Some new hardy type inequalities and their limiting. Inequalities by hardy g h littlewood j e polya g abebooks. Use features like bookmarks, note taking and highlighting while reading inequalities cambridge mathematical library. But, be aware that every line in most of the proofs takes nonnegligible time and effort on the part of the reader, in order to outguess why in heavens, hardy wrote it. In this paper, we derive a generalized multiplicative hardylittlewoodpolya type inequality, as well as several related additive. Hardys inequality is an inequality in mathematics, named after g. Hardy and littlewood type inequalities springerlink. Thomas of the following problem was recently presented in 1, p. On exact inequalities of hardylittlewoodpolya type. Extensions of the hardylittlewood inequalities for schwarz symmetrization h. Citeseerx rearrangement, majorization and stochastic scheduling. The best constant for the centered hardylittlewood maximal inequality by antonios d. On an inequality of hardy, littlewood and polya by hoffman, a. Polya this classic of the mathematical literature forms a comprehensive study of the inequalities used throughout mathematics.
This is obviously a generalization of the hlp inequality only when p 2. Dec 16, 2009 in this section, two inequalities are provided. Oct 21, 2016 this chapter considers time scale versions of classical hardy type inequalities and time scale versions of hardy and littlewood type inequalities. Polya, pdfsam 2 2 1port how to solve it will show anyone in any. If the righthand side is finite, equality holds if and only if for all n an integral version of hardys inequality states the following. An older book with incredible depth is g hardy, j e littlewood and g polya, inequalities, second edition, cambridge university press, 1952. Also, the boundedness of hardy littlewood maximal function is much more straightforward than the general marcinkiewicz interpolation theorem. The resulting inequality has been dubbed the help hardyeverittlittlewoodpolya inequality. These dynamic inequalities not only contain the integral and discrete inequalities but can be extended to different types of time scales. Interpolation of hardy inequality and trudingermoser. Inequalities cambridge mathematical library kindle edition by hardy, g. Also, the boundedness of hardylittlewood maximal function is much more straightforward than the general marcinkiewicz interpolation theorem.
Find the largest number c, in terms of a and b, such that a xb1. Inequalities of hardylittlewoodpolya type for functions. To explain this brie y, we recall that hardy and littlewood hl27 showed that for 1 hardy, littlewood, and polya s inequalities, but as much as i admire these authors for their other works, i never got much out of their inequalities book. In retrospect one sees that hardy, littlewood and polya has been one of the. Inequalities of hardylittlewoodpolya type for functions of operators. In 1934, hardy, littlewood and polya established the exact inequality between the norms of the derivatives of a function in a space.
The best constant for the centered hardylittlewood maximal. To explain this brie y, we recall that hardy and littlewood hl27 showed that for 1 inequalities free ebook download as pdf file. It states that if,, is a sequence of nonnegative real numbers, then for every real number p 1 one has. Inequalities arithmetic mean geometric mean inequal. We prove hardy littlewood sobolev inequality on product spaces by using strong maximal functions. Hardys inequalities for sobolev functions juha kinnunen and olli martio abstract. Inequalities cambridge mathematical library 2, hardy, g. In this paper, we derive a generalizedmultiplicative hardylittlewood polya type inequality, as well as several related additive inequalities, for func tions of operators in hilbert spaces. This classic of the mathematical literature forms a comprehensive study of the inequalities used throughout mathematics.
Optimal hardylittlewood inequalities uniformly bounded by a. In this talk we will deal with weighted inequalities for the hardy littlewood maximal operator, m. It is well known that the hardy littlewood maximal function plays an important role in many parts of analysis. Indeed, many majorization concepts have been reinvented and used in di. In this paper, we derive a generalized multiplicative hardylittlewood polya type inequality, as well as several related additive. On the other hand, the hardy littlewoodpolya hlp inequality1, inequality 381, p. The basic realvariable construct was introduced by hardy and littlewood for \n1\, and by wiener for \n\ge2\. More than a classic, this book is an encyclopedia of inequalities.
Mean values with an arbitrary function and the theory of convex functions. In a 1947 lecture, their admirer, the eminent danish mathematician harald bohr, brother of the nobel prize winning physicist niels bohr, reported a colleague as saying. Generalization of a hardylittlewoodpolya inequality. A hardylittlewood integral inequality on finite intervals with a.
Dimensionsthe field to which polya made his most numer ous contributions. Increasing rearrangement and hardylittlewood inequality. The authors were well known for their powers of exposition and were able here to make the subject accessible to a wide audience of. I made an attempt to get an html version of the references, but the adobe tool i. A note on hardylittlewood maximal operators journal of. The additional terms involve the dual counterparts, i. Hardylittlewoodsobolev inequality on product spaces. These inequalities can be viewed as a noncommutative su2version of the hardy littlewood inequalities hl27. Various putnam exam problems involving inequalities.
Hardylittlewoodpolya inequalities and hausdorff operators. The hardylandaulittlewood inequalities with less smoothness constantin p. This is the definitive and monumental text on inequality the little bent lines you see in all fields of mathematics from analysis to number theory to topology to combinatorics i would say all fields of science but i am not a math apologist, mind you, so i would stay with those things. Generalized hardylittlewoodsobolev inequality mathoverflow. We prove that c is the largest root of the quadratic equation 12c2. Determination of the best constant in an inequality of hardy, littlewood, and polya enticott, c. We first collect some basic concept on the schwarz rearrangement of functions. I nd the typesetting in this book hard to read it was done in the days before don knuths invention of tex but it contains a huge amount of material, some of it. Synopsis this classic of the mathematical literature forms a comprehensive study of the inequalities used throughout mathematics. As far as the rightmost inequality is concerned, there is a vast literature on the subject. Kolm 1969 was the very rst one, followed by dasgupta, sen and starrett 1973, to point out the relevance of this result in establishing the foundations of inequality measurement. Jan 20, 2016 the basic realvariable construct was introduced by hardy and littlewood for \n1\, and by wiener for \n\ge2\. Optimal exponents for hardylittlewood inequalities for m. Hardy and john edensor littlewood, states that if f and g are nonnegative measurable real functions vanishing at infinity that are defined on ndimensional euclidean space r n then.
This inequality will be referred to as a generalized hardylittlewood polya inequality since it reduces to 1. The fractional maximal function of the gradient gives a pointwise interpretation of hardys inequality for functions u. These inequalities can be viewed as a noncommutative su2version of the hardylittlewood inequalities hl27. We present extensions of hardy type inequalities on time scales. Inequalities of hardylittlewoodpolya type for functions of. Inequalities cambridge mathematical library kindle edition by g.
For example, the arithmeticgeometric mean inequality, jensens inequality, hermitehadamards inequality and hardylittlewoodpolyas inequality can be easily derived by using an argument based on schurconvex functions theory. Such inequalities are the continuous versions of the classical rearrangement inequalities for discrete sets of numbers. One is the hardylittlewood inequality, the other is of cherrier type. Download it once and read it on your kindle device, pc. Inequalities mathematics publisher national bureau of standards collection. Isabel, aguilar canestro and pedro, ortega salvador, proceedings of the japan academy, series a. Hardy little wood polya inequalities free ebook download as pdf file. Due to its large file size, this book may take longer to download. Single machine slack duewindow assignment and scheduling of linear timedependent deteriorating jobs and a deteriorating maintenance activity. Hardy and john edensor littlewood, states that if f and g are nonnegative. Hardylittlewoodpaley inequalities and fourier multipliers. This is a study of the inequalities used throughout mathematics. The resulting inequality has been dubbed the help hardyeverittlittlewood polya inequality. Hausdorff operators on block spaces kwokpun ho abstract.
Abstract this paper is devoted to improvements of sobolev and onofri inequalities. To obtain the best constant in the weighted hardylittlewoodsobolev whls inequality 2, one can maximize the functional jf. Rearrangement inequalities, such as the classical hardy littlewood polya inequality and the more general days inequality, and related majorization results are often useful in solving scheduling problems. A well written, classic text written by three larger than life math legends hardy, littlewood, polya. In this paper, we derive a generalizedmultiplicative hardy littlewood polya type inequality, as well as several related additive inequalities, for func tions of operators in hilbert spaces. Hardy, littlewood, cartwright and ramanujan transcript. In mathematical analysis, the hardylittlewood inequality, named after g. On the hardylittlewoodpolya inequality mathematisches institut. The cauchy, bunyakovskil, holder, and minkowski inequalities 2327 3554 6777 miscellaneousproblems 1. Stuart 2 1 department of mathematics, university of virginia, usa. On exact inequalities of hardylittlewoodpolya type article in journal of mathematical analysis and applications 2452.
Inequalities second edition, with changes and three new appendices by hardy, g. Extensions of the hardylittlewood inequalities for. The hardy, littlewood and polyas theorem is the key mathematical result in the area of inequality measurement. First published in 1934, it presents clearly and exhaustively both the statement and proof of all the standard inequalities of analysis. First published in 1934, it presents both the statement and proof of all the standard inequalities of analysis.
Get your kindle here, or download a free kindle reading app. In this paper we investigate in detail the exponents involved in hardylittlewood type inequalities and provide several optimal results that were not achieved by the previous approaches. Scribd is the worlds largest social reading and publishing site. There is a direct and selfcontained proof of hls inequality in analysis by lieb and loss, theorem 4.
Introduction in last years, a lot of papers was dedicated to majorization theory, that was scattered in journals in a wide variety of. First published in 1934, it presents clearly and lucidly both the statement and proof of all the standard inequalities of analysis. Polya and a great selection of related books, art and collectibles available now at. It is well known that the hardylittlewood maximal function plays an important role in many parts of analysis. Hardy littlewood polya s inequality is generalized.
Optimal hardylittlewood inequalities uniformly bounded. You can get a useful view from amazons search inside which has the cool. Polya was born in budapest on december, 1887, and died in palo alto. Littlewood, elementary theorems concerning power series with positive coefficients and moment constants of positive functions,j. The classic work in this field is hardy, littlewood, and polyas inequalities, but as. Convolution inequalities and applications saburou saitoh, vu kim tuan and masahiro yamamoto. The hardylittlewoodpolya majorization theorem is extended to. Among other things, they are essential for pairwise interchange arguments. An extension of the hardylittlewoodpolya inequality.
1097 589 577 1394 1243 1121 257 83 118 1114 349 366 45 1162 797 351 580 614 1148 232 659 442 1198 516 205 928 1291 346 267 1452