Jensen's inequality for filtration consistent nonlinear expectation without domination condition

In this paper, the general filtration consistent nonlinear expectation defined on the integrable variable space is considered, based on the results in [F. Coquet, Y. Hu, J. Memin, S. Peng, Filtration consistent nonlinear expectations and related g-expectation, Probab. Theory Related Fields 123 (2002) 1-27]. Under a natural continuous assumption for the nonlinear expectation, which weakens the domination assumption in [F. Coquet, Y. Hu, J. Memin, S. Peng, Filtration consistent nonlinear expectations and related g-expectation, Probab. Theory Related Fields 123 (2002) 1-27], the author obtains the necessary and sufficient conditions under which Jensen's inequality for filtration consistent nonlinear expectation holds in general, respectively on scalar function and bivariate function. These two results generalize the known results on Jensen's inequality for g-expectation in [Z. Chen, R. Kulperger, L. Jiang, Jensen's inequality for g-expectation: Part 1, C. R. Acad. Sci. Paris Ser. I 337 (11) (2003) 725-730; Z. Chen, R. Kulperger, L. Jiang, Jensen's inequality for g-expectation: Part 2, C. R. Acad. Sci. Paris Ser. I 337 (12) (2003) 797-800; L. Jiang, On Jensen's inequality of bivariate function for g-expectation, J. Shandong Univ. 38 (5) (2003) 13-22 (in Chinese); L. Jiang, Z. Chen, On Jensen's inequality for g-expectation, Chinese Ann. Math. Ser. B 25 (3) (2004) 401-412; L. Jiang, Jensen's inequality for backward stochastic differential equation, Chinese Ann. Math. Ser. B 27 (5) (2006) 553-564; S. Fan, Jensen's inequality for g-expectation on convex (concave) function, Chinese Ann. Math. Ser. A 27 (5) (2006) 635-644 (in Chinese)].     

On Bohr's Inequality

Bohr's inequality says that if is a bounded analytic function on the closed unit disc, then for 0 leq r 1/3 and that1/3 is sharp. In this paper we give an operator-theoretic proofof Bohr's inequality that is based on von Neumann's inequality.Since our proof is operator-theoretic, our methods extend toseveral complex variables and to non-commutative situations. We obtain Bohr type inequalities for the algebras of boundedanalytic functions and the multiplier algebras of reproducingkernel Hilbert spaces on various higher-dimensional domains,for the non-commutative disc algebra A_n, and for the reduced(respectively full) group C*-algebra of the free group on ngenerators. We also include an application to Banach algebras. We provethat every Banach algebra has an equivalent norm in which itsatisfies a non-unital version of von Neumann's inequality. 2000 Mathematical Subject Classification: 47A20, 47A56.     

关于Banach空间中的向量值的Bochner积分型的广义Jensen不等式

李泽民(1990)将R^n中的极值问题的Kuhu-Tucker条件推广到了线性拓扑空间中的向量极值问题．本文作者从另一角度，以锥为工具，把在概率论与鞅论等学科有着广泛应用的R中的著名的Jensen不等式推广到序Banach空间，导出向量值的Bochner积分型的广义Jensen不等式，从而推广了前人的工作．     

关于几何凸函数的Jensen不等式的加细

利用几何凸函数的Jensen不等式建立一个由{1,2,…,n}到(0,+∞)上的一个映射,研究了这个映射的单调性,获得一个该Jensen不等式的加细,并得到几何凸函数的一些新的不等式.     

ON JENSEN’S INEQUALITY FOR g-EXPECTATION

Briand et al.gave a counterexample showing that given g, Jensen‘s inequality for g-expectation usually does not hold in general. This paper proves that Jensen‘s inequality for g-expectation holds in general if and only if the generator g(t,z) is super-homogeneous in z. In particular, g is not necessarily convex in z.     

ON JENSEN’S INEQUALITY FOR g-EXPECTATION

Briand et al. gave a counterexample showing that given g, Jensen's inequalityfor g-expectation usually does not hold in general. This paper proves that Jensen'sinequality for g-expectation holds in general if and only if the generator g(t,z) issuper-homogeneous in z. In particular, g is not necessarily convex in z.     

Jensen's Inequality for Backward Stochastic Differential Equations

Under the Lipschitz assumption and square integrable assumption on g, the author proves that Jensen's inequality holds for backward stochastic differential equations with generator g if and only if g is independent of y, g(t, 0) = 0 and g is super homogeneous with respect to z. This result generalizes the known results on Jensen's inequality for g-expectation in [4, 7-9].     

Jensen's inequality relative to matrix-valued measures

A Bochner-integral formulation of Jensen's inequality is presented for Hermitian matrix-valued functions and measures.     

涉及Jensen函数的不等式

引入了Jensen函数及Jensen平均的概念,借助于数学分析和代数工具给出了Jensen函数的分解公式,利用这个公式给出了推广和加强Jensen不等式的一种崭新的思路,作为应用,给出了Jensen不等式成立的一个有趣的充分条件．旨在为数学研究提供一些有用的解析不等式．     

More Bounds on Norms of Generalized Characters with Applications to p-Local Bounds and Blocks

A new inequality concerning generalized characters of p-groupsis obtained and applications to bounding the number of irreduciblecharacters in blocks of finite groups are given. 2000 MathematicsSubject Classification 20C20.     

A relation between Jensen's inequality and a geometrical problem

A relation is established between Jensen's inequality and a problem suggested by H. Jung concerning the size of the smallest sphere containing a set of given diameter. An estimate is obtained of the size of this sphere in terms of the absolute value of the convexity of the space.Translated from Matematicheskie Zametki, Vol. 3, No. 3, pp. 327–338, March, 1968.     

Jensen's operator inequality for functions of two variables

The operator convex functions of two variables are characterized in terms of a non-commutative generalization of Jensen's inequality.

     

Trace Inequalities of Sobolev Type in the Upper Triangle Case

We give a new non-capacitary characterization of positive Borelmeasures µ on Rⁿ such that the potential space I*L^p isimbedded in L^q(dµ) for $1qp+, that is, the trace inequality holds, for Riesz potentials I = (- )². A weak-type trace inequality is also characterized. A non-isotropic version on the unit sphere Sⁿ is studied,as well as the holomorphic case for Hardy–Sobolev spaces in the ball. 1991 MathematicsSubject Classification: primary 31C15, 42B20; secondary 32A35.     

Generalisation of the Bogomolov-Miyaoka-Yau Inequality to Singular Surfaces

The paper considers pairs (X, B) where X is a normal projectivesurface over C, and B is a Q-divisor whose coefficients are1 or 1–1/m for some natural number m. A log canonicalsingularity on such a pair is a quotient by a finite or infinitegroup, so if (X, B) has log canonical singularities, the orbifoldEuler number e_orb(X, B) can be defined. The main result is aBogomolov-Miyaoka-Yau-type inequality which implies that if(X, B) has log canonical singularities and (X, K_X + B) 0 then(K_X+B)² 3e_orb(X, B). The actual inequality proved is somewhatstronger and it also implies all the previously published versionsof the Bogomolov-Miyaoka-Yau inequality. The proof involvesthe Log Minimal Model Program, Q-sheaves when K_X+B is nef, anda study of the changes in the two sides of the inequality undera contraction. The paper also contains a further generalisationwhere the coefficients of B can be arbitrary rational numbersin [0, 1], a different condition is imposed on the singularitiesand K_X+B is required to be nef. Some applications of the inequalitiesare also given, for example, estimating the number of singularitiesor certain kinds of configurations of curves on surfaces. 1991Mathematics Subject Classification: 14J17, 14J60, 14C17.     

Superquadratic functions in several variables

The concept of superquadratic functions in several variables, as a generalization of the same concept in one variable is introduced. Analogous results to results obtained for convex functions in one and several variables are presented. These include refinements of Jensen's inequality and its counterpart, and of Slater-Pe?ari?'s inequality.     

Some Remarks on the Elliptic Harnack Inequality

Three short results are given concerning the elliptic Harnackinequality, in the context of random walks on graphs. The firstis that the elliptic Harnack inequality implies polynomial growthof the number of points in balls, and the second that the ellipticHarnack inequality is equivalent to an annulus-type Harnackinequality for Green's functions. The third result uses thelamplighter group to give a counter-example concerning the relationof coupling with the elliptic Harnack inequality. 2000 MathematicsSubject Classification 31B05 (primary), 60J35, 31C25 (secondary).     

Jensen's inequality for multivariate medians

Given a probability measure μ on Borel sigma-field of Rd, and a function f:Rd?R, the main issue of this work is to establish inequalities of the type f(m)?M, where m is a median (or a deepest point in the sense explained in the paper) of μ and M is a median (or an appropriate quantile) of the measure μf=μ○f−1. For the most popular choice of halfspace depth, we prove that the Jensen's inequality holds for the class of quasi-convex and lower semi-continuous functions f. To accomplish the task, we give a sequence of results regarding the “type D depth functions” according to classification in [Y. Zuo, R. Serfling, General notions of statistical depth function, Ann. Statist. 28 (2000) 461-482], and prove several structural properties of medians, deepest points and depth functions. We introduce a notion of a median with respect to a partial order in Rd and we present a version of Jensen's inequality for such medians. Replacing means in classical Jensen's inequality with medians gives rise to applications in the framework of Pitman's estimation.     

How and Why to Solve the Operator Equation AX-XB = Y

The entities A, B, X, Y in the title are operators, by whichwe mean either linear transformations on a finite-dimensionalvector space (matrices) or bounded (= continuous) linear transformationson a Banach space. (All scalars will be complex numbers.) Thedefinitions and statements below are valid in both the finite-dimensionaland the infinite-dimensional cases, unless the contrary is stated.1991 Mathematics Subject Classification 15A24, 47A10, 47A62,47B47, 47B49, 65F15, 65F30.     

On an operator inequality

Jensen's operator inequality characterizes operator convex functions of two variables (F. Hansen, Proc. Amer. Math. Soc. 125 (1997) 2093–2102). We give a simplified proof of this theorem formulated for matrices.     

The Trace in Banach Jordan Pairs

The algebraic trace form (as defined by O. Loos) of an element(x, y) of a (complex) Banach Jordan pair V, where x or y isin the socle, is equal to the sum of the products of all spectralvalues and their multiplicity. The trace form is calculatedfor two examples, the Banach Jordan pair of bounded linear operatorsbetween two Banach spaces, and the Banach Jordan pair of a quadraticform. Using analytic multifunctions, it is also shown that thecomplement of the socle of a Banach Jordan pair V is eitherdense or empty. In the last case, V has finite capacity. 1991Mathematics Subject Classification 17C65, 46H70.