Practice questions for mathematical induction inequality. The lower bound follows by applying the upper bound to. How to show amgm inequality using rearrangement inequality. Higher summability and discrete weighted muckenhoupt and. Some eigenvalue inequalities for a class of jacobi matrices mark s. Symmetric rearrangement, schwarz symmetrization, polarization, twopoint rearrangement, po. The rearrangement inequality by dragos hrimiuc in this note we reveal a nice result that provides a very simple but powerfulinequality that canbeused forproving many other inequalities. One application that rearrangement inequalities have found in probability is in. For all these, besides giving the proof, we present several examples that show how to use them in mathematical olympiad problems.
The inequality for an integral of a product of functions of one variable is further extended to the case of functions of several variables. Cases of equality in the riesz rearrangement inequality jstor. Luttinger, a general rearrangement inequality for multiple. In this paper we get the estimate of rearrangement in homogeneous space that is. Convolution, rearrangement and related inequalities. We introduce a special class of kernels for which the product inequality holds, and then we prove that it also holds when the product is replaced by a socalled. Beyond the entropy power inequality, via rearrangements arxiv. In contrast, we prove that for matrices, the general rearrangement inequality holds for all disordered words. Kinyin li, rearrangement inequality, mathematical excalibur, volume 4, number 3, january, 1999 march, 1999 yue kwok choy, rearrangement inequality samin riasat, basics of olympiad inequalities. 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. We also show that for larger matrices, the general rearrangement inequality holds for all disordered words, for most in a sense of full measure that are sufficiently small perturbations of the identity. A volumeinvariant packing problem and a combinatorial isoperimetric problem are addressed, as the geometric interpretation of the. We have also obtained discrete analogues of several of the results of this paper.
Rearrangement inequalities in the discrete setting and some. Lets consider two triplets a1,a2,a3 and b1,b2,b3 of real numbers. Wu south china normal university, china andy liu university of alberta, canada we will introduce our subject via an example, taken from a chinese competition in 1978. Pdf this paper presents an analogue of the rearrangement inequality, namely the circular rearrangement inequality.
But, by laxmilgrams and rellichs theorems, the set of such. Discrete convolution rearrangement inequalities and the faberkrahn inequality on regular trees. If we take all rearrangements permutations x1,x2,x3 of b1,b2,b3 we can generate 3. Rearrangement inequalities for functionals with monotone integrands almut burchard. The rearrangement inequality states that, if is a permutation of a finite set in fact, multiset of real numbers and is a permutation of another finite set of real numbers, the quantity is maximized when and are similarly sorted that is, if is greater than or equal to exactly of the other members of, then is also greater than or equal to exactly of the other members of. Linear matrix inequalities in system and control theory. It is quite often applied for the subtraction andor greatness, using the assumption at the step 2. The riesz rearrangement inequality can be used to prove the polyaszego inequality. You may take 2 bills from one box, 3 bills from another, 4 bills from another, and 5 bills from the remaining box.
Vince department of mathematics, university of florida, gainesville, fl 32611 one chapter of the classic book inequalities by hardy, littlewood, and polya 3 is dedicated to inequalities involving sequences with terms rearranged. Short proof of a discrete gronwall inequality sciencedirect. A lot of open challenging questions can be raised about discrete symmetrization, as for example, hardy and sobolev inequalities. To our knowledge there are few results available in literature concerning discrete symmetrization see. I inequality valid on any locally compact abelian group. Changsen yang, yu li refinements and reverses of young type inequalities. In this paper we prove a rearrangement inequality that generalizes in equalities given in the book by hardy, littlewood and polyai and by luttinger and friedberg. Several applications are developed, including to discrete entropy power inequalities, the littlewoodo. This inequality has a very broad and easy use in proving other inequalities. While a wide range of inequality indices of inequality have been developed, some general properties apply to. The rearrangement inequality is a statement about the pairwise products of two sequences. Discrete applied mathematics 16 1987 279281 northholland 279 note short proof of a discrete gronwall inequality dean s. Neither discrete operations such as the addition, deletion, or replacement of parts nor discrete changes in object shapes or spatial relationships can be accommodated at all withinthephysicallybased manipulation framework. Some rearrangement inequalities on space of homogeneous.
Lecture 23 rearrangement inequality holden lee 6411 1 the inequalities we start with an example. It can be extended to chebyshevs inequality, and illustrates the practical power of greedy algorithms. The riesz rearrangement inequality states that the functional. It seems easier to fool oneself by constructing a false proof of an inequality than of any other. Okay so i have two questions i think they are pretty simple which was why i put them together, both relating to inequalities that are proving to be challenging. Wave packets interactions with potential wells and barriers inequality 4 will be directly applicable to the program you write there will be a parameter a that youll need to make sure respects this inequality. Interactive physicallybased manipulation of discretecontinuous models mikako harada department of architecture. The proofs are obtained by employing new discrete weighted hardys type inequalities and their converses for nonincreasing sequences, which, for completeness, we prove in our special setting. Best examples of mathematical induction inequality iitutor. Rearrangement inequalities in the discrete setting and some applications.
Some eigenvalue inequalities for a class of jacobi matrices. Various rearrangement techniques for functions defined on. Then the inequality can be extended to characteristic functions of measurable sets. As motivation, let us start with the following random walk exit problem. Rearrangement inequalities in the discrete setting and. The rearrangement inequality states that, for two sequences. I have learned the amgmhm inequalities, the rearrangement inequality and the cauchyschwarz inequality. Consider, for example, a random variable x with standard normal distribution n0,1.
In the onedimensional case, the inequality is first proved when the functions, and are characteristic functions of a finite unions of intervals. We also emphasize how the substitution strategy is used to deduce several inequalities. A survey on cauchybunyakowskyschwarz type discrete inequalities, j. While a wide range of inequality indices of inequality have been developed, some general properties apply to all. Clark department of mathematics, university of rhode island, kingston, ri 02881, usa received 27 august 1985 revised 30 june 1986 we give an elementary proof of a generalization of the classical discrete gronwall inequality xn inequality called the rearrangement inequality. Pdf circular rearrangement inequality researchgate. Rearrangement inequalities in the discrete setting and some applications article in nonlinear analysis 723. A sharpened hausdor young inequality michael christ university of california, berkeley ipam workshop kakeya problem, restriction problem, sumproduct theory. Abstract the inequalities of hardylittlewood and riesz say that certain integrals involving. A general rearrangement inequality cristina draghici communicated by david preiss abstract. The proof follows from rearrangement inequalities going back at least to hardy, littlewood, and pdlya and is the onedimensional discrete analogue of a wellknown. Publications rima alaifari assistant professor for. We also establish cases of equality in these inqualities and give some applications of our result.
Rearrangement inequalities for functionals with monotone. To the best of the authors knowledge, these higher summability results have not been considered before. Further generalisations of discrete inequalities of wirtingers type for higher di. Defining and measuring inequality odi inequality briefings 1. We prove a general rearrangement inequality for multiple integrals, using polarization. Holte variations of gronwalls lemma gronwalls lemma, which solves a certain kind of inequality for a function, is useful in the theory of di. Some rearrangement inequalities on space of homogeneous type author. I hausdor young inequality kbfk lq kfk lp whenever 1 p 2, q p0 p p 1 conjugate exponent.
Schwarz inequality, the rearrangement inequality, the jensen inequality, the muirhead theorem, among others. A notable difference from the rearrangement inequality is that the variables are now. The mle, or grenander estimator, has been studied extensively, and much is known about. Dedicated to albert baernstein, ii on the occasion of his 65th birthday. In contrast, the riesz rearrangement inequality is speci. A generalization of the rearrangement inequality jan holstermann abstract in this article we present a generalization of the rearrangement inequal. Thisisanunfortunate limitation because a great many problems of interest involve both continuous and discrete parameters.
It has an uncanny resemblance to the famous rearrangement inequality, which is about the sum of product of terms, hence its namesake. The rearrangement inequality or the permutation inequality is an elementary inequality and at the same time a powerful inequality. Hausdor young inequality i fourier transform normalized to be unitary on l2rd. Geometric tomography western washington university. Reverse rearrangement inequalities via matrix technics gwdg. Omc 2011 rearrangement inequality lecture 23 clet a x 1 3 y 1 3 z 2 3, b x 3z y 2 3, and c y 1 3 z 1 3 x 2 3.
Circular rearrangement inequality haiyangyu abstract. In fact, we use it to give a new, independent, and simple proof of the entropy power inequality in the case when the summands are identically distributed. A discrete existence rcrjiilt discrete manipulative steps, arralogoiis to those wed in the continmiis proidem, are employed to derive a discrete version of theorem 2. To get rearrangement inequalities onx, polarization proved to be an e. If one is increasing and the other is decreasing they are said to be inversely or, oppositely ordered.
Discrete convolutionrearrangement inequalities and. Interactive physicallybased manipulation of discrete. Volume 72, issues 34, 1 february 2010, pages 11401148. We prove extended hardylittlewood, riesz and polyaszego inequalities in the discrete case. Many important inequalities can be proved by the rearrangement inequality, such as the arithmetic mean geometric mean inequality, the cauchyschwarz inequality, and chebyshevs sum inequality. Recent papers pdf files for published versions of most of my earlier papers are available on request. Gardner, paolo gronchi, and markus kiderlen, rearrangement and polarization, submitted. I optimal constant 1 for many groups, including td, zd. Mathematical induction inequality is being used for proving inequalities.
Computational solution to discrete timedependent schrodingers equation. Introduction and preliminaries in the wellknown monograph written by hardy, littlewood and po. We give a reverse inequality to the most standard rearrangement inequality for. A rearrangement inequality and the permutahedron a. An early version of my paper the brunnminkowski inequality, bull. A study of variational inequalities and finite element methods n. H ajaiej, rearrangement inequalities in the discrete. Lets take a look at the following handpicked examples. H ajaiej, rearrangement inequalities in the discrete setting and so me. Two sequences that are both increasing or both decreasing are said to be similarly ordered. A notable difference from the rearrangement inequality is that the variables are now required to be non. Hajaiej,rearrangement inequalities in the discrete setting and some applications, nonlinear. If we calculate the probability of the normal using a table of the normal law or using the computer, we obtain. The reverse rearrangement inequality allows us to compare the product of sums of terms in an inequality.