On some inequalities for the Bernoulli numbers

In modo elementare si migliorano sensibilmente alcune diseguaglianze recentemente stabilite da D. J. Leeming sui numeri di Bernoulli [3].     

Nordhaus-Gaddum inequalities for the fractional and circular chromatic numbers

For a graph G on n vertices with chromatic number χ(G), the Nordhaus-Gaddum inequalities state that , and . Much analysis has been done to derive similar inequalities for other graph parameters, all of which are integer-valued. We determine here the optimal Nordhaus-Gaddum inequalities for the circular chromatic number and the fractional chromatic number, the first examples of Nordhaus-Gaddum inequalities where the graph parameters are rational-valued.     

Bell numbers, determinants and series

In this article, we study Bell numbers and Uppuluri Carpenter numbers. We obtain various expressions and relations between them. These include polynomial recurrences and expressions as determinants of certain matrices of binomial coefficients.     

Some inequalities for the gamma function

Several inequalities involving gamma function are obtained. They are established using elementary properties of logarithmically convex functions.     

Two recursive inequalities for crossing numbers of graphs

In this paper, two recursive inequalities for crossing numbers of graphs are given by using basic counting method. As their applications, the crossing numbers of K _1,3,n and K _4,n \ e are determined, respectively.     

Norms and inequalities for condition numbers,III

The main results provide comparisons between condition numbers (based on unitarily invariant norms) of (i) positive definite (Hermitian) matrices A, B and of A + B, (ii) a positive definite matrix and its principal submatrix, and (iii) a matrix and an augmented form of the matrix.     

Some bounds for the Ramsey-Paris-Harrington numbers

It has recently been discovered that a certain variant of Ramsey's theorem cannot be proved in first-order Peano arithmetic although it is in fact a true theorem. In this paper we give some bounds for the “Ramsey-Paris-Harrington numbers” associated with this variant of Ramsey's theorem, involving coloring of pairs. In the course of the investigation we also study certain weaker and stronger partition relations.     

Some inequalities for partial orders

We establish some inequalities connecting natural parameters of a partial order P. For example, if every interval [a,b] contains at most maximal chains, if some antichain has cardinality v, and if there are ₁ chains whose union is cofinal and coinitial in P, then the chain decomposition number for P is ₁v (Theorem 2.2), and the inequality is sharp in a certain sense (Section 3).This paper was written while the authors were visitors at the Laboratoire d'algèbre ordinale, Département de Mathématiques, Université Claude Bernard, Lyon 1, France.Research supported by NSERC grant # A5198.     

Some inequalities for starlike functions

For normalized starlike univalent functions in the unit disc we derive upper and lower estimates for certain differential expressions ofn-th order (including |zf′(z)/f(z)| forn=1) in terms of |f(z)|. Our results generalize and/or improve earlier ones by Twomey and Singh. The operators and methods applied come from the theory of the Peschl-Bauer differential equation.     

Some inequalities for Laplace transforms

Recently we established Matysiak and Szablowski's conjecture [V. Matysiak, P.J. Szablowski, Some inequalities for characteristic functions, Theory Probab. Appl. 45 (2001) 711-713] about a lower bound of real-valued characteristic functions. In this paper, we investigate the counterparts for Laplace transforms of non-negative random variables. Surprisingly, the resulting inequalities hold true on the right half-line. Besides, we show some more inequalities by applying the convex/concave properties of the remainder in Taylor's expansion for the exponential function.     

Some determinant inequalities for operators

The aim of this note is to study the determinant inequalities for operators in a finite von Neumann algebra. In particular, some determinant inequalities due to Hua, Lin and Zhang are extended to the case of operators in a finite von Neumann algebra.     

Some inequalities for Franklin series

Обозначим через {f _k (t)} _k=0 ^∞ систему Франклина на отрезке [0, 1]. Доказано, чт о еслиg∈L(0,1), то $$mes\{ t:|\sum\limits_{\kappa = 0}^\infty {\varepsilon _\kappa a_\kappa (g)f_\kappa (t)| > y} \} \leqq \frac{B}{y}\int\limits_0^1 {|g(t)|dt} $$      

Some inequalities for characteristic functions

In a recent paper, Matysiak and Szablowski [V. Matysiak, P.J. Szablowski, Theory Probab. Appl. 45 (2001) 711-713] posed an interesting conjecture about a lower bound of real-valued characteristic functions. Under a suitable moment condition on distributions, we prove the conjecture to be true. The unified approach proposed here enables us to obtain new inequalities for characteristic functions. We also show by example that the improvement in the bounds is significant if more information about the distribution is available.     

Some congruences for the Apery numbers

In 1979 R. Apéry introduced the numbers a_n = Σ₀ⁿ(_kⁿ)²(_k^{n + k}) and u_n = Σ₀ⁿ(_kⁿ)²(_k^{n + k})² in his irrationality proof for ζ(2) and ζ(3). We prove some congruences for these numbers which generalize congruences previously published in this journal.     

Some inequalities for absolute moments

In this note, we obtain several inequalities for absolute moments of sums and differences of independent random variables, using one representation of absolute moments in terms of the characteristic function and inequalities for characteristic functions.     

Some inequalities derived from log-Sobolev inequalities

We show how a log-Sobolev inequality can be manipulated to give an improved version of itself.