On some generalizations of the Hilbert–Hardy type discrete inequalities

New Hilbert-type discrete inequalities are presented by using new techniques in proof. By specializing the weight coefficient functions in the hypothesis and the parameters, we obtain many special cases which include, in particular, the discrete inequality derived by Hilbert and Hardy. Many improvements and generalizations of known results are given in this paper.     

Hilbert–Pachpatte type general integral inequalities

In this article we present very general weighted Hilbert–Pachpatte type integral inequalities. These are regarding ordinary derivatives and fractional derivatives of Riemann–Liouiville and Canavati types. Also regarding general derivatives of Widder type and linear differential operators. Our results apply to continuous functions and some to integrable functions.     

On Hilbert’s solution of Waring’s problem

In 1909, Hilbert proved that for each fixed k, there is a number g with the following property: Every integer N ≥ 0 has a representation in the form N = x ₁ ^k + x ₂ ^k + … + x _g ^k , where the x _i are nonnegative integers. This resolved a conjecture of Edward Waring from 1770. Hilbert’s proof is somewhat unsatisfying, in that no method is given for finding a value of g corresponding to a given k. In his doctoral thesis, Rieger showed that by a suitable modification of Hilbert’s proof, one can give explicit bounds on the least permissible value of g. We show how to modify Rieger’s argument, using ideas of F. Dress, to obtain a better explicit bound. While far stronger bounds are available from the powerful Hardy-Littlewood circle method, it seems of some methodological interest to examine how far elementary techniques of this nature can be pushed.     

On general Bernstein and Nikol’ski type inequalities

The goal of this paper is to establish the relations between general Bernstein and Nikol’ski type inequalities under some weak conditions. From these relations some known classical inequalities are implied. Also, a family of functions equipped with Bernstein type inequality which satisfies Nikol’ski type inequality is found.     

On Stanley’s inequalities for character multiplicities

Let G be a group of automorphisms of a ranked poset \({{\mathcal Q}}\) and let N _k denote the number of orbits on the elements of rank k in \({{\mathcal Q}}\). What can be said about the N _k for standard posets, such as finite projective spaces or the Boolean lattice? We discuss the connection of this question to the representation theory of the group, and in particular to the inequalities of Livingstone-Wagner and Stanley. We show that these are special cases of more general inequalities which depend on the prime divisors of the group order. The new inequalities often yield stronger bounds depending on the order of the group.     

New kinds of Hardy–Hilbert’s integral inequalities

New kinds of Hardy–Hilbert’s integral inequalities are presented.     

On some problems involving Hardy’s function

Some problems involving the classical Hardy function

Refined Young inequalities with Specht’s ratio

In this paper, we show that the ν-weighted arithmetic mean is greater than the product of the ν-weighted geometric mean and Specht’s ratio. As a corollary, we also show that the ν-weighted geometric mean is greater than the product of the ν-weighted harmonic mean and Specht’s ratio. These results give the improvements for the classical Young inequalities, since Specht’s ratio is generally greater than 1. In addition, we give an operator inequality for positive operators, applying our refined Young inequality.     

On some higher order Hardy–Rellich type inequalities with boundary terms

We determine boundary remainder terms for some higher order Hardy–Rellich inequalities involving the polyharmonic operator (−Δ)^m${(-\Delta )}^m$. The results are proved by studying suitable auxiliary boundary eigenvalue problems, the optimal constants found may not be the classical Hardy–Rellich ones.     

A generalization of weyl’s inequalities with implications

This paper suggests a generalization of the additive Weyl inequalities to the case of two square matrices of different orders. As a consequence of the generalized Weyl inequalities, a theorem describing the location of eigenvalues of a Hermitian matrix in terms of the eigenvalues of an arbitrary Hermitian matrix of smaller order is derived. It is demonstrated that the latter theorem provides a generalization of Kahan’s theorem on clustered eigenvalues. It is also shown that the theorem on extended interlacing intervals is another consequence of the generalized additive Weyl inequalities suggested. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 49–59. Translated by L. Yu. Kolotilina.     

Hilbert’s irreducibility theorem for prime degree and general polynomials

Letf (X, t)εℚ[X, t] be an irreducible polynomial. Hilbert’s irreducibility theorem asserts that there are infinitely manyt ₀εℤ such thatf (X, t ₀) is still irreducible. We say thatf (X, t) isgeneral if the Galois group off (X, t) over ℚ(t) is the symmetric group in its natural action. We show that if the degree off with respect toX is a prime ≠ 5 or iff is general of degree ≠ 5, thenf (X, t ₀) is irreducible for all but finitely manyt ₀εℤ unless the curve given byf (X, t)=0 has infinitely many points (x ₀,t ₀) withx ₀εℚ,t ₀εℤ. The proof makes use of Siegel’s theorem about integral points on algebraic curves, and classical results about finite groups, going back to Burnside, Schur, Wielandt, and others. Supported by the DFG.     

On some Turán-type inequalities for q-special functions

Intrinsic inequalities involving Turán-type inequalities for some q-special functions are established. A special interest is granted to q-Dunkl kernel. The results presented here would provide extensions of those given in the classical case.     

On some new results related to Gram’s law

The paper contains the formulations of some new results related to Gram??s law in the theory of the Riemann zeta-function and describing the irregularity in the distribution of complex zeros of this function. Namely, we obtain some results related to the distribution of pairs, triples, quadruples etc. of the neighbouring ordinates of such zeros that simultaneously do not satisfy Gram??s law.     

On some eigenvalue inequalities for Schatten–von Neumann operators

Michael Gil' recently obtained some bounds for eigenvalues in [J. Funct. Anal. 267 (2014) 3500–3506] and [Commun. Contemp. Math. 18 (2016) 1550022], which improve some classical results related to this aspect. We revisit these results by providing genuinely different arguments (e.g., using Aluthge transform, majorization). New results are derived along our discussions.