Journal of mathematical analysis and applications 361. In this paper the complex representation of quaternion matrices is applied to establish the inverse young inequality for. We give a number of subadditivity results and conjectures for symmetric norms, matrices and block matrices. The aim of this note is to give a new proof of the inequality m k. It is also known as legendrefenchel transformation or fenchel transformation after adrienmarie legendre and werner fenchel.
Request pdf improved young and heinz inequalities for matrices we give refinements of the classical young inequality for positive real numbers and we use these refinements to establish. Furthermore, we give some inequalities dealing with operator means. In mathematics, young s inequality for products is a mathematical inequality about the product of two numbers. In this section, we deal with reverses of the young inequality for the hilbertschmidt norm. The individual values in the matrix are called entries. In this paper we obtain some trace inequalities for positive operators via recent refinements and reverses of youngs inequality due to kittanehmanasrah, liaowuzhao, zuoshifujii, tominaga and furuichi. Improved young and heinz inequalities for matrices core. Norm inequalities related to the matrix geometric mean 7 the. Young type inequalities for matrices east china normal. Moreover, there is equality in these inequalities for gaussian functions. Pdf some extensions of the young and heinz inequalities. The new reverses of young type inequalities for numbers. Part of the point is to mention the original papers young 1912a and young 1912b.
Improved young and heinz inequalities for matrices sciencedirect. Here is a matrix of size 2 3 2 by 3, because it has 2 rows and 3 columns. The resulting sums replace the column elements of row b while row a remains unchanged. Two matrices a and b are said to be equal, written a b, if they have the same dimension and their corresponding elements are equal, i. Pdf inverse young inequality in quaternion matrices. In general, an m n matrix has m rows and n columns and has mn entries.
Improved young and heinz inequalities for matrices request pdf. Some extensions of the young and heinz inequalities for matrices. In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the legendre transformation which applies to nonconvex functions. M n c such that a and b are positive semidefinite and if 0lessorequalslant. Then, we re ne matrix young type inequalities for the case of hilbertschmidt norm, which hold under certain assumptions on positive semide nite matrices appearing therein. Pdf we present some reverse youngtype inequalities for the hilbertschmidt norm as well as any unitarily invariant norm. Lecture 15 symmetric matrices, quadratic forms, matrix. This process is experimental and the keywords may be updated as the learning algorithm improves. Reverses of the young inequality for matrices and operators article pdf available in rocky mountain journal of mathematics 464.
Pdf reverses of the young inequality for matrices and. It is also widely used to estimate the norm of nonlinear terms in pde theory, since it allows one to estimate a product of two terms by a sum of the same. These include, in particular, bernoullis, youngs, holders, cauchyschwarz, and minkowskis inequalities of course, this is not a complete list. Matrix young inequalities for the hilbertschmidt norm. The quaternion matrixvalued youngs inequality emis. We give refinements of the classical young inequality for positive real numbers and we use these refinements to establish improved young and heinz inequalities for matrices. Young s convolution inequality, bounding the convolution product of two functions. Inequality regarding ranks of matrices problems in mathematics. Improved young and heinz inequalities for matrices. In some instances it is convenient to think of vectors as merely being special cases of matrices. For more information on heinz inequality for matrices the reader is referred to.
Youngs inequality in compact operators ftp directory listing. To prove the triangle inequality requires the following classical result. Youngs inequality for products can be used to prove holders inequality. Then, for all measurable real or complexvalued functions f and g on s. Unitary matrix complex matrice hermitian matrix matrix anal young inequality these keywords were added by machine and not by the authors. The result has been so well assimilated that already the extensive bibliography of riesznagy 1952 did not list these papers, although others of youngs did appear.
Usually they will have defined what they mean earlier in the text. Linear matrix inequalities in system and control theory. Improved variants of young, heinz, and h older inequalities. By ruans theorem 22, this matricial structure leads us to consider an operator space structure on the vector space where we are taking. M 1 sciences, isfahan university of 2 department of mathematical the heinz mean for every nonnegative real numbers a, b and every 0. Provided by the academic center for excellence 3 solving systems of linear equations using matrices summer 2014 3 in row addition, the column elements of row a are added to the column elements of row b. We use operator monotone and operator convex functions to prove an inverse to the young inequality for eigenvalues of positive definite. A note on reverses of young type inequalities journal of.
The second inequality follows from a result by araki 6. Andomajorizations and inequalities in matrix theory. Improved young and heinz inequalities for matrices based on the refined young inequality 2. Therefore, just to start talking about the hausdor. The inequality m k above is a considerable re nement to the wellknown young inequality and has many nice applications. We use operator monotone and operator convex functions to prove an inverse to the young inequality for eigenvalues of positive definite matrices and then apply it to obtain a matrix inverse young inequality which can be considered as a complement of a result of t. In this paper, we will show some improvements of heron mean and the refinements of youngs inequalities for operators and matrices with a different method based on others results. Unitary matrix complex matrice hermitian matrix matrix anal young inequality.
Relations between the given inequalities and earlier results are discussed. A matrix majorization version of the general young s inequality xy. By using young s inequality, we can obtain some results of heinz mean. The sharp constant in youngs inequality has been determined independently by beckner and. Heron mean, youngs inequality, operator inequality, unitarily invariant norm. Some extensions of the young and heinz inequalities for matrices article pdf available in bulletin of the iranian mathematical society 4434 may 2017 with 165 reads how we measure reads. For comprehensivediscussions of matrix arithmeticgeometric mean and related inequalities, we refer to 3,5,8,10,11,12and the references therein. Matrices and matrix operations in matlab the identity matrix and the inverse of a matrix the n nidentity matrix is a square matrix with ones on the diagonal and zeros everywhere else.
Mn such that a and b are positive semide nite, and for 0. We conjecture that for all nonnegative concave function f on 0. Further improved young inequalities for operators and matrices. Pdf in this article, we interpolate the wellknown young s inequality for numbers and matrices, when equipped with the hilbertschmidt norm, then. Pdf matrix young inequalities for the hilbertschmidt. Pdf reverses of the young inequality for matrices and operators. Prove the cauchyschwarz inequality problems in mathematics. Then we use these inequalities to establish corresponding inequalities for matrices. The paper concentrates on the associated mathematics. Norms of matrices, numerical range, applications of functional analysis to matrix theory see also 65f35, 65j05 47a30.
These keywords were added by machine and not by the authors. Thompson 6 proved that, the equality in the matrixvalued triangle inequality 1. Mahmoud manjegani journal of inequalities and applications tracial and majorisation heinz meantype inequalities for matrices seyed mahmoud manjegani 0 1 2 0 theorem. For more information on matrix versions of the young inequality the reader is referred to 49. We present some reverse young type inequalities for the hilbertschmidt norm as well as any unitarily invariant norm. The sharp constant in youngs inequality has been determined independently by beckner and brascamplieb. The matrix mean inequality and the matrix young inequality were considered with the numerical radius norm by salemi and sheikhhosseini in 14.
Among other inequalities, it is shown that if a and b are positive semidefinite matrices then for any positive integer k. Norm inequalities related to the matrix geometric mean. Our proof is based on the arithmeticgeometric mean inequality and will shorten considerably the lines of proof given by y. The classical young inequality for two scalars is the. Keywords young inequality positive operator operator mean unitarily invariant norm determinant trace. Operators and matrices, 12, 4 2018 10631071 keywords. We prove the cauchyschwarz inequality in the ndimensional vector space rn. Example here is a matrix of size 2 2 an order 2 square matrix. In this paper, we establish an analogue for the case of equality in the setting of operators. In this paper, we study the further improvements of the reverse young and heinz inequalities for the wider range of v, namely \v\in \mathbb r\. It is called the identity because it plays the same role that 1 plays in multiplication, i. By using youngs inequality, we can obtain some results of heinz mean.
Let a, b, z be matrices of same size and suppose that a, b are normal and z is expansive. The classical young inequality for scalar says that if a. The last inequality above follows from the matrix version of young s inequality by ando 2. Reverses of the young inequality for the hilbertschmidt norm.
Clearly, the 1norm and 2 norms are special cases of the pnorm. Contents contents notation and nomenclature a matrix a ij matrix indexed for some purpose a i matrix indexed for some purpose aij matrix indexed for some purpose an matrix indexed for some purpose or the n. Matrix form of the inverse young inequalities sciencedirect. As an application, a determinant inequality for positive definite matrices is obtained. Some classical inequalities among all inequalities, there is a number of wellknown classical inequalities. Youngs inequality and its generalizations introduction. Notes on vector and matrix norms university of texas at. These modified inequalities are used to establish corresponding operator inequalities on a hilbert space. It is also widely used to estimate the norm of nonlinear terms in pde theory, since it allows.
In this short note, we prove that the conjecture of singular value young inequality holds when j n. In a recent paper, young s inequality has been seen in a di. The inequality is named after william henry young and should not be confused with youngs convolution inequality. Convexity, inequalities, and norms 9 applying the same reasoning using the integral version of jensens inequality gives p q z x fpd 1p x fqd 1q for any l1 function f.
Let a, b, z be matrices of same size and suppose that a, b are normal and z is expansive, i. A matrix young inequality was obtained by ando in 1. Trace inequalities for positive operators via recent. Many of them have been proved by famous mathematicians and named after them. It seems that matrix versions of the young inequality have aroused considerable interest. Trace inequalities for matrices bulletin of the australian. Also, we give a necessary and sufficient condition for the equality.
Characterization of the trace by youngs inequality rims, kyoto. Note on young and arithmeticgeometric mean inequalities for matrices yanqiu wu school of mathematics and statistics chongqing three gorges university chongqing, 404100 p. Since the trace of hvanishes, this is the case if all the j s vanish and hence his the zero matrix. Multivariate young s inequality is extended to the matrix setting by means of the geometric mean of positive. Proving that the pnorm is a norm is a little tricky and not particularly relevant to this course. Youngs inequality for vectors and matrices mathematics. Finally, we present some singular value inequalities of young type involving trace and determinant. First we give an improvement of the matrix heinz inequality for the case of the hilbertschmidt norm. As a matter of fact, the proof of the main result in 1 relies on a parametrization of functions which was used in and was suggested by brunns proof of the brunnminkowski inequality. The main purpose of this paper is to give improved reverses of young type inequalities and. Reverses of the young inequality for matrices and operators. Finally, we give the re nement and the reverse of the matrix h older inequality which holds for every unitarily invariant norm. The main purpose of this paper is to give some young type inequalities for matrices. Trace inequalities for sums and products of matrices are presented.
We note that the product ab of two positive matrices a and b is not necessarily positive. The new reverses of young type inequalities for numbers, operators and matrices author. This linear matrix inequality specifies a convex constraint on y. Also, some operator inequalities around the young s inequality for semidefinite invertible matrices are. Characterization of the trace, matrix youngs inequality. M nc is a non scalar strictly positive matrix such that 1. Further improved young inequalities for operators and matrices author. University of california berkeley, linear algebra qualifying exam problem. Multivariate youngs inequality is extended to the matrix setting by means of the geometric mean of positive semide. William henry young, english mathematician 18631942 hausdorff young inequality, bounding the coefficient of fourier series. The heinz norm inequality, which is one of the essential inequalities in operator theory, states that for any positive operators a,b 2mn, any operator x 2mn and v 20,1, the following double inequality holds. Matrix monotone functions, matrix means, and majorization.
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