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New identities and inequalities are given for weighted majorization theorem for n-convex functions by using extension of the Montgomery identity and Green function. Various bounds for the reminders in new generalizations of weighted majorization formulae are provided using Čebyšev type inequalities. Mean value theorems are also discussed for functional related to new results. [ABSTRACT FROM AUTHOR]

In the paper, the authors investigate properties, including the monotonicity, logarithmic concavity, concavity, and inequalities, of a sequence arising from geometric probability for pairs of hyperplanes intersecting with a convex body. [ABSTRACT FROM AUTHOR]

In this paper, a new general identity for differentiable mappings via k-fractional integrals is derived. By using the concept of $(h,m)$ -convexity, $(\alpha,m)$ -convexity and the obtained equation, some new trapezium-like integral inequalities are established. The results presented provide extensions of those given in earlier works. [ABSTRACT FROM AUTHOR]

We provide the monotonicity and convexity properties and sharp bounds for the generalized elliptic integrals $\mathscr{K}_{a}(r)$ and $\mathscr {E}_{a}(r)$ depending on a parameter $a\in(0,1)$ , which contains an earlier result in the particular case $a=1/2$ . [ABSTRACT FROM AUTHOR]

In this paper, we formulate new Abel-Gontscharoff type identities involving new Green functions for the 'two-point right focal' problem. We use Fink's identity and a new Abel-Gontscharoff-type Green's function for a 'two-point right focal' to generalize the refinement of Jensen's inequality given in (Horváth and Pečarić in Math. Inequal. Appl. 14: 777-791, 2011) from convex function to higher order convex function. Also we formulate the monotonicity of the linear functional obtained from these identities using the recent theory of inequalities for n-convex function at a point. Further we give the bounds for the identities related to the generalization of the refinement of Jensen's inequality using inequalities for the Cebyšev functional. Some results relating to the Grüss and Ostrowski-type inequalities are constructed. [ABSTRACT FROM AUTHOR]

The inequality of Popoviciu, which was improved by Vasić and Stanković (Math. Balk. 6:281-288, 1976), is generalized by using new identities involving new Green's functions. New generalizations of an improved Popoviciu inequality are obtained by using generalized Montgomery identity along with new Green's functions. As an application, we formulate the monotonicity of linear functionals constructed from the generalized identities, utilizing the recent theory of inequalities for n-convex functions at a point. New upper bounds of Grüss and Ostrowski type are also computed. [ABSTRACT FROM AUTHOR]

By making use of the kernel given in [11], some new Ostrowski-type inequalities on time scales for double integrals are established. The results obtained here can be used in the estimation of the error. [ABSTRACT FROM AUTHOR]

In this paper, we obtain two refinements of the ordering relations among Heinz means with different parameters via the Taylor series of some hyperbolic functions and by the way, we derive new generalizations of Heinz operator inequalities. Moreover, we establish a matrix version of Heinz inequality for the Hilbert-Schmidt norm. Finally, we introduce a weighted multivariate geometric mean and show that the weighted multivariate operator geometric mean possess several attractive properties and means inequalities. [ABSTRACT FROM AUTHOR]

In the paper, the authors introduce a new concept "extended ((s1, m1)-(s2, m2))-convex function on co-ordinates" and establish some new inequalities of the Hermite-Hadamard type for extended ((s1, m1)-(s2, m2))-convex functions of 2 variables on co-ordinates. [ABSTRACT FROM AUTHOR]

The authors first introduce the concepts of generalized $(\alpha,m)$ -preinvex function, generalized quasi m-preinvex function and explicitly $(\alpha, m)$ -preinvex function, and then provide some interesting properties for the newly introduced functions. The more important point is that we give a necessary and sufficient condition respecting the relationship between the generalized $(\alpha, m)$ -preinvex function and the generalized quasi m-preinvex function. Second, a new Riemann-Liouville fractional integral identity involving twice differentiable function on m-invex is found. By using this identity, we establish the right-sided new Hermite-Hadamard-type inequalities via Riemann-Liouville fractional integrals for generalized $(\alpha,m)$ -preinvex mappings. These inequalities can be viewed as generalization of several previously known results. [ABSTRACT FROM AUTHOR]

In this paper, we define some integral transforms and obtain suitable bounds for them in order to introduce a main class of integral inequalities including Ostrowski and Ostrowski-Grüss inequalities and various kinds of new integral inequalities. In this sense, we also introduce a three point quadrature formula and obtain its error bounds. [ABSTRACT FROM PUBLISHER]

In the paper, the authors establish three kinds of inequalities for the trigamma function in terms of the exponential function to powers of the digamma function. These newly established inequalities extend some known results. The method in the paper utilizes some facts from the asymptotic theory and is a natural way to solve problems for approximating some quantities for large values of the variable. [ABSTRACT FROM AUTHOR]

The present paper focuses on three types of fractional convolutions. First consideration is to extend the famous Young's inequality in the context of fractional convolution. Then for these convolutions, weighted iteratedinequalities have been investigated. Finally, the notion of fractional convolutions between functions and distributions and also between distributions has been defined and various corresponding properties have been studied. [ABSTRACT FROM AUTHOR]

In this article we establish a sharp two-sided inequality for bounding the Wallis ratio. Some best constants for the estimation of the Wallis ratio are obtained. An asymptotic formula for the Wallis ratio is also presented. [ABSTRACT FROM AUTHOR]

We give a discrete Grüss type inequality on fractional calculus. [ABSTRACT FROM AUTHOR]

We obtain some new Grüss type inequalities for the general corrected three-point quadrature formulae of Euler type. As special cases, we derive some new bounds for the corrected Euler Simpson formula, the corrected dual Euler Simpson formula and the corrected Euler Maclaurin formula. Also, applications for the corrected Euler Bullen-Simpson formula are considered. [ABSTRACT FROM AUTHOR]

We present several new Young-type inequalities for positive real numbers and we apply our results to obtain the matrix analogues. Among others, for real numbers,and, withand, we prove the inequalitieswhereandare, respectively, the (weighted) arithmetic and geometric means of the positive real numbersandwith. In addition, we show that both bounds are sharp. An example of a matrix analogue for the caseis the double-inequalityfor positive definite matrices. Our results extend some fresh inequalities established by Kittaneh, Manasrah, Hirzallah and Feng. Estimates for the quotientand its matrix analogues given by Furuichi and Minculete are also improved. [ABSTRACT FROM AUTHOR]

New Čebyšev type inequalities for fractional Riemann–Liouville integrals are established. Our results are based on some Montgomery identities. [ABSTRACT FROM AUTHOR]

A certain class of weighted approximations, which extends the results of Masjed-Jamei [6] is introduced for integrable functions and some of upper bounds are obtained for the absolute value of the errors of such approximations in two L1[a, b] and L∞[a, b] spaces. As the main motivation for introducing the aforesaid class, it is shown that many new inequalities can be generated from the given error bounds. Some illustrative examples are presented in this sense. Moreover, by using the obtained error bounds, a nonstandard type of three-point weighted quadrature rules is introduced and its error bounds are computed. [ABSTRACT FROM AUTHOR]

We generalize several monotonicity and convexity properties as well as sharp inequalities for the complete elliptic integrals to the complete p-elliptic integrals. [ABSTRACT FROM AUTHOR]

By using a Lidstone interpolation, Green’s function and Montogomery’s identity, we prove a new generalization of Steffensen’s inequality. Some related inequalities providing generalizations of certain results given in (J. Math. Inequal. 9(2):481-487, 2015) have also been obtained. Moreover, from these inequalities, we formulate linear functionals and describe their properties. [ABSTRACT FROM AUTHOR]

In this paper, we first prove an integral majorization theorem related to integral inequalities for functions defined on rectangles. We then apply the result to establish some new integral inequalities for functions defined on rectangles. The results obtained are generalizations of weighted Favard’s inequality, which also provide a generalization of the results given by Maligranda et al. (J. Math. Anal. Appl. 190:248-262, 1995) in an earlier paper. [ABSTRACT FROM AUTHOR]

In this paper we give extensions of Sherman’s inequality considering the class of convex functions of higher order. As particular cases, we get an extended weighted majorization inequality as well as Jensen’s inequality which have direct connection to information theory. We use the obtained results to derive new estimates for Shannon’s and Rényi’s entropy, information energy, and some well-known measures between probability distributions. Using the Zipf-Mandelbrot law, we introduce new functionals to derive some related results. [ABSTRACT FROM AUTHOR]