Jensen-type Inequalities Research Articles

On γ-quasiconvexity, superquadracity and two-sided reversed Jensen type inequalities

In this paper we deal with γ -quasiconvex functions when −1γ 0, to derive sometwo-sided Jensen type inequalities. We also discuss some Jensen-Steffensen type inequalitiesfor 1-quasiconvex function ...

A new order-preserving average function on a quotient space of strictly monotone functions and its applications

We introduce an order in a quotient space of strictly monotone continuous functions on a real interval and show that a new average function on this quotient space is order-preserving. We also apply this new order-preserving function to derive a finite form of Jensen type inequality with negative weights.MSC:39B62, 26B25, 26A51.

A Simple Proof of the Jensen-Type Inequality of Fink and Jodeit

We discuss the extension of Jensen’s inequality to the framework of quasiconvex functions. Moreover, it is proved that our results work for a class of signed measures larger than the class of probability measures.

Regular operator mappings and multivariate geometric means

We introduce the notion of regular operator mappings of several variables generalising the notion of spectral function. This setting is convenient for studying maps more general than what can be obtained from the functional calculus, and it allows for Jensen type inequalities and multivariate non-commutative perspectives.As a main application of the theory we consider geometric means of k operator variables extending the geometric mean of k commuting operators and the geometric mean of two arbitrary positive definite matrices. We propose different types of updating conditions that seems natural in many applications and prove that each of these conditions, together with a few other natural axioms, uniquely defines the geometric mean for any number of operator variables. The means defined in this way are given by explicit formulas and are computationally tractable.

Comments on Jensen’s Inequalities

The paper gives generalizations of some Jensen type inequalities for convex functions of one variable. The work is based on the methods which use convex combinations in deriving inequalities. The main inequality is applied to the quasi-arithmetic means.

On the Jensen type inequality for generalized Sugeno integral

We prove necessary and sufficient conditions for the validity of Jensen type inequalities for generalized Sugeno integral. Our proofs make no appeal to the continuity of neither the fuzzy measure nor the operators. For several choices of operators, we characterize the classes of functions for which the corresponding inequalities are satisfied.

Reverses of the Jensen-Type Inequalities for Signed Measures

In this paper we derive refinements of the Jensen type inequalities in the case of real Stieltjes measuredλ, not necessarily positive, which are generalizations of Jensen's inequality and its reverses for positive measures. Furthermore, we investigate the exponential and logarithmic convexity of the difference between the left-hand and the right-hand side of these inequalities and give several examples of the families of functions for which the obtained results can be applied. The outcome is a new class of Cauchy-type means.

On some relative convexities

We study two types of relative convexities of convex functions f and g. We say that f is convex relative to g in the sense of Palmer (2002, 2003), if f=h(g), where h is strictly increasing and convex, and denote it by f≻(1)g. Similarly, if f is convex relative to g in the sense studied in Rajba (2011), that is if the function f−g is convex then we denote it by f≻(2)g. The relative convexity relation ≻(2) of a function f with respect to the function g(x)=cx2 means the strong convexity of f. We analyze the relationships between these two types of relative convexities. We characterize them in terms of right derivatives of functions f and g, as well as in terms of distributional derivatives, without any additional assumptions of twice differentiability. We also obtain some probabilistic characterizations. We give a generalization of strong convexity of functions and obtain some Jensen-type inequalities.

Convex geometric means

A class of multivariable weighted geometric means of positive definite matrices admitting Jensen-type inequalities for geodesically convex functions is considered. It is shown that there are infinitely many such geometric means including the weighted inductive, Bini–Meini–Poloni and Karcher means and each of these means provides a geometric mean majorization on the space of positive definite matrices. Some connections between our geometric mean majorizations and classical results of the standard majorization of real numbers are discussed. In particular, we establish the Hardy–Littlewood–Pólya majorization theorem and also Rado’s theorem and Schur’s convexity theorem for the weighted Karcher mean.

Operator-valued Bochner integrable functions and Jensen's inequality

In this paper we obtain a Jensen's type inequality for operator-valued integrable functions, which generalizes some of the previous results in this regard. More precisely, if is a probability measure space and if ν is an operator convex function, then, under suitable conditions, we show that , where is assumed to be Bochner integrable and is a measurable function with .

$\mathbf{n}$-exponential convexity for Jensen-type inequalities

Starting from the results given in [J. Pecaric, I. Peric, M. Rodic Lipanovic: Uniform treatment of Jensen type inequalities, to appear in Math. Rep. (Bucur.)] where the uniform treatment of the Jensen type inequalities and its converses is given, we investigate the exponential convexity of differences of the left-hand and the right-hand side of these inequalities. Using these differences, we produce new exponentially convex functions. Finally, we give several examples of the families of functions for which the obtained results can be applied, and we get some generalized Cauchy means. Results from this paper present the generalization of the results from [M. Anwar, J. Pecaric: Cauchy's means for signed measures, Bull. Malays. Math. Sci. Soc. (2) 34(1) (2011), 31-38.].

Riemannian Distances between Geometric Means

In this paper we study the problem of finding a weighted geometric mean sufficiently close to the weighted Karcher mean of positive definite matrices. This problem arises in various numerical algorithms for computing the weighted Karcher mean and in a physical problem of averaging on the Riemannian manifold of positive definite matrices. We prove that this is possible for any “convex" geometric means satisfying the Jensen-type inequality for geodesically convex functions varying over an open set of weights in the simplex of positive probability vectors. This follows from a useful estimation of the Riemannian distance between the Karcher mean and any convex geometric means. A better upper bound between the Karcher mean and the Riemannian convex combination admitting an explicit formula is presented.

New Jensen-type inequalities

We develop a new framework for the Jensen-type inequalities that allows us to deal with functions that are not necessarily convex and Borel measures that are not necessarily positive.

Jensen type inequalities for twice dierentiable functions

In this paper, we give some Jensen-type inequalities for \(\varphi: I\rightarrow\mathbb{R}, I=[\alpha,\beta ]\subset\mathbb{R}\) where \(\varphi\) is a continuous function on \(I\); twice differentiable on \(I^°=(\alpha,\beta )\) and there exists \(m = \inf _{x\in I^°} \varphi ''(x)\) or \(M = \sup_{x\in I^°}\varphi '' (x)\). Furthermore, if \(\varphi ''\) is bounded on \(I^°\) ; then we give an estimate, from below and from above of Jensen inequalities.

An extension of Jensen's discrete inequality to half convex functions

We extend the right and left convex function theorems to weighted Jensen's type inequalities, and then combine the new theorems in a single one applicable to a half convex function f(u), defined on a real interval and convex for u ≤ s or u ≥ s, where . The obtained results are applied for proving some open relevant inequalities.

JENSEN TYPE INEQUALITIES FOR Q-CLASS FUNCTIONS

AbstractSome inequalities of Jensen type for Q-class functions are proved. More precisely, a refinement of the inequality f((1/P)∑ ni=1pixi)≤P∑ ni=1(f(xi)/pi) is given in which p1,…,pn are positive numbers, P=∑ ni=1pi and f is a Q-class function. The notion of the jointly Q-class function is introduced and some Jensen type inequalities for these functions are proved. Some Ostrowski and Hermite–Hadamard type inequalities related to Q-class functions are presented as well.

Lower Semicontinuity for Integral Functionals in the Space of Functions of Bounded Deformation Via Rigidity and Young Measures

We establish a general weak* lower semicontinuity result in the space $\BD(\Omega)$ of functions of bounded deformation for functionals of the form $$\Fcal(u) := \int_\Omega f \bigl(x, \Ecal u \bigr) \dd x + \int_\Omega f^\infty \Bigl(x, \frac{\di E^s u}{\di \abs{E^s u}} \Bigr) \dd \abs{E^s u} + \int_{\partial \Omega} f^\infty \bigl(x, u|_{\partial \Omega} \odot n_\Omega \bigr) \dd \Hcal^{d-1}$$, $u \in \BD(\Omega)$. The main novelty is that we allow for non-vanishing Cantor-parts in the symmetrized derivative $Eu$. The proof is accomplished via Jensen-type inequalities for generalized Young measures and a construction of good blow-ups, which is based on local rigidity arguments for some differential inclusions involving symmetrized gradients. This strategy allows us to establish the lower semicontinuity result without an Alberti-type theorem in $\BD(\Omega)$, which is not available at present. We also include existence and relaxation results for variational problems in $\BD(\Omega)$, as well as a complete discussion of some differential inclusions for the symmetrized gradient in two dimensions.

Jensen type inequalities for positive bilinear operators

A general Jensen type inequality for positive bilinear operators between uniformly complete vector lattice is proved. Then some new inequalities for linear and bilinear operators and an interpolation result for positive bilinear operators between Calderon–Lozanovskiĭ spaces are deduced. The proof of the main result relies upon homogeneous functional calculus on vector lattices and the Fremlin tensor product of Archimedean vector lattices.

Stabilization of nonlinear networked systems with sensor random packet dropout and time-varying delay

In this paper, a nonlinear stochastic system model is proposed to describe the networked control systems (NCSs) with both random packet dropout and network-induced time-varying delay. Based on this more general nonlinear NCSs model, by choosing appropriate Lyapunov functional and employing new discrete Jensen type inequality, a sufficient condition is derived to establish the quantitative relation of maximum allowable delay upper bound, packet dropout rate and the nonlinear level to the exponential stability of the nonlinear NCSs. Design procedures for output feedback controller are also presented in terms of utilizing cone complementarities linearization algorithm or solving corresponding linear matrix inequalities (LMIs). Illustrative examples are provided to demonstrate the effectiveness of the proposed method.

Properties of isoperimetric, functional and Transport-Entropy inequalities via concentration

Various properties of isoperimetric, functional, Transport-Entropy and concentration inequalities are studied on a Riemannian manifold equipped with a measure, whose generalized Ricci curvature is bounded from below. First, stability of these inequalities with respect to perturbation of the measure is obtained. The extent of the perturbation is measured using several different distances between perturbed and original measure, such as a one-sided L ∞ bound on the ratio between their densities, Wasserstein distances, and Kullback–Leibler divergence. In particular, an extension of the Holley–Stroock perturbation lemma for the log-Sobolev inequality is obtained, and the dependence on the perturbation parameter is improved from linear to logarithmic. Second, the equivalence of Transport-Entropy inequalities with different cost-functions is verified, by obtaining a reverse Jensen type inequality. The main tool used is a previous precise result on the equivalence between concentration and isoperimetric inequalities in the described setting. Of independent interest is a new dimension independent characterization of Transport-Entropy inequalities with respect to the 1-Wasserstein distance, which does not assume any curvature lower bound.