially convex function. Monotonicity and exponentially convexity of functions are used with some traditional and forthright inequalities. In the application part, we give examples and new inequalities for the special means.

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 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 Generalized Steffensen Type Inequalities Involving Convex Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Josip Pečarić ◽  
Ksenija Smoljak Kalamir
Keyword(s):  
Convex Functions ◽  
Linear Functionals ◽  
Left Hand ◽  
Exponentially Convex Functions ◽  
The Difference ◽  
The Right ◽  
Class Of Functions ◽  
Right And Left Hand

In this paper generalized Steffensen type inequalities related to the class of functions that are “convex at pointc” are derived and as a consequence inequalities involving the class of convex functions are obtained. Moreover, linear functionals from the difference of the right- and left-hand side of the obtained generalized inequalities are constructed and new families of exponentially convex functions related to constructed functionals are derived.

scholarly journals Estimation of divergence measures on time scales via Taylor’s polynomial and Green’s function with applications in q-calculus

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Iqrar Ansari ◽  
Khuram Ali Khan ◽  
Ammara Nosheen ◽  
Ðilda Pečarić ◽  
Josip Pečarić
Keyword(s):  
Green’S Function ◽  
Time Scales ◽  
Green's Function ◽  
Convex Functions ◽  
Higher Order ◽  
Quantum Calculus ◽  
Divergence Measures ◽  
New Inequalities

AbstractTaylor’s polynomial and Green’s function are used to obtain new generalizations of an inequality for higher order convex functions containing Csiszár divergence on time scales. Various new inequalities for some divergence measures in quantum calculus and h-discrete calculus are also established.

scholarly journals Stolarsky type means related to an extension of H ¨older-type inequality

2014 ◽  
Vol 23 (1) ◽  
pp. 107-114
Author(s):  
KSENIJA SMOLJAK ◽  
Keyword(s):  
Convex Functions ◽  
Type Inequality ◽  
Monotonicity Property ◽  
Linear Functionals ◽  
Exponential Convexity ◽  
Exponentially Convex Functions

In this paper linear functionals related to an extension of Holder-type inequality are defined and their n−exponential convexity is proved. Furthermore, new Stolarsky type means, using families of exponentially convex functions, are defined and their monotonicity property is proved.

scholarly journals m-Convex Functions of Higher Order

2020 ◽  
Vol 34 (2) ◽  
pp. 241-255
Author(s):  
Teodoro Lara ◽  
Nelson Merentes ◽  
Edgar Rosales
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Divided Difference ◽  
Higher Order

AbstractIn this research we introduce the concept of m-convex function of higher order by means of the so called m-divided difference; elementary properties of this type of functions are exhibited and some examples are provided.

scholarly journals Inequalities Involving Conformable Approach for Exponentially Convex Functions and Their Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-17 ◽  
Author(s):  
Jun-Feng Li ◽  
Saima Rashid ◽  
Jia-Bao Liu ◽  
Ahmet Ocak Akdemir ◽  
Farhat Safdar
Keyword(s):  
Convex Functions ◽  
Exponentially Convex Functions ◽  
New Inequalities

In the article, we present several Hermite–Hadamard-type inequalities for the exponentially convex functions via conformable integrals. As applications, we give new inequalities for certain bivariate means.

Higher-Order and Mixed Discrete Derivatives such as a Novel Graph- Theoretical Invariant for Generating New Molecular Descriptors

2019 ◽  
Vol 19 (11) ◽  
pp. 944-956 ◽  
Author(s):  
Oscar Martínez-Santiago ◽  
Yovani Marrero-Ponce ◽  
Ricardo Vivas-Reyes ◽  
Mauricio E.O. Ugarriza ◽  
Elízabeth Hurtado-Rodríguez ◽  
...  
Keyword(s):  
Molecular Descriptors ◽  
3D Qsar ◽  
Higher Order ◽  
Molecular Structures ◽  
Qsar Modeling ◽  
Statistical Parameters ◽  
Molecular Graphs ◽  
New Family ◽  
Mixed Derivatives ◽  
Discrete Derivative

Background: Recently, some authors have defined new molecular descriptors (MDs) based on the use of the Graph Discrete Derivative, known as Graph Derivative Indices (GDI). This new approach about discrete derivatives over various elements from a graph takes as outset the formation of subgraphs. Previously, these definitions were extended into the chemical context (N-tuples) and interpreted in structural/physicalchemical terms as well as applied into the description of several endpoints, with good results. Objective: A generalization of GDIs using the definitions of Higher Order and Mixed Derivative for molecular graphs is proposed as a generalization of the previous works, allowing the generation of a new family of MDs. Methods: An extension of the previously defined GDIs is presented, and for this purpose, the concept of Higher Order Derivatives and Mixed Derivatives is introduced. These novel approaches to obtaining MDs based on the concepts of discrete derivatives (finite difference) of the molecular graphs use the elements of the hypermatrices conceived from 12 different ways (12 events) of fragmenting the molecular structures. The result of applying the higher order and mixed GDIs over any molecular structure allows finding Local Vertex Invariants (LOVIs) for atom-pairs, for atoms-pairs-pairs and so on. All new families of GDIs are implemented in a computational software denominated DIVATI (acronym for Discrete DeriVAtive Type Indices), a module of KeysFinder Framework in TOMOCOMD-CARDD system. Results: QSAR modeling of the biological activity (Log 1/K) of 31 steroids reveals that the GDIs obtained using the higher order and mixed GDIs approaches yield slightly higher performance compared to previously reported approaches based on the duplex, triplex and quadruplex matrix. In fact, the statistical parameters for models obtained with the higher-order and mixed GDI method are superior to those reported in the literature by using other 0-3D QSAR methods. Conclusion: It can be suggested that the higher-order and mixed GDIs, appear as a promissory tool in QSAR/QSPRs, similarity/dissimilarity analysis and virtual screening studies.

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