Related papers: Generalizations of Popoviciu's inequality
In this paper, a general integral identity for convex functions is derived. Then, we establish new some inequalities of the Simpson and the Hermite-Hadamard's type for functions whose absolute values of derivatives are convex. Some…
We bound the variance and other moments of a random vector based on the range of its realizations, thus generalizing inequalities of Popoviciu (1935) and Bhatia and Davis (2000) concerning measures on the line to several dimensions. This is…
Fourier series in orthogonal polynomials with respect to a measure $\nu$ on $[-1,1]$ are studied when $\nu$ is a linear combination of a generalized Jacobi weight and finitely many Dirac deltas in $[-1,1]$. We prove some weighted norm…
In this article we give some improvements and generalizations of the famous Jensen's and Jensen-Mercer inequalities for twice differentiable functions, where convexity property of the target function is not assumed in advance. They…
In this article, we prove that convex functions and log-convex functions obey certain general refinements that lead to several refinements and reverses of well known inequalities for matrices, including Young's inequality, Heinz inequality,…
In this paper we discuss convexity, its average principle, an extrinsic average variational method in the Calculus of Variations, an average method in Partial Differential Equations, a link of convexity to $p$-subharmonicity, subsolutions…
In this paper we present new versions of the classical Brunn-Minkowski inequality for different classes of measures and sets. We show that the inequality \[ \mu(\lambda A + (1-\lambda)B)^{1/n} \geq \lambda \mu(A)^{1/n} +…
We introduce a new operation between nonnegative integrable functions on $\mathbb{R} ^n$, that we call geometric combination; it is obtained via a mass transportation approach, playing with inverse distribution functions. The main feature…
We use different approaches to study a generalization of a result of Levin and Ste\v{c}kin concerning an inequality analogous to Hardy's inequality. Our results lead naturally to the study of weighted remainder form of Hardy-type…
The upper bound inequality for variance of weighted sum of correlated random variables is derived according to Cauchy-Schwarz's inequality, while the weights are non-negative with sum of 1. We also give a novel proof with positive…
In this paper, the authors establish a new type integral inequalities for differentiable s-convex functions in the second sense. By the well-known H\"older inequality and power mean inequality, they obtain some integral inequalities related…
In this preprint we consider generalizations of discrete and integral Cauchy--Bunyakovskii inequalities by the method of mean values with some applications. Mostly the material is compiled as a short survey but some results are proved. Main…
The parametric geometry of numbers has allowed to visualize the simultaneous approximation properties of a collection of real numbers through the combined graph of the related successive minima functions. Several inequalities among…
An inequality providing some bounds for the integral mean via Pompeiu's mean value theorem and applications for quadrature rules and special means are given.
In this note, we present two general classes of integral inequalities motivated by their applications to infinite dimensional systems. The inequalities possess general structures in terms of weight functions and lower quadratic bounds. Many…
We extend the definitions of $\nabla-$convex and completely monotonic functions for two variables. Some general identities of Popoviciu type for sum $\sum \sum p_{ij} f(y_i, z_j)$ and integrals $\int P(y)f(y) dy$, $\int \int P(y,z) f(y,z)…
We give necessary and sufficient conditions for Borel measures to satisfy the inequality introduced by Komisarski, Rajba (2018). This inequality is a generalization of the convex order inequality for binomial distributions, which was proved…
It is established that general s-convex functions are a new class of generalized convex functions. In a similar vein, a new class of general s-convex sets is introduced, which are generalizations of s-convex sets. Additionally, certain…
In this paper, an integral identity for twice differentiable functions is generalized. Then, by using convexity of |f''| or q-th power of |f''| and with the aid of power mean and Holder's inequalities we achieved some new results. We also…
The notion of $n$th order convexity in the sense of Hopf and Popoviciu is defined via the nonnegativity of the $(n+1)$st order divided differences of a given real-valued function. In view of the well-known recursive formula for divided…