Related papers: Optimal evaluations for the S\'{a}ndor-Yang mean b…

In this paper, we establish some nontrivial and effective upper bounds for the least common multiple of consecutive terms of a finite arithmetic progression. Precisely, we prove that for any two coprime positive integers $a$ and $b$, with…

The arithmetic properties of the second order mock theta function $\mathcal{B}(q)$, introduced by McIntosh, defined by \begin{equation*} \mathcal{B}(q) := \sum_{n \geq 0} \frac{q^n (-q;q^2)_n}{(q;q^2)_{n+1}} = \sum_{n \geq 0}b(n)q^n,…

It is known that if $M,\,N$ are continuous two-variable means such that $|M(x,y)-N(x,y)| < |x-y|$ for every $x,\ y$ with $x\ne y$, then there exists a unique invariant mean (which is continuous too). We are looking for invariant means for…

Let $n,k,a$ and $c$ be positive integers and $b$ be a nonnegative integer. Let $\nu_2(k)$ and $s_2(k)$ be the 2-adic valuation of $k$ and the sum of binary digits of $k$, respectively. Let $S(n,k)$ be the Stirling number of the second kind.…

We calculate the general planar dual-conformally invariant double-pentagon and pentabox integrals in four dimensions. Concretely, we derive one-fold integral representations for these elliptic integrals over polylogarithms of weight three.…

In this paper, we obtain some new inequalities for ({\alpha},m)-convex functions. The analysis used in the proofs is fairly elementary and based on the use of Power-mean inequality.

Using recently evaluated contributions (including a novel one calculated here), we present updated values for the pole mass and $\bar{MS}$ mass of the $b$ quark: $m_b=5022\pm58$ MeV, for the pole mass, and $\bar{m}_b(\bar{m}_b)=4286\pm36$…

In the present paper we initiate the study of a certain kind of partition inequality, by showing, for example, that if $M\geq 5$ is an integer and the integers $a$ and $b$ are relatively prime to $M$ and satisfy $1\leq a<b<M/2$, and the…

We give a necessary and sufficient mean condition for the quotient of two Jensen functionals and define a new class $\Lambda_{f,g}(a, b)$ of mean values where $f, g$ are continuously differentiable convex functions satisfying the relation…

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. Suppose that $a$ and $q$ are positive integers satisfying $(a,q)=1$. Denote by $\mathcal{P}_2(a,q)$ the least almost-prime…

For two coprime positive integers $a,b$, let $T(a,b)=\{ ax+by : x,y\in \mathbb{Z}_{\ge 0} \} $ and let $s(a,b)=ab-a-b$. It is well known that all integers which are greater than $s(a,b)$ are in $T(a,b)$. Let $\pi (a, b)$ be the number of…

In this article we consider a modification of the Stein's spherical maximal operator of complex order $\alpha$ on ${\mathbb R^n}$: $$ {\mathfrak M}^\alpha_{[1,2]} f(x) =\sup\limits_{t\in [1,2]} \big| {1\over \Gamma(\alpha) } \int_{|y|\leq…

We consider a two weight $L^{p}(\mu) \to L^{q}(\nu)$-inequality for well localized operators as defined and studied by F. Nazarov, S. Treil and A. Volberg when $p=q=2$. A counterexample of F. Nazarov shows that the direct analogue of these…

We prove that if p is a prime with a primitive root 2 then S_p(2^p)=p and give a sufficient condition for an equality of kind S_p(2^p)=+or-p.

We consider the nonplanar universal anomalous dimension of twist-two operators at four loops in N=4 supersymmetric Yang-Mills theory and push its direct diagrammatic calculation through Lorentz spin j=20, one unit beyond the state of the…

Let $M_\gamma$ be precise a priori majorant of first eigenvalues of Sturm--Liouville problems $-y"+qy=\lambda y,\quad y(0)=y(1)=0$, where $q\leqslant 0$ and $\int_0^1 |q|^\gamma\,dx=1$, $\gamma\in (0,1/2)$. It is shown that the inequality…

In this paper, we prove that the double inequality \begin{equation*} 1+\alpha r'^2<\frac{\mathcal{K}_{a}(r)}{\sin(\pi a)\log(e^{R(a)/2}/r')}<1+\beta r'^2 \end{equation*} holds for all $a\in (0, 1/2]$ and $r\in (0, 1)$ if and only if…

For $0\leq \alpha<1$ and prime number $p$ let $L(\alpha,p)$ be the sum of the first $[\alpha p]$ values of Legendre symbol modulo $p$. We study positivity of $L(\alpha,p)$ and prove that for $|\alpha-\frac13|<2\cdot 10^{-6}$ and for…

We prove some refinements of a reverse AM-GM operator inequality due to M. Lin [Studia Math. 2013;215:187-194]. In particular, we show the operator inequality \begin{eqnarray*} \Phi^p\left(A\nabla_\nu B+2rMm(A^{-1}\nabla B^{-1}-A^{-1}\sharp…

Let $f(x, y) \in \mathbb{Z}[x, y]$ be a cubic form with non-zero discriminant, and for each integer $m \in \mathbb{Z}$, let, $N_{f}(m)=\#\left\{(x, y) \in \mathbb{Z}^{2}: f(x, y)=m\right\} $. In 1983, Silverman proved that…