Sharp bounds for the partition function of integer sequences

By means of a new criterion for higher multiplicities of additive representations of integers as sums of distinct terms of a fixed integer sequence sharp bounds are determined for the partition function of many sequences with the aid of a computer.     

Sharp power mean bounds for the Gaussian hypergeometric function

Sharp inequalities are established between the Gaussian hypergeometric function and the power mean. These results extend known inequalities involving the complete elliptic integral and the hypergeometric mean.     

Computing the Lambert W function in arbitrary-precision complex interval arithmetic

Numerical Algorithms - We describe an algorithm to evaluate all the complex branches of the Lambert W function with rigorous error bounds in arbitrary-precision interval arithmetic or ball...     

Sharp bounds for the Landau constants

The Landau constants are defined for all integers n≥0 by $$ G_{n}=\sum_{k=0}^{n} \frac{1}{16^k}\binom{2k}{k}^2. $$ We establish sharp bounds for the Landau constants G _n . Relevant connections of the results presented here with those derived in earlier work are also pointed out.     

Sharp bounds for the Bernoulli numbers

We determine the best possible real constants a\alpha and b\beta such that the inequalities [(2(2n)!)/((2p)²ⁿ)] [1/(1-2^a-2n)] \leqq |B_2n| \leqq [(2(2n)!)/((2p)²ⁿ)] [1/(1-2^b-2n)]{2(2n)! \over(2\pi)^{2n}} {1 \over 1-2^{\alpha -2n}} \leqq |B_{2n}| \leqq {2(2n)! \over (2\pi )^{2n}}\, {1 \over 1-2^{\beta -2n}}hold for all integers n\geqq 1n\geqq 1. Here, B₂, B₄, B₆,... are Bernoulli numbers.     

Continued fraction expansions for the Lambert $$varvec{W}$$ function

Aequationes mathematicae - In the first part of the paper we give a new integral representation for the principal branch of the Lambert W function. Then we deduce two continued fraction expansions...     

Sharp bounds for NBUE distributions

Let F be a new better than used in expectation (NBUE) distribution function with mean μ. In a previous paper (Brown in Probab. Eng. Inf. Sci. 20:195–230, 2006), the author derived the following bound. For any t≥μ, $$\overline{F}(t) = \mathit{Pr}(X \ge t) \le e^{-[{t\over\mu}-1]}. $$ The main result of this paper is to show that this bound is sharp. Other sharp bounds for NBUE distributions are also derived.     

Sharp bounds for integral means

We introduce a bound M of f, ‖f‖_∞?M?2‖f‖_∞, which allows us to give for 0?p<∞ sharp upper bounds, and for −∞<p<0 sharp lower bounds for the average of |f|^p over E if the average of f over E is zero. As an application we give a new proof of Grüss's inequality estimating the covariance of two random variables. We also give a new estimate for the error term in the trapezoidal rule.     

Sharp bounds for harmonic numbers

In the paper, we collect some inequalities and establish a sharp double inequality for bounding the n-th harmonic number.     

Sharp deviation bounds for quadratic forms

This paper presents sharp inequalities for deviation probability of a general quadratic form of a random vector ξ with finite exponential moments. The obtained deviation bounds are similar to the case of a Gaussian random vector. The results are stated under general conditions and do not suppose any special structure of the vector ξ. The obtained bounds are exact (non-asymptotic), all constants are explicit and the leading terms in the bounds are sharp.     

Sharp bounds for the spectral radius of digraphs

Let $G=(V\text{,}E)$ be a digraph with n vertices and m arcs without loops and multiarcs. The spectral radius $\rho (G)$ of G is the largest eigenvalue of its adjacency matrix. In this paper, the following sharp bounds on $\rho (G)$ have been obtained.$\min \{\sqrt{t_i^{+}{t}_j^{+}}:(v_i\text{,}{v}_j)\in E\}?\rho (G)?\max \{\sqrt{t_i^{+}{t}_j^{+}}:(v_i\text{,}{v}_j)\in E\}$where G is strongly connected and $t_i^{+}$ is the average 2-outdegree of vertex $v_i$. Moreover, each equality holds if and only if G is average 2-outdegree regular or average 2-outdegree semiregular.     

Sharp bounds for fractional one-sided operators

In this paper,we characterize the sharp boundedness of the one-sided fractional maximal function for one-weight and two-weight inequalities.Also a new two-weight testing condition for the one-sided fractional maximal function is introduced extending the work of Martín-Reyes and de la Torre.Improving some extrapolation result for the one-sided case,we get weak sharp bounded estimates for one-sided fractional maximal function and weak and strong sharp bounded estimates for one-sided fractional integral.     

Sharp error bounds for Newton's process

Summary The method of nondiscrete mathematical induction is applied to the Newton process. The method yields a very simple proof of the convergence and sharp apriori estimates; it also gives aposteriori bounds which are, in general, better than those given in [1].     

Sharp bounds for the first non-zero Stekloff eigenvalues

Let (M,,) be an n(2)-dimensional compact Riemannian manifold with boundary and non-negative Ricci curvature. Consider the following two Stekloff eigenvalue problems

where Δ is the Laplacian operator on M and ν denotes the outward unit normal on ∂M. The first non-zero eigenvalues of the above problems will be denoted by p₁ and q₁, respectively. In the present paper, we prove that if the principle curvatures of the second fundamental form of ∂M are bounded below by a positive constant c, then with equality holding if and only if Ω is isometric to an n-dimensional Euclidean ball of radius , here λ₁ denotes the first non-zero eigenvalue of the Laplacian of ∂M. We also show that if the mean curvature of ∂M is bounded below by a positive constant c then q₁nc with equality holding if and only if M is isometric to an n-dimensional Euclidean ball of radius . Finally, we show that q₁A/V and that if the equality holds and if there is a point x₀∂M such that the mean curvature of ∂M at x₀ is no less than A/{nV}, then M is isometric to an n-dimensional Euclidean ball, being A and V the area of ∂M and the volume of M, respectively.     

Sharp bounds on the density,Green function and jumping function of subordinate killed BM

Subordination of a killed Brownian motion in a domain D^d via an /2-stable subordinator gives rise to a process Z_t whose infinitesimal generator is –(–|_D)^/2, the fractional power of the negative Dirichlet Laplacian. In this paper we establish upper and lower estimates for the density, Green function and jumping function of Z_t when D is either a bounded C^1,1 domain or an exterior C^1,1 domain. Our estimates are sharp in the sense that the upper and lower estimates differ only by a multiplicative constant.Mathematics Subject Classification (2000):Primary 60J45, Secondary 60J75, 31C25     

Sharp bounds for the valence of certain harmonic polynomials

In Khavinson and Swiatek (2002) it was proved that harmonic polynomials , where is a holomorphic polynomial of degree , have at most complex zeros. We show that this bound is sharp for all by proving a conjecture of Sarason and Crofoot about the existence of certain extremal polynomials . We also count the number of equivalence classes of these polynomials.