相关论文: The best bounds of harmonic sequence

The upper estimates for the optimal constants of the multilinear Bohnenblust--Hille inequality obtained in [J. Funct. Anal. 264 (2013), 429--463] are here improved to: {0.1cm} {enumerate} For real scalars:…

The generalized hyperharmonic numbers $h_n^{(m)}(k)$ are defined by means of the multiple harmonic numbers. We show that the hyperharmonic numbers $h_n^{(m)}(k)$ satisfy certain recurrence relation which allow us to write them in terms of…

The EKG or electrocardiogram sequence is defined by a(1) = 1, a(2) = 2 and, for n >= 3, a(n) is the smallest natural number not already in the sequence with the property that gcd {a(n-1), a(n)} > 1. In spite of its erratic local behavior,…

In 1974, Gautschi proved an intriguing inequality involving the gamma function $\Gamma$. Precisely, he proved that, for $z>0$, the harmonic mean of $\Gamma(z)$ and $\Gamma(1/z)$ can never be less than 1. In 2017, Alzer and Jameson extended…

The solutions of a kind of second-order homogeneous partial differential equation are called (real kernel) alpha-harmonic functions. The alpha-harmonic functions and their first-order partial derivative functions on unit disk are estimated…

The main aim of this paper is to prove that the double inequality \frac{(k-1)!}{\Bigl\{x+\Bigl[\frac{(k-1)!}{|\psi^{(k)}(1)|}\Bigr]^{1/k}\Bigr\}^k}…

Let $d(n)$ be the number of divisors of $n$, let $$ \Delta(x) := \sum_{n\le x}d(n) - x(\log x + 2\gamma -1) $$ denote the error term in the classical Dirichlet divisor problem, and let $\zeta(s)$ denote the Riemann zeta-function. Several…

We consider the sum $\sum 1/\gamma$, where $\gamma$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in an interval $(0,T]$, and consider the behaviour of the sum as $T \to\infty$. We show that, after subtracting a…

Given $N\geq 3,$ we consider the critical Hardy-Sobolev equation $-\Delta u-\frac{\gamma}{|x|^2}u=\frac{|u|^{2^*(s)-2}u}{|x|^s}$ in $\mathbb{R}^N\setminus \{0\},$ where $0<\gamma<\gamma_{H}:=\left(\frac{N-2}{2}\right)^2,\,s\in (0,2)$ and…

Assuming the Generalized Riemann Hypothesis, we provide uniform upper bounds with explicit main terms for moduli of $\left(\cL'/\cL\right)(s)$ and $\log{\cL(s)}$ for $1/2+\delta\leq\sigma<1$, fixed $\delta\in(0,1/2)$ and for functions in…

The Ulam sequence is defined as $a_1 =1, a_2 = 2$ and $a_n$ being the smallest integer that can be written as the sum of two distinct earlier elements in a unique way. This gives $$1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47,…

Neither the Euler-Mascheroni constant, $\gamma=0.577215\ldots$, nor the Euler-Gompertz constant, $\delta=0.596347\ldots$, is currently known to be irrational. However, it has been proved that these two numbers are disjunctively…

For $t \in [-1, 1)$, a set of points on the $(n-1)$-dimensional unit sphere is called $t$-almost equiangular if among any three distinct points there is a pair with inner product $t$. We propose a semidefinite programming upper bound for…

Let $C=\inf (k/n)\sum_{i=1}^n x_i(x_{i+1}+\dots+x_{i+k})^{-1}$, where the infimum is taken over all pairs of integers $n\geq k\geq 1$ and all positive $x_1,\dots,x_{n+k}$ subject to cyclicity assumption $x_{n+i}=x_i$, $i=1,\dots,k$. We…

We establish the best (minimum) constant for Ulam stability of first-order linear $h$-difference equations with a periodic coefficient. First, we show Ulam stability and find the Ulam stability constant for a first-order linear equation…

We improve the classical discrete Hardy inequality \begin{equation*}\label{1} \sum _{{n=1}}^{\infty }a_{n}^{2}\geq \left({\frac {1}{2}}\right)^{2} \sum _{{n=1}}^{\infty }\left({\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}\right)^{2},…

We establish the optimal regularity of viscosity solutions to \begin{equation*} u_t - x_n^\gamma \Delta u = f, \end{equation*} which arises in the regularity theory for the porous medium equation. Specifically, we prove that under the zero…

Let $[x]$ be the greatest integer not exceeding $x$. In the paper we introduce the sequence $\{U_n\}$ given by $U_0=1$ and $U_n=-2\sum_{k=1}^{[n/2]}\binom n{2k}U_{n-2k}\quad(n\ge 1)$, and establish many recursive formulas and congruences…

Optimal pointwise estimates are derived for the biharmonic Green function under Dirichlet boundary conditions in arbitrary $C^{4,\gamma}$-smooth domains. Maximum principles do not exist for fourth order elliptic equations and the Green…

We find the sharp constants $C_p$ and the sharp functions $C_p=C_p(x)$ in the inequality $$|u(x)|\leq \frac{C_p}{(1-|x|^2)^{(n-1)/p}}\|u\|_{h^p(B^n)}, u\in h^p(B^n), x\in B^n,$$ in terms of Gauss hypergeometric and Euler functions. This…