On Schur inequality and Schur functions (original) (raw)
Related papers
Some positive differences of products of Schur functions
2004
The product s µ s ν of two Schur functions is one of the most famous examples of a Schur-positive function, i.e. a symmetric function which, when written as a linear combination of Schur functions, has all positive coefficients. We ask when expressions of the form s λ s ρ − s µ s ν are Schur-positive. This general question seems to be a difficult one, but a conjecture of Fomin, Fulton, Li and Poon says that it is the case at least when λ and ρ are obtained from µ and ν by redistributing the parts of µ and ν in a specific, yet natural, way. We show that their conjecture is true in several significant cases. We also formulate a skew-shape extension of their conjecture, and prove several results which serve as evidence in favor of this extension. Finally, we take a more global view by studying two classes of partially ordered sets suggested by these questions.
Some refinements and generalizations of I. Schur type inequalities
2014
Recently, extensive researches on estimating the value of e have been studied. In this paper, the structural characteristics of I. Schur type inequalities are exploited to generalize the corresponding inequalities by variable parameter techniques. Some novel upper and lower bounds for the I. Schur inequality have also been obtained and the upper bounds may be obtained with the help of Maple and automated proving package (Bottema). Numerical examples are employed to demonstrate the reliability of the approximation of these new upper and lower bounds, which improve some known results in the recent literature.
Journal of Mathematical Analysis and Applications, 1997
Denote by the set of all real algebraic polynomials of degree at most n. The n
Some Remarks on Various Schur Convexity
Kragujevac journal of mathematics, 2022
The aim of this work is to investigate the Schur convexity, Schur geometrically convexity, Schur harmonically convexity and Schur power convexity of some special functions. Some sufficient conditions are obtained to guarantee the above-mentioned properties satisfy. We attain some special inequalities. Also, we obtain some applications of main results.
Necessary conditions for Schur-positivity
Journal of Algebraic Combinatorics, 2008
In recent years, there has been considerable interest in showing that certain conditions on skew shapes A and B are sufficient for the difference s A − s B of their skew Schur functions to be Schur-positive. We determine necessary conditions for the difference to be Schur-positive. Our conditions are motivated by those of Reiner, Shaw and van Willigenburg that are necessary for s A = s B , and we deduce a strengthening of their result as a special case.
Schur polynomials and matrix positivity preservers
Discrete Mathematics & Theoretical Computer Science, 2020
International audience A classical result by Schoenberg (1942) identifies all real-valued functions that preserve positive semidefi- niteness (psd) when applied entrywise to matrices of arbitrary dimension. Schoenberg's work has continued to attract significant interest, including renewed recent attention due to applications in high-dimensional statistics. However, despite a great deal of effort in the area, an effective characterization of entrywise functions preserving positivity in a fixed dimension remains elusive to date. As a first step, we characterize new classes of polynomials preserving pos- itivity in fixed dimension. The proof of our main result is representation theoretic, and employs Schur polynomials. An alternate, variational approach also leads to several interesting consequences including (a) a hitherto unexplored Schubert cell-type stratification of the cone of psd matrices, (b) new connections between generalized Rayleigh quo- tients of Hadamard powers and Sc...
Refinements of the Schur inequality for principal characters
Linear algebra and its applications, 1996
For this matrix function the Schur inequality asserts that dc(A) b det A if A is positive definite. This article contains a result which refines the above inequality and improves on some results of Bapat, Bunce, and Marcus and Sandy.
Schur-Convex functions related to Hadamard-type inequalities
Journal of Mathematical Inequalities, 2007
The Schur-convexity on the upper and the lower limit of the integral for a mean of the convex function is researched. As applications, a generalized logarithmic mean with a parameter is obtained and a relevant double inequality that is a extension of the known inequality is established.