Using 1, inequalities can be established for the length of a. The theorem, of course, covers the original holder inequality 10 as a special case. Defining and measuring inequality odi inequality briefings 1. Income inequality metrics, used to measure income and economic inequality among participants in a particular economy.
A proof as well as historical comments on are given in milovanovic et al. William henry young, english mathematician 18631942 hausdorff young inequality, bounding the coefficient of fourier series. While a wide range of inequality indices of inequality have been developed, some general properties apply to all. You can multiply an inequality by a nonzero number but if the number you multiply by is negative, the inequality is reversed. Here are a couple of basic rules which ill use constantly. Normally to use youngs inequality one chooses a speci c p, and a and b are freeoating quantities. The proof is in the picture steps for writing a proof 1. The isoperimetric inequality 1 is valid also for a twodimensional manifold of bounded curvature, which is a more general type of manifold than a riemannian manifold. Fortunately there is a strong correlation between the complex gini coef. Youngs convolution inequality, bounding the convolution product of two functions.
Very recently, an even simpler proof using entropy inequalities was given by corderoerausquin and ledoux 21. Equality in 1 is attained for a nonregular object a domain isometric to the lateral surface of a right circular cone with complete angle about the vertex. It is very useful in real analysis, including as a tool to prove holders inequality. Solving inequalities mctyinequalities20091 inequalities are mathematical expressions involving the symbols, youngs inequality can be found in 2. According to barthe, we rewrite the inequality slightly. Lesson 6 12 597 getting started math message tell whether each inequality is true or false. Our proof is simpler and gives youngs inequality and its converse altogether. The latter also proved a sharp reverse inequality in the case of exponents less than 1. The sine counterpart of youngs inequality states that for all n. Development strategy and policy analysis unit department. For deriving the inequality, we use the taylor series expansion and the hermite. Young s convolution inequality, bounding the convolution product of two functions. Yet another proof titu andreescu and bogdan enescu give an elegant and memorable proof of the cauchyschwarz inequality among the gems in their mathematical olympiad treasures birkhauser, 2003. Youngs, minkowskis, and holders inequalities penn math.
Before proving youngs inequality, we require a certain fact about the exponential function. In particular, where can i find the proof of the discrete version i. Although both of these capture the whole distribution of a given indicator, inequality is independent of the mean of the distribution or at least this is a desirable property of an inequality measure, as is discussed. International inequality, economic differences between countries.
Sharp inequalities in harmonic analysis summer school, kopp august 30th september 4th, 2015 organizers. Nominally, the proof is inductive, but what i like so much about it is that the induction step comes as close to being. Youngs inequality, which is a version of the cauchy inequality that lets the power of 2 be replaced by the power of p for any. More explicitly, j z t fx ygydyj z t jfx yjjgyjdy kfk l2tkgk l2t taking the supremum of the inequality we just got, we get. The nondecreasing sequence two puzzle whose most elegant solution relies on youngs inequality. This high level of income inequality is the outcome of a longterm trend increase. Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. Youngs inequality and its generalizations 3 remark 2. Clearly mark the beginning of the proof with the word proof 3.
Isoperimetric inequality encyclopedia of mathematics. The typical hispanic household has lost 66% of its wealth. Clearly identify and define the variables used in the proof 5. In this note we offer two short proofs of youngs inequality and prove its reverse. One can show that equality in the sharp young inequality implies that the functions have to be gaussian. Click here for a proof as mentioned in the introduction, young s inequality is essential in the proof of holders inequality. The sharp constant in youngs inequality has been determined independently by beckner and. Youngs inequality is a special case of the weighted amgm inequality. The inherent inequality a s t b t sp1 ab extra a s t b t sp1 ab extra since f2 lp.
The most familiar form of young s inequality, which is frequently used to prove the wellknown h. Inequality is also a much narrower concept than welfare. The subject of inequalities is vast, so our discussion will barely scratch the surface. Youngs inequality yue kwok choy question 1 let f be a realvalued function which is continuously differentiable and strictly increasing on the interval i 0. The aim of our work is to discuss the di erent kinds of youngs inequality in di ernt spaces with di erent. We give a new proof of the sharp form of youngs inequality for convolutions, first proved by beckner be and brascamplieb brli. In particular, since the sum telescopes, we have f. You can add a number to both or all sides of an inequality. Inequality in such variables is generally summarised by an inequality index, which in terms of figure 1 can be expressed as the degree of dispersion or width of the distribution.
It is also a special case of a more general inequality known as young s inequality for increasing functions. Apparently young s inequality actually is a special case of the weighted amgm inequality. Assume further that the following inequality is called the young inequality since both are nonzero, we can divide both side of the above inequality by and obtain the following equivalent form let us define a new variable using the relation we can see that thus the inequality can be rewritten as. Income inequality is greater in the united states than in any other democracy in the developed world. So jensens inequality gives that log 1 n xn k1 a k logex e logx 1 n xn k1 loga k. Income inequality in the united states increased significantly in the recent decades and at a stronger pace than the oecd average. Note that the p q 2 case has an even simpler proof. Apparently youngs inequality actually is a special case of the weighted amgm inequality. Improved young and heinz inequalities for matrices core. The rst thing to note is young s inequality is a farreaching generalization of cauchys inequality. It is also widely used to estimate the norm of nonlinear terms in pde theory, since it allows. The function gx jxjp is convex, because away from 0. Concepts, measures and policies frances stewart professor of development economics and director of the centre for research on inequality, human security and ethnicity, university of oxford abstract inequality, in various forms, is. One side of a triangle is longer than another side of a triangle if and only if the measure of the angle opposite the longer side is greater than the angle opposite the shorter side.
It is also a special case of a more general inequality known as youngs inequality for increasing functions. Inequalitythe state of not being equal, especially in status, rights, and opportunities1is a concept very much at the heart of social justice theories. It is so straightforward that its proof is often omitted. This is pretty simple, since is a concave function, is convex, and we can use jensens inequality, writing and we have also notice that if so the inequality is sharp, and moreover, since we are dealing with a strictly convex function the divergence is zero if and only if. Holder inequality 3 may be rewritten as since we integrate with respect to a probability measure, we can set then set.
Rupert frank california institute of technology, pasadena, usa diogo oliveira e silva university of bonn, germany christoph thiele university of bonn, germany supported by hausdor center for mathematics, bonn. Give a reason or principle that supports the assertion being made. Between the mid1980s and 2012, inequality measured by the. Economic inequality, difference in economic wellbeing between population groups. Youngs inequality and its generalizations introduction. In rn, we use the cauchyschwarz inequality or simply the schwarz inequality, which states that for all v,w. Moreover, there is equality in these inequalities for gaussian functions. Multidimensional extension of lc youngs inequality emis. Proof of youngs convolutions inequality for a general. Now, if f, and g 2ct then our result follows right away by applying holders inequality, we get the desired result. Acrossgroup inequality appears to be stable or decreasing from 2000 to 2008. We shall establish the equality case of 16 followed by a proof of the inequality. Inequality and the mean inequalities107 11 newtons inequality, maclaurins inequality.
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