The idea of proofs of Theorem . and Theorem . are essentially the same as that of Theorem A. The point is that a variant of the methods of Wu in  works for the case of multiple sequences. The process will be done in Section and Section after the preparing lemmas in Section . In Section , as an application of Theorem ., we reﬁne a reversed Hölder inequality diﬀerent from Theorem .. We refer to [, ] for the general theory of inequalities.
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The Jensen inequality for convex functions plays a crucial role in the Theory of Inequalities due to the fact that other inequalities such as that arithmetic mean- geometric mean inequality, Hölder and Minkowski inequalities, Ky Fan’s inequality etc. can be obtained as particular cases of it.
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With the development of the theory of inequalities for a two-dimensional case, more attention has also been paid to weakly singular integral inequalities in two variables and their applications to the partial diﬀerential equation with singular kernel. Upon the re- sults in  and , Cheung and Ma  investigated some new weakly singular integral inequalities of Wendroﬀ type
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The Jensen inequality for convex functions plays a crucial role in the Theory of Inequalities due to the fact that other inequalities such as that arithmetic mean- geometric mean inequality, H¨ older and Minkowski inequalities, Ky Fan’s inequality etc. can be obtained as particular cases of it.
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The Jensen inequality for convex functions plays a crucial role in the Theory of Inequalities due to the fact that other inequalities such as the generalised triangle inequality, the arithmetic mean-geometric mean inequality, Hölder and Minkowski inequalities, Ky Fan’s inequality etc. can be obtained as particular cases of it.
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Abstract. Sharp bounds for the deviation of a real-valued function f defined on a compact interval [a, b] to the chord generated by its end points (a, f (a)) and (b, f (b)) under various assumptions for f and f 0 including absolute conti- nuity, convexity, bounded variation, monotonicity etc., are given. Some appli- cations for weighted means and f-divergence measures in Information Theory are also provided.
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Quantum mechanics cannot be described by local hidden variable theories. In quantum theory, the tests of local realism are based on Bell-type inequalities. Original Bell inequality d i d n o t h a v e a n y c a p a b i l i t i e s t o b e s t u d i e d e mp i r i c a l l y i n t h e l a b o r a t o r i e s . Since then, many attempts have been made to obtain Bell-type inequalities which are violated by a higher factor so that it would be experimentally easy to test the non-locality feature of quantum theory. A s t h e n o n - l o c a l i t y f e a t u r e o f q u a n t u m t h e o r y i s i n t e n s i v e l y u s e d i n q u a n t u m i n f o r ma t i o n , B e l l t yp e i n e q u a l i t i e s h a v e r e c e i v e d mo r e a t t e n t i o n i n r e c e n t ye a r s [ 2 ] .
We can get Jensen’s inequality (1.7) directly from (1.5) by setting l = 1 and b = (1). The concept of majorization has a large number of appearances in many diﬀerent ﬁelds of applications, particular in many branches of mathematics. A complete and superb refer- ence on the subject is the monograph , and many results from the theory of majorization are directly or indirectly inspired by it. In this paper we give extensions of Sherman’s in- equality by considering the class of convex functions of higher order. As a particular case, we get an extension of weighted majorization inequality and Jensen’s inequality which can be used to derive some new estimates for some entropies and measures between probabil- ity distributions. Also, we use the Zipf–Mandelbrot law to illustrate the obtained results.
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This paper is the ﬁrst part of a work devoted to the study of generalized variational inequalities on non-convex sets. It describes the constrained inequalities umbrella frame- work for variational and quasi-variational inequalities. The main existence results on general systems of constrained inequalities (Theorems , below) are derived from new topological generalizations of the ﬁxed point theorem of Kakutani without convexity (Theorems and ). The domains considered are spaces modeled on locally ﬁnite poly- hedra having non-trivial Euler-Poincaré characteristic which are not necessarily compact. Rather, compactness is imposed on the maps. Solvability of generalized variational in- equalities expressed as co-equilibria problems for non-self non-convex set-valued maps deﬁned on Lipschitzian retracts is established in the last section (Theorem and Corol- lary ). The paper also illustrates how the general results apply to particular situations in the theories of variational inequalities, complementarity, and optimal control.
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In the last decade, many authors see, e.g., 6 and the references therein have extended the above result for diﬀerent classes of functions defined on a compact interval, including func- tions of bounded variation, monotonic functions, convex functions, n-time diﬀerentiable func- tions whose derivatives are absolutely continuous or satisfy diﬀerent convexity properties, and so forth, and they pointed out sharp inequalities for the absolute value of the diﬀerence
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assessment tool developed in Wales. This 12-question tool enables rapid assessment of health policy, pro- grammes or services for their current or future impact on health inequalities. The HEAT tool questions are pre- sented in Appendix 1. During the workshops the ques- tions in the tool were applied to a range of health issues, demonstrating the use of the tool in multiple contexts. The HEAT tool includes an Intervention Framework to Improve Health and Reduce Inequalities outlined in Figure 1. This Intervention Framework describes a comprehen- sive approach at four levels: structural, intermediary path- ways, health and disability services, and impact. Approximately 160 people participated in the workshops, including members of the senior management team of the MoH and most of the 21 DHBs. Some senior staff found it difficult to make themselves available for two-day work- shops. As a result other staff members were able to attend. Results
We study quasilinear elliptic variational-hemivariational inequalities involving general Leray- Lions operators. The novelty of this paper is to provide existence and comparison results whereby only a local growth condition on Clarke’s generalized gradient is required. Based on these results, in the second part the theory is extended to discontinuous variational-hemivariational inequalities. Copyright q 2009 S. Carl and P. Winkert. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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The integral inequalities play a fundamental role in the theory of diﬀerential equations. Signiﬁcant development in this area has been achieved for the last two decades. For details, we refer to [12, 13, 16, 22, 18, 19] and the references therein. Moreover, the study of the the fractional q-integral inequalities is also of great importance. We refer the reader to [3, 15] for further information and applications. Now we shall introduce some important results that have motivated our work. We begin by , where Ngo et al. proved that for any positive continuous function f on [0, 1] satisfying
One of the important issues in many applications of Probability Theory is finding an appropriate measure of distance (or difference or discrimination ) between two probability distributions. A number of divergence measures for this purpose have been proposed and extensively studied by Jeffreys , Kullback and Leibler , R´ enyi , Havrda and Charvat , Kapur , Sharma and Mittal , Burbea and Rao , Rao , Lin , Csisz´ ar , Ali and Silvey , Vajda , Shioya and Da-te  and others (see for example  and the references therein).
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One of the important issues in many applications of Probability Theory is finding an appropriate measure of distance (or difference or discrimination) between two probability distributions. A number of divergence measures for this purpose have been proposed and extensively studied by Jeffreys , Kullback and Leibler , R´enyi , Havrda and Charvat , Kapur , Sharma and Mittal , Burbea and Rao , Rao , Lin , Csisz´ar , Ali and Silvey , Vajda , Shioya and Da-te  and others (see for example  and the references therein).
We will prove some inequalities among trigonometric quantities of two and three operators. In particular, we will establish an inequality among joint trigonometric quantities of two operators and trigonometric quantities of each operator. As a corollary, we will find an upper bound and a lower bound for the total joint antieigenvalue of two positive operators in terms of the smallest and largest eigenvalues of these operators.
We establish some new nonlinear Gronwall-Bellman-Ou-Iang type integral inequalities with two variables. These inequalities generalize former results and can be used as handy tools to study the qualitative as well as the quantitative properties of solutions of diﬀerential equations. Example of applying these inequalities to derive the properties of BVPs is also given.
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Proof. The inequalities P > √ LA and L > √ GP are proved in 10. We will see, that further refinements of these inequalities are true. Now, the second inequality of 3.1 follows by the first inequality of 2.3, while the second inequality of 3.2 follows by the first inequality of 2.4. The last inequality is in fact an inequality by Carlson 11. For the inequalities on AG/P, we use 2.3 and 2.8.
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There are several reasons for this ambivalence as to the significance of class for the debates on intersectionality. Class is not a justicable inequality under EU legislation, while US writings have often (though not always) focused on ethnicity and race. In an EU context, the six inequalities that are the subject of legislation have been subject to the most analysis. The EU Treaty of Amsterdam in 1997 and the consequent Directives to implement it name six grounds for legal action on illegal discrimination: gender, ethnic- ity, disability, age, religion/belief and sexual orientation (Council Directives 2000/43/ EC, 2000/78/EC, 2004/113/EC; European Commission, 2009). Class is not a ‘justicable’ inequality in the same way as the other six inequalities and has some important ontologi- cal dissimilarity with them. The attempt to include ‘socio-economic’ grounds in the UK Equality Act in 2010 failed. However, class is an important aspect of the structuring of inequalities, intersecting in complex ways with all inequalities (Hills et al, 2010). It is important in the structuring of the employment laws and institutional machinery of tribu- nals and courts that implement these laws. The implementation of the laws on non-class justicable inequalities takes place in institutions that were originally established to secure justice and good relations for class-based relations between employers and employees. The institutions of tribunals and courts are still primarily shaped by class in the composi- tion of the decision-makers which includes representatives of employers and workers as well as independent legal experts; they are not composed of representatives of men and women, black and white, disabled and able-bodied people. In Britain and elsewhere, issues of discrimination in pay and working conditions are still central to legal interven- tions in inequalities, despite their extension to the supply of goods and services. Class- based oppositional institutions, such as trade unions, have developed complex internal committees and practices to address the intersection of class with other inequalities. Class has continuing effects on the equality architecture.
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Abstract. In this paper we survey some recent results obtained by the au- thor related to Pompeiu’s mean value theorem and inequality. Natural appli- cations to Ostrowski type inequalities that play an important role in Numerical Analysis, Approximation Theory, Probability Theory & Statistics, Information Theory and other fields, are given as well.
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