An upper bound on the characteristic polynomial of a nonnegative matrix leading to a proof of the Boyle–Handelman conjecture (original) (raw)

On a strong form of a conjecture of Boyle and Handelman

The Electronic Journal of Linear Algebra, 2002

Let ρr,m(x, λ) := (x − λ) r m i=0 r+i−1 i x m−i λ i. In this paper it is shown that if λ 1 ,. .. , λn are complex numbers such that λ 1 = λ 2 =. .. = λr > 0 and 0 ≤ n i=1 λ k i ≤ nλ k 1 , for 1 ≤ k ≤ m := n − r, then n i=1 (λ − λ i) ≤ ρr,m(λ, λ 1), for all λ ≥ 6.75λ 1. (*) Moreover, if r ≥ m, then (*) holds for all λ ≥ λ 1 , while if r < m, but r is close to m, and n is large, one can lower the constant of 6.75 in the inequality (*). The inequality (*) is inspired by, and related to, a conjecture of Boyle and Handelman on the nonzero spectrum of a nonnegative matrix.

Two and a Half Remarks on the Marica-Schönheim Inequality

Journal of the London Mathematical Society, 1993

The Marica-Schonheim inequality states that the number of distinct differences of the form A\B, with A, B taken from a given finite family si of sets is at least \st\. We prove that equality occurs essentially if and only if si is the product of an ideal and a filter. We also prove an infinite version of the theorem, conjectured (in weaker form) by Daykin and Lovasz. Finally, we note that a generalization (due to Ahlswede and Daykin) of the inequality which considers two families si and & holds under a weaker assumption on the relation between si and 28. 0. The Marica-Schonheim inequality Given two families of sets $0 and 38, we write (We should remark that by a ' family of sets' we really mean a set of sets, that is, repetitions are not allowed.) Motivated by a problem of Graham in combinatorial number theory, Marica and Schonheim proved the following inequality. THEOREM 0.1 [5]. For every finite family stf we have that \s& ~ s#\ ^ \s#\. This simple inequality turned out to be closely related to several more sophisticated correlation inequalities in combinatorics, all subsumed by the Four Functions Theorem of Ahlswede and Daykin [1]. In this paper we make a few separate observations connected to the Marica-Schonheim inequality. The background for each of them will be presented in the respective section of the paper. However, all our remarks are based on a proof of the inequality which was given by Daykin and Lovasz [4]. For the reader's convenience, and in order to establish notation, we produce that proof here. Let V be the ground set of the family j^, that is, A £ V for all Aestf. For xeV we write Given xe V, we consider two families of subsets of V\{x} derived from J / as follows: Once x is fixed, we shall omit the subscript and write simply s/ + , s#~. We also form the following two families:

On inequalities of Korn, Friedrichs and Babu�ka-Aziz

Arch Ration Mech Anal, 1983

... 174 where We may also write where CO HORGAN & LE PAYNE [f'(0) ] 2 (6.14) Q2(~ = 17(071 " zrz p2 1 -- nr 2 . . . . 2 p2 on ~R, (6.15) nr ... This choice of/3 satisfies (6.19), by virtue of (6.17) and (6.22). On substitution from (6.21) in (6.20) one finds that h 2 dA < max h .2 dA. (6.23) ...

Inequalities Involving Lower-Triangular Matrices

Proceedings of the London Mathematical Society, 1980

The constant at the right of (1) was fixed by Landau [14] who showed that it is the best possible for each p. Since then many alternative proofs of Hardy's inequality have been given (cf. Broadbent [1], Elliot [5], Grandjot [7], Knopp [13], Kaluza and Szego [12]). Copson [2] generalized Theorem A by replacing the arithmetic mean of sequence u by a weighted arithmetic mean. We shall consider Copson's generalization in § 10. If A = {a mn) is the Cesaro matrix a mn = rnr 1 , where m ^ n, then Hardy's inequality can be written as (2) for 1 < p < oo. Petersen [18] and subsequently Davies and Petersen [4] produced sufficient conditions on a matrix A and an auxiliary sequence/ £ co for the existence of an inequality of the form \\A\x\\\ p^K \\f.a.x\\ p (l^p<co) for some K with a = {a mm }, the main diagonal sequence of A. Our object in this paper is to study inequalities of the form or, indeed, of the form (3) M i z i u , with (A, || • || A), (/x, || • 11^) being quasinormed FK spaces satisfying certain hypotheses. Henceforward, inequalities of the form (3) will* be referred to as HPD inequalities (for Hardy, Petersen, and Davies). Particular attention will be paid to the existence of best possible HPD inequalities, for fixed A, (A, ||*|| A), and (/x, || 1 1^); best possible not by virtue of the smallness of the number K, but of the sequence b, with respect to a particular definition of 'smallness' of sequences (see § 3). It is clear that the smaller the K, the better is the inequality (3). Just so, the 'smaller' the 6, the better is the inequality (3). We shall obtain conditions on A } (^>II'IIA)> a n d 0*ilHI/») for * n © existence of a 'smallest' b for which (3) holds for some K and will apply our results with A,/u among the l r and A specified or confined to some special class of matrices.

Note on an inequality of M.A. Malik

Applicable Analysis and Discrete Mathematics

Let P(z):= ?nv=0 avzv be a univariate complex coefficient polynomial of degree n. It was shown by Malik [J London Math Soc, 1 (1969), 57-60] that if P(z) has all its zeros in |z| ? k, k ? 1, then max|z|=1 |P?(z)| ? n 1 + k max |z|=1 |P(z)|. In this paper, we prove an inequality for the polar derivative of a polynomial which besides give extensions and refinements of the above inequality also produce various inequalities that are sharper than the previous ones known in very rich literature on this subject.