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 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.

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 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 A New Proof for Lyapunov-Type Inequality on the Fractional Boundary Value Problem

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 29
Author(s):  
Yumei Zou ◽  
Xin Zhang ◽  
Hongyu Li
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Green’S Function ◽  
Boundary Value Problems ◽  
Green's Function ◽  
Boundary Value ◽  
Type Inequality ◽  
Fractional Boundary Value Problem ◽  
Lyapunov Type Inequality ◽  
Fractional Boundary Value Problems

In this article, some new Lyapunov-type inequalities for a class of fractional boundary value problems are established by use of the nonsymmetry property of Green’s function corresponding to appropriate boundary conditions.

scholarly journals Weighted averages of n-convex functions via extension of Montgomery’s identity

2019 ◽  
Vol 9 (2) ◽  
pp. 381-392
Author(s):  
Asif R. Khan ◽  
Josip E. Pečarić ◽  
Marjan Praljak
Keyword(s):  
Convex Function ◽  
Green’S Function ◽  
Green's Function ◽  
Convex Functions ◽  
Weighted Averages

Abstract Using an extension of Montgomery’s identity and the Green’s function, we obtain new identities and related inequalities for weighted averages of n-convex functions, i.e. the sum $$\sum _{i=1}^m \rho _i h(\lambda _i)$$ ∑ i = 1 m ρ i h ( λ i ) and the integral $$\int ^{b}_{a} \rho (\lambda ) h(\gamma (\lambda ))d\lambda $$ ∫ a b ρ ( λ ) h ( γ ( λ ) ) d λ where h is an n-convex function.

Sharp Green’s Function Estimates on Hadamard Manifolds and Adams Inequality

2020 ◽  
Author(s):  
Jerome Bertrand ◽  
Kunnath Sandeep
Keyword(s):  
Green’S Function ◽  
Sectional Curvature ◽  
Green's Function ◽  
Type Inequality ◽  
Hadamard Manifold ◽  
Hadamard Manifolds ◽  
Negative Constant

Abstract In this article, we establish estimates on Riesz-type kernels and prove the Adams-type inequality for $W^{k,p}(M)$ functions, where $(M,g)$ is an $n$-dimensional Hadamard manifold with sectional curvature bounded from below and above by a negative constant and $k$ is an integer satisfying $kp=n$.

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