Related papers: Sharp Bohr-Rogosinski radii for Schwarz functions …

A starlike univalent function $f$ is characterized by the function $zf'(z)/f(z)$; several subclasses of these functions were studied in the past by restricting the function $zf'(z)/f(z)$ to take values in a region $\Omega$ on the right-half…

Quantitative estimates are obtained for the (finite) valence of functions analytic in the unit disk with Schwarzian derivative that is bounded or of slow growth. A harmonic mapping is shown to be uniformly locally univalent with respect to…

Let $f$ be a complex-valued harmonic mapping defined in the unit disk $\mathbb D$. We introduce the following notion: we say that $f$ is a Bloch-type function if its Jacobian satisfies $$ \sup_{z\in\mathbb D}(1-|z|^2)\sqrt{|J_f(z)|}<\infty.…

We prove several improved versions of Bohr's inequality for the harmonic mappings of the form $f=h+\overline{g}$, where $h$ is bounded by 1 and $|g'(z)|\le|h'(z)|$. The improvements are obtained along the lines of an earlier work of Kayumov…

If $\Omega \subset \R^n$ is a smooth bounded domain and $q \in (0, \frac{n}{n-1})$ we consider the Poincare-Sobolev inequality \[ c \Bigl(\int_{\Omega} \abs{u}^\frac{n}{n-1}\Bigr)^{1-\frac{1}{n}} \le \int_{\Omega} \abs{Du}, \] for every $u…

A holomorphic function $f$ on the unit disc $\mathbb{D}$ belongs to the class $\mathcal{U}_A(\mathbb{D})$ of Abel universal functions if the family $\{f_r: 0\leq r<1\}$ of its dilates $f_r(z):=f(rz)$ is dense in the space of continuous…

Let ${\mathcal H}$ denote the class of all normalized complex-valued harmonic functions $f=h+\bar{g}$ in the unit disk ${\mathbb D}$, and let $K=H+\bar{G}$ denote the harmonic Koebe function. Let $a_n,b_n, A_n, B_n$ denote the Maclaurin…

We revisit the J-matrix method for the one dimensional radial harmonic oscillator (RHO) and construct its tridiagonal matrix representation within an orthonormal basis phi(z)n of L2 (R+);parametrized by a fixed z in the complex unit disc D…

Tight lower and upper bounds for the radius of univalence of some normalized Bessel, Struve and Lommel functions of the first kind are obtained via Euler-Rayleigh inequalities. It is shown also that the radius of univalence of the Struve…

Let $f$ be a nonzero holomorphic function in the unit ball $\mathbb B$ of the $n$-dimensional complex Euclidean space $\mathbb C^n$ such that the function $f$ vanishes on the set ${\sf Z}\subset \mathbb B$ and satisfies the constraint…

In this paper, we study the Bohr phenomenon for functions that are defined on a general simply connected domain of the complex plane. We improve known results of R. Fournier and St. Ruscheweyh for a class of analytic functions. Furthermore,…

There is a known generalization of the classical Schwarz lemma to holomorphic functions from the polydisk to the disk. In this paper, we characterize those functions which satisfy equality everywhere in this generalized inequality: they are…

This paper introduces the second Bohr radius for vector-valued holomorphic functions defined on arbitrary complete Reinhardt domains. We aim to establish the lower and upper bounds of the second Bohr radius in both finite and…

We establish certain fine properties for functions of bounded $\mathscr A$-variation known in the classical $BV$ setting. Here, $\mathscr A$ is a $k$th order constant-coefficient homogeneous linear differential operator with a…

In this paper we prove an isoperimetric inequality for holomorphic functions in the unit polydisc $\mathbf U^n$. As a corollary we derive an inclusion relation between weighted Bergman and Hardy spaces of holomorphic functions in the…

We describe the strong dual space $({\mathcal O} (D))^*$ for the space ${\mathcal O} (D)$ of holomorphic functions of several complex variables over a bounded Lipschitz domain $D$ with connected boundary $\partial D$ (as usual, ${\mathcal…

We study the distribution of families of multiplicative functions among the coprime residue classes to moduli varying uniformly in a wide range, obtaining analogues of the Siegel--Walfisz Theorem for large classes of multiplicative…

We establish a new description of the Schur-Agler norm of a holomorphic function on the polydisc as the solution of a convex optimization problem. Consequences of this description are explored both from a theoretical and from a practical…

Let $\Omega$ be a convex domain in the complex plane ${\mathbb C}$ with $\Omega \not= {\mathbb C}$, and $P$ be a conformal map of the unit disk ${\mathbb D}$ onto $\Omega$. Let ${\mathcal F}_\Omega$ be the class of analytic functions $g$ in…

For a function $f$ from the Sobolev space $W^{1,p}(C)$ ($C\subset\mathbb{R}^d$ is an open convex cone), a sharp inequality that estimates $\| f\|_{L_{\infty}}$ via the $L_{p}$-norm of its gradient and a seminorm of the function is obtained.…