scholarly journals Schurm-Power Convexity of a Class of Multiplicatively Convex Functions and Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Wen Wang ◽  
Shiguo Yang
Keyword(s):  
Convex Functions ◽  
Symmetric Functions ◽  
Power Mean ◽  
Harmonic Means

We investigate the conditions under which the symmetric functionsFn,k(x,r)=∏1≤i1<i2<⋯<ik≤n ‍f(∑j=1k‍xijr)1/r,  k=1,2,…,n,are Schurm-power convex forx∈R++nandr>0. As a consequence, we prove that these functions are Schur geometrically convex and Schur harmonically convex, which generalizes some known results. By applying the theory of majorization, several inequalities involving thepth power mean and the arithmetic, the geometric, or the harmonic means are presented.

scholarly journals Global Bounds for the Generalized Jensen Functional with Applications

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2105
Author(s):  
Slavko Simić ◽  
Bandar Bin-Mohsin
Keyword(s):  
Power Mean ◽  
Jensen Functional ◽  
Exact Bounds ◽  
Harmonic Means ◽  
Stolarsky Means ◽  
Generalized Arithmetic

In this article we give sharp global bounds for the generalized Jensen functional Jn(g,h;p,x). In particular, exact bounds are determined for the generalized power mean in terms from the class of Stolarsky means. As a consequence, we obtain the best possible global converses of quotients and differences of the generalized arithmetic, geometric and harmonic means.

scholarly journals Some Fejér type integral inequalities for geometrically-arithmetically-convex functions with applications

Filomat ◽  
2018 ◽  
Vol 32 (6) ◽  
pp. 2193-2206 ◽  
Author(s):  
Muhammad Latif ◽  
Sever Dragomir ◽  
Ebrahim Momoniat
Keyword(s):  
Integral Inequality ◽  
Convex Functions ◽  
Symmetric Functions ◽  
Integral Inequalities ◽  
Real Numbers ◽  
Positive Real ◽  
Special Means ◽  
Type Integral

In this paper, the notion of geometrically symmetric functions is introduced. A new identity involving geometrically symmetric functions is established, and by using the obtained identity, the H?lder integral inequality and the notion of geometrically-arithmetically convexity, some new Fej?r type integral inequalities are presented. Applications of our results to special means of positive real numbers are given as well.

scholarly journals Fractional Integral Inequalities for Some Convex Functions

2021 ◽  
Vol 104 (4) ◽  
pp. 14-27
Author(s):  
B.R. Bayraktar ◽  
◽  
A.Kh. Attaev ◽  
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Arithmetic Mean ◽  
Integral Inequalities ◽  
Power Mean ◽  
Logarithmic Mean ◽  
Real Numbers ◽  
Hadamard Inequality ◽  
Special Means

In this paper, we obtained several new integral inequalities using fractional Riemann-Liouville integrals for convex s-Godunova-Levin functions in the second sense and for quasi-convex functions. The results were gained by applying the double Hermite-Hadamard inequality, the classical Holder inequalities, the power mean, and weighted Holder inequalities. In particular, the application of the results for several special computing facilities is given. Some applications to special means for arbitrary real numbers: arithmetic mean, logarithmic mean, and generalized log-mean, are provided.

Exponential trigonometric convex functions and Hermite-Hadamard type inequalities

2021 ◽  
Vol 71 (1) ◽  
pp. 43-56
Author(s):  
Mahir Kadakal ◽  
İmdat İşcan ◽  
Praveen Agarwal ◽  
Mohamed Jleli
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Integral Inequalities ◽  
Power Mean ◽  
Hadamard Inequality ◽  
Absolute Value ◽  
First Derivative ◽  
Algebraic Properties ◽  
Class Of Functions

Abstract In this manuscript, we introduce and study the concept of exponential trigonometric convex functions and their some algebraic properties. We obtain Hermite-Hadamard type inequalities for the newly introduced class of functions. We also obtain some refinements of the Hermite-Hadamard inequality for functions whose first derivative in absolute value, raised to a certain power which is greater than one, respectively at least one, is exponential trigonometric convex function. It has been shown that the result obtained with Hölder-İşcan and improved power-mean integral inequalities give better approximations than that obtained with Hölder and improved power-mean integral inequalities.

scholarly journals Hermite–Hadamard-type inequalities for geometrically r-convex functions in terms of Stolarsky’s mean with applications to means

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Muhammad Amer Latif
Keyword(s):  
Integral Inequality ◽  
Convex Functions ◽  
Power Mean ◽  
Power Mean Inequality

AbstractIn this paper, we obtain new Hermite–Hadamard-type inequalities for r-convex and geometrically convex functions and, additionally, some new Hermite–Hadamard-type inequalities by using the Hölder–İşcan integral inequality and an improved power-mean inequality.

scholarly journals Sharp Power Mean Bounds for the One-Parameter Harmonic Mean

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Yu-Ming Chu ◽  
Li-Min Wu ◽  
Ying-Qing Song
Keyword(s):  
Harmonic Mean ◽  
Power Mean ◽  
Double Inequality ◽  
The One ◽  
Harmonic Means

We present the best possible parametersα=α(r)andβ=β(r)such that the double inequalityMα(a,b)<Hr(a,b)<Mβ(a,b)holds for allr∈(0, 1/2)anda, b>0witha≠b, whereMp(a, b)=[(ap+bp)/2]1/p  (p≠0)andM0(a, b)=abandHr(a, b)=2[ra+(1-r)b][rb+(1-r)a]/(a+b)are the power and one-parameter harmonic means ofaandb, respectively.

scholarly journals A Geometric Mean of Parameterized Arithmetic and Harmonic Means of Convex Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Sangho Kum ◽  
Yongdo Lim
Keyword(s):  
Convex Functions ◽  
Convex Analysis ◽  
Geometric Mean ◽  
Positive Semidefinite ◽  
Dual Operator ◽  
Positive Semidefinite Matrices ◽  
Essential Properties ◽  
Harmonic Means ◽  
Further Development ◽  
Interesting Generalization

The notion of the geometric mean of two positive reals is extended by Ando (1978) to the case of positive semidefinite matricesAandB. Moreover, an interesting generalization of the geometric meanA # BofAandBto convex functions was introduced by Atteia and Raïssouli (2001) with a different viewpoint of convex analysis. The present work aims at providing a further development of the geometric mean of convex functions due to Atteia and Raïssouli (2001). A new algorithmic self-dual operator for convex functions named “the geometric mean of parameterized arithmetic and harmonic means of convex functions” is proposed, and its essential properties are investigated.

scholarly journals Boundedness of Fractional Integral Operators Containing Mittag-Leffler Function via Exponentially s-Convex Functions

2020 ◽  
Vol 2020 ◽  
pp. 1-7 ◽  
Author(s):  
Gang Hong ◽  
G. Farid ◽  
Waqas Nazeer ◽  
S. B. Akbar ◽  
J. Pečarić ◽  
...  
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Symmetric Functions ◽  
Integral Operators ◽  
Arbitrary Point ◽  
Fractional Integrals ◽  
Hadamard Inequality ◽  
Operator Inequalities ◽  
Fractional Integral Operators ◽  
Exponentially Convex Functions

The main objective of this paper is to obtain the fractional integral operator inequalities which provide bounds of the sum of these operators at an arbitrary point. These inequalities are derived for s-exponentially convex functions. Furthermore, a Hadamard inequality is obtained for fractional integrals by using exponentially symmetric functions. The results of this paper contain several such consequences for known fractional integrals and functions which are convex, exponentially convex, and s-convex.

scholarly journals KETAKSAMAAN HERMITE-HADAMARD TERHADAP INTEGRAL RIEMANN-STIELTJES

2012 ◽  
Vol 4 (1) ◽  
pp. 59
Author(s):  
Denny Ivanal Hakim ◽  
Hendra Gunawan
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Closed Interval ◽  
Mean Value ◽  
Stieltjes Integral ◽  
Power Mean ◽  
Real Numbers ◽  
Positive Real ◽  
Hadamard Inequality

The Hermite-Hadamard inequality is an inequality for convex functions that gives an estimate for the integral mean value of a convex function on a closed interval by its value at the middle of interval and the average of its values at the endpoints. The Hermite-Hadamard inequality can be generalized by using the Riemann-Stieltjes integral mean value.  An application of the Hermite-Hadamard inequality with respect to Riemann-Stieltjes integral  for estimating the power mean of   positive real numbers by the aritmethic mean is given at the end of discussion.

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