Related papers: Growth estimates for meromorphic solutions of high…
We obtain Schwarz-Pick lemma for $(\alpha, \beta)$-harmonic functions u in the disc, where $\alpha$ and $\beta$ are complex parameters satisfying $\Re \alpha + \Re \beta > -1$. We prove sharp estimate of derivative at the origin for such…
The full ``classical" Dirac-Maxwell equations are considered under various simplifying assumptions. A reduction of the equations is performed in the case when the Dirac field is {\em static} and a further reduction is performed in the case…
Let G be a piecewise constant $n\times n$ matrix function which is defined on a smooth closed curve $\Gamma$ in the complex sphere and which has m jumps. We consider the problem of determining the partial indices of the factorization of the…
Fixing an arithmetic lattice $\Gamma$ in an algebraic group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $\Delta$ with $[\Gamma : \Gamma \cap \Delta] [\Delta: \Gamma \cap \Delta] =…
In this paper, we are concerned with the following equation involving higher-order fractional Lapalacian \begin{equation*} \left\{\begin{aligned} &(-\Delta)^{p+{\frac{\alpha}{2}}}u(x)=u_+^\gamma~~ \mbox{ in }\mathbb{R}^n,\\…
Fractional vector calculus is discussed in the spherical coordinate framework. A variation of the Legendre equation and fractional Bessel equation are solved by series expansion and numerically. Finally, we generalize the hypergeometric…
This paper studies the upper and lower bounds of the lifespan for the classical solutions to the initial value problems of one dimensional wave equations with non-autonomous semilinear terms including the space-derivative of the unknown…
Let $U\not\equiv \pm\infty$ be a $\delta$-subharmonic function on a closed disc of radius $R$ centered at zero. In the previous two parts of our paper, we obtained general and explicit estimates of the integral of the positive part of the…
We prove square function estimates for certain conical regions. Specifically, let $\{\Delta_j\}$ be regions of the unit sphere $\mathbb{S}^{n-1}$ and let $S_j f$ be the smooth Fourier restriction of $f$ to the conical region…
We establish the existence of a deterministic exponential growth rate for the norm (on an appropriate function space) of the solution of the linear scalar stochastic delay equation dX(t) = X(t-1) dW(t) which does not depend on the initial…
We prove maximum and comparison principles for fractional discrete derivatives in the integers. Regularity results when the space is a mesh of length $h$, and approximation theorems to the continuous fractional derivatives are shown. When…
Given a two-variable function $f$ without critical points and a compact region $R$ bounded by two level curves of $f$, this note proves that the integral over $R$ of the second-order directional derivative of $f$ in the tangential…
The purpose of this paper is to investigate the non-constant entire as well as meromorphic solutions of the Fermat-type partial differential-difference equation: \[\left(\sum_{j=1}^m\frac{\partial f(z_1, z_2, \ldots, z_m)}{\partial…
A polynomial-in-time growth bound is established for global Sobolev $H^s(\mathbb T)$ solutions to the derivative nonlinear Schr\"odinger equation on the circle with $s>1$. These bounds are derived as a consequence of a nonlinear smoothing…
Estimates are obtained for the initial coefficients of a normalized analytic function $f$ in the unit disk $\mathbb{D}$ such that $f$ and the analytic extension of $f^{-1}$ to $\mathbb{D}$ belong to certain subclasses of univalent…
Recently in [M. Hairer, M. Hutzenthaler, and A. Jentzen, Ann. Probab. 43, 2 (2015), 468--527] and [A. Jentzen, T. M\"uller-Gronbach, and L. Yaroslavtseva, Commun. Math. Sci. 14, 6 (2016), 1477--1500] stochastic differential equations (SDEs)…
We study the growth of the quantity $\int_{\mathbb{T}}|R'(z)|\,dm(z)$ for rational functions $R$ of degree $n$, which are bounded and univalent in the unit disk, and prove that this quantity may grow as $n^\gamma$, $\gamma>0$, when…
We determine the optimal universal growth scale for the integral means of normalized logarithmic derivatives in the Carath\'eodory class. This resolves a problem of D.~K.~Thomas.
This article aims at finding sufficient conditions for a family of meromorphic functions to be normal by involving partial sharing of sets with differential polynomials. Moreover, corresponding results for normal meromorphic functions are…
We obtain a stochastic differential equation (SDE) satisfied by the first $n$ coordinates of a Brownian motion on the unit sphere in $\mathbb{R}^{n+\ell}$. The SDE has non-Lipschitz coefficients but we are able to provide an analysis of…