Related papers: Sharp double inequality for complete elliptic inte…

We study the symmetry/asymmetry of functions providing sharp constants in the embedding theorems ${\stackrel{\circ}{W}}\vphantom{W}_2^r(-1,1)\hookrightarrow{\stackrel{\circ}{W}}\vphantom{W}_\infty^k(-1,1)$ for various $r$ and $k$. The sharp…

We find two series expansions for Legendre's second incomplete elliptic integral $E(\lambda, k)$ in terms of recursively computed elementary functions. Both expansions converge at every point of the unit square in the $(\lambda, k)$ plane.…

We prove the conjecture stated in F. Qi and R. Agarwal, \textit{On complete monotonicity for several classes of functions related to ratios of gamma functions}, J. Inequal. Appl. (2019), 1-42, that the function $1/\arctan$ is…

A closed-form formula is derived for the generalized Clebsch-Gordan integral $ \int_{-1}^1 {[}P_{\nu}(x){]}^2P_{\nu}(-x)\D x$, with $ P_\nu$ being the Legendre function of arbitrary complex degree $ \nu\in\mathbb C$. The finite Hilbert…

This is the first in our series of papers concerning some Hardy-Littlewood-Sobolev type inequalities. In the present paper, the main objective is to establish the following sharp reversed HLS inequality in the whole space $\mathbb R^n$…

Let $E$ be an elliptic curve defined over a number field $K$, let $\alpha \in E(K)$ be a point of infinite order, and let $N^{-1}\alpha$ be the set of $N$-division points of $\alpha$ in $E(\bar{K})$. We prove strong effective and uniform…

In the present article we prove second-order and Lipschitz regularity for quasilinear elliptic equations in metric spaces endowed with a lower bound on the Ricci curvature. The estimates we obtain are quantitative and cover a large class of…

A two-parameter generalization of the complete elliptic integral of second kind is expressed in terms of the Appell function $F_{4}$. This function is further reduced to a quite simple bilinear form in the complete elliptic integrals $K$…

In this paper, exact rate of approximation of functions by linear means of Fourier series and Fourier integrals and corresponding $K$-functionals are expressed via special moduli of smoothness. . Introduction is given in $\S 1$. In $\S2$…

We consider divergence form elliptic operators L = - div A(x)\nabla, defined in the half space R^{n+1}_+, n \geq 2, where the coefficient matrix A(x) is bounded, measurable, uniformly elliptic, t-independent, and not necessarily symmetric.…

We implement a probe counterpart of Newman-Janis algorithm, which Wick rotates the all-orders geodesic deviation equation into a part of exact spinning-particle equations of motion. Consequently, the gravitational dynamics of the Kerr black…

Given $N\geq 3$, $1<p<N$, two measurable functions $V\left(r \right)\geq 0$, $K\left(r\right)> 0$ and a continuous function $A(r) >0$ ($r>0$), we study the quasilinear elliptic equation \[ -\mathrm{div}\left(A(|x| )|\nabla u|^{p-2} \nabla…

We compute the optimal constant for a generalized Hardy-Sobolev inequality, and using the product of two symmetrizations we present an elementary proof of the symmetries of some optimal functions. This inequality was motivated by a…

We prove $q$-analogues of identities that are equivalent to the functional equation of the arithmetic-geometric mean. We also present $q$-analogues of $F(\sqrt{k},\frac{\pi}{2})$, the complete elliptical integral of the first kind, and its…

We study the radius of absolute monotonicity R of rational functions with numerator and denominator of degree s that approximate the exponential function to order p. Such functions arise in the application of implicit s-stage, order p…

We establish arithmeticity in the sense of A. Katok and F. Rodriguez Hertz of smooth actions $\alpha$ of $\mathbb{R}^k$ on an anonymous manifold $M$ of dimension $2k+1$ provided that there is an ergodic invariant Borel probability measure…

We embark on a systematic study of the $(k+1)$-th derivative of $x^{k-r}H(x^r)$, where $H(x):=-x\log x-(1-x)\log(1-x)$ is the binary entropy and $k>r\geq 1$ are integers. Our motivation is the conjectural entropy inequality $\alpha_k…

A generalization of Jacobi's elliptic functions is introduced as inversions of hyperelliptic integrals. We discuss the special properties of these functions, present addition theorems and give a list of indefinite integrals. As a physical…

We consider the following question: if a function of the form $\int_0^{\infty}\varphi(t)\, e^{-xt}dt$ is completely monotonic, is it then $\varphi\ge0$? It turns out that the question is related to a moment problem. In the end we apply…

This is the second in our series of papers concerning some reversed Hardy--Littlewood--Sobolev inequalities. In the present work, we establish the following sharp reversed Hardy--Littlewood--Sobolev inequality on the half space $\mathbb…