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 [1]. 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 [4], 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[21]. 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[7]. 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 [14], 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 [1], Kullback and Leibler [2], R´ enyi [3], Havrda and Charvat [4], Kapur [5], Sharma and Mittal [6], Burbea and Rao [7], Rao [8], Lin [9], Csisz´ ar [10], Ali and Silvey [12], Vajda [13], Shioya and Da-te [40] and others (see for example [5] 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 [22], Kullback and Leibler [31], R´enyi [42], Havrda and Charvat [20], Kapur [25], Sharma and Mittal [44], Burbea and Rao [5], Rao [41], Lin [34], Csisz´ar [10], Ali and Silvey [1], Vajda [52], Shioya and Da-te [45] and others (see for example [25] 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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