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.     

Several polynomials associated with the harmonic numbers

We develop polynomials in z∈C for which some generalized harmonic numbers are special cases at z=0. By using the Riordan array method, we explore interesting relationships between these polynomials, the generalized Stirling polynomials, the Bernoulli polynomials, the Cauchy polynomials and the Nörlund polynomials.     

Isoperimetric type inequalities for harmonic functions

For 0<p<+∞ let hp be the harmonic Hardy space and let bp be the harmonic Bergman space of harmonic functions on the open unit disk U. Given 1?p<+∞, denote by ‖⋅bp_‖ and ‖⋅hp_‖ the norms in the spaces bp and hp, respectively. In this paper, we establish the harmonic hp-analogue of the known isoperimetric type inequality ‖fb2p_‖?‖fhp_‖, where f is an arbitrary holomorphic function in the classical Hardy space Hp. We prove that for arbitrary p>1, every function f∈hp satisfies the inequality

‖fb2p_‖?ap‖fhp_‖,     

Sharp integral inequalities for harmonic functions

Motivated by Carleman's proof of isoperimetric inequality in the plane, we study some sharp integral inequalities for harmonic functions on the upper half‐space. We also derive the regularity for nonnegative solutions of the associated integral system and some Liouville‐type theorems. © 2007 Wiley Periodicals, Inc.     

On the largest prime divisor of an odd harmonic number

A positive integer is called a (Ore's) harmonic number if its positive divisors have integral harmonic mean. Ore conjectured that every harmonic number greater than is even. If Ore's conjecture is true, there exist no odd perfect numbers. In this paper, we prove that every odd harmonic number greater than must be divisible by a prime greater than .

     

Generalized harmonic numbers,Jacobi numbers and a Hankel determinant evaluation

In this paper we will introduce a sequence of complex numbers that are called the Jacobi numbers. This sequence generalizes in a natural way several sequences that are known in the literature, such as Catalan numbers, central binomial numbers, generalized catalan numbers, the coefficient of the Hilbert matrix and others. Subsequently, using a study of the polynomial of Jacobi, we give an evaluation of the Hankel determinants that associated with the sequence of Jacobi numbers. Finally, by finding a relationship between the Jacobi numbers and generalized harmonic numbers, we determine the evaluation of the Hankel determinants that are associated with generalized harmonic numbers.     

The Dattoli-Srivastava conjectures concerning generating functions involving the harmonic numbers

In a recent paper Dattoli and Srivastava [3], by resorting to umbral calculus, conjectured several generating functions involving harmonic numbers. In this sequel to their work our aim is to rigorously demonstrate the truth of the Dattoli-Srivastava conjectures by making use of simple analytical arguments. In addition, one of these conjectures is stated and proved in more general form.     

Certain summation formulas involving harmonic numbers and generalized harmonic numbers

Harmonic numbers and generalized harmonic numbers have been studied since the distant past and involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics and theoretical physics. Here we aim at presenting further interesting identities about certain finite or infinite series involving harmonic numbers and generalized harmonic numbers by applying an algorithmic method to a known summation formula for the hypergeometric function ₅F₄(1).     

Sharp estimates for the covering numbers of the Weierstrass fractal kernel

In this paper, we present sharp estimates for the covering numbers of the embedding of the reproducing kernel Hilbert space (RKHS) associated with the Weierstrass fractal kernel into the space of continuous functions. The method we apply is based on the characterization of the infinite-dimensional RKHS generated by the Weierstrass fractal kernel and it requires estimates for the norm operator of orthogonal projections on the RKHS.     

Summation formulas involving generalized harmonic numbers

By means of the derivative operator and three hypergeometric series identities, several interesting summation formulas involving generalized harmonic numbers are established.     

Sums of products of Cauchy numbers

In this paper, we consider a kind of sums involving Cauchy numbers, which have not been studied in the literature. By means of the method of coefficients, we give some properties of the sums. We further derive some recurrence relations and establish a series of identities involving the sums, Stirling numbers, generalized Bernoulli numbers, generalized Euler numbers, Lah numbers, and harmonic numbers. In particular, we generalize some relations between two kinds of Cauchy numbers and some identities for Cauchy numbers and Stirling numbers.     

Euler-Seidel method for certain combinatorial numbers and a new characterization of Fibonacci sequence

In this paper we use the Euler-Seidel method for deriving new identities for hyperharmonic and r-Stirling numbers. The exponential generating function is determined for hyperharmonic numbers, which result is a generalization of Gosper’s identity. A classification of second order recurrence sequences is also given with the help of this method.      

Product inequalities for norms of linear factors

We give an inequality which bounds the product of the Lp norms of the linear factors of a polynomial by a multiple of the Lp norm of that polynomial. This result generalizes two inequalities of Króo and Pritsker.     

Minton-Karlsson identities and summation formulae involving generalized harmonic numbers

By applying the derivative operator to Minton and Karlsson's hypergeometric series identities, several interesting summation formulae involving generalized harmonic numbers are established.     

Oracle inequalities for support vector machines that are based on random entropy numbers

In this paper, we present a new technique for bounding local Rademacher averages of function classes induced by a loss function and a reproducing kernel Hilbert space (RKHS). At the heart of this technique lies the observation that certain expectations of random entropy numbers can be bounded by the eigenvalues of the integral operator associated with the RKHS. We then work out the details of the new technique by establishing two new oracle inequalities for support vector machines, which complement and generalize previous results.     

Sharp thresholds for the phase transition between primitive recursive and Ackermannian Ramsey numbers

We compute the sharp thresholds on g at which g-large and g-regressive Ramsey numbers cease to be primitive recursive and become Ackermannian.We also identify the threshold below which g-regressive colorings have usual Ramsey numbers, that is, admit homogeneous, rather than just min-homogeneous sets.     

Approximation numbers of Sobolev embeddings—Sharp constants and tractability

We investigate optimal linear approximations (approximation numbers) in the context of periodic Sobolev spaces H^s(T^d)$H^s\left({\mathbb{T}}^d\right)$ of fractional smoothness s>0$s>0$ for various equivalent norms including the classical one. The error is always measured in _L2(T^d)$L_2\left({\mathbb{T}}^d\right)$. Particular emphasis is given to the dependence of all constants on the dimension d$d$. We capture the exact decay rate in n$n$ and the exact decay order of the constants with respect to d$d$, which is in fact polynomial. As a consequence we observe that none of our considered approximation problems suffers from the curse of dimensionality. Surprisingly, the square integrability of all weak derivatives up to order three (classical Sobolev norm) guarantees weak tractability of the associated multivariate approximation problem.     

Sharp inequalities for convolution-type operators

Let μ be a probability measure on [− a, a], a > 0, and let x₀ε[− a, a], f ε Cⁿ([−2a, 2a]), n 0 even. Using moment methods we derive best upper bounds to ¦∫_−a^a ([f(x₀ + y) + f(x₀ − y)]/2) μ(dy) − f(x₀)¦, leading to sharp inequalities that are attainable and involve the second modulus of continuity of f⁽ⁿ⁾ or an upper bound of it.     

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.