Related papers: A note on subordination

We consider the averages of a function $ f$ on $ \mathbb R ^{n}$ over spheres of radius $ 0< r< \infty $ given by $ A_{r} f (x) = \int_{\mathbb S ^{n-1}} f (x-r y) \; d \sigma (y)$, where $ \sigma $ is the normalized rotation invariant…

We relate two different solutions of a Mahler equation; one solution is only defined at certain roots of unity, while the other is an analytic function inside the unit disk.

Let $\mathbb{D}:=\{z\in \mathbb{C}: |z|<1\}$ be the unit disk. For $0<\alpha <1$, let $f_{\alpha}(z)=z/(1-z^\alpha)$ for $z \in \mathbb{D}$. We consider the class $\mathcal{F}$ of analytic functions $f_{\alpha}$ which satisfy $\Re…

Let $\A$ be the operator which assigns to each $m \times n$ matrix-valued function on the unit circle with entries in $H^\infty + C$ its unique superoptimal approximant in the space of bounded analytic $m \times n$ matrix-valued functions…

Normality arguments are applied to study the oscillation of solutions of $f''+Af=0$, where the coefficient $A$ is analytic in the unit disc $\mathbb{D}$ and $\sup_{z\in\mathbb{D}} (1-|z|^2)^2|A(z)| < \infty$. It is shown that such…

We obtain sharp upper bounds for three-term segments of a bounded power series. Along the way we show that the Taylor polynomials of a certain algebraic function do not vanish in the unit disk.

Submodular Functions are a special class of set functions, which generalize several information-theoretic quantities such as entropy and mutual information [1]. Submodular functions have subgradients and subdifferentials [2] and admit…

There is a universal constant $0<r_0<1$ with the following property. Suppose that $f$ is an analytic function on the unit disk $\D$, and suppose that there exists a constant $M>0$ so that the Euclidean area, counting multiplicity, of the…

Using unitary equivalence of magnetic translation operators, explicit upper and lower convex bounds on the partition function of the Hofstadter model are given for any rational ``flux" and any value of Bloch momenta. These bounds (i)…

Let $A$ be a square matrix and let $\Omega$ be an open set in the plane containing the spectrum of $A$. We consider the problem of maximizing the operator norm $\|f(A)\|$ amongst all holomorphic functions $f$ from $\Omega$ into the closed…

We introduce extremal affine surface areas in a functional setting. We show their main properties. Among them are linear invariance, isoperimetric inequalities and monotonicity properties. We establish a new duality formula, which shows…

Let $z_0$ and $w_0$ be given points in the open unit disk $\mathbb{D}$ with $|w_0| < |z_0|$. Let $\mathcal{H}_0$ be the class of all analytic self-maps $f$ of $\mathbb{D}$ normalized by $f(0)=0$, and $\mathcal{H}_0 (z_0,w_0) = \{ f \in…

This work concerns the study of the subdifferential of the integral functional $$ E_f(x)=\int_{T} f(t,x)d\mu(t), $$ where $f$ is a (not necessarily convex) normal integrand, $({T},\mathcal{A},\mu)$ is a $\sigma$-finite measure space, while…

Let $Co(\alpha)$ denote the class of concave univalent functions in the unit disk $\ID$. Each function $f\in Co(\alpha)$ maps the unit disk $\ID$ onto the complement of an unbounded convex set. In this paper we find the exact disk of…

We analyze Bergman spaces A p f (D) of generalized analytic functions of solutions to the Vekua equation $\partial$w = ($\partial$f /f)w in the unit disc of the complex plane, for Lipschitz-smooth non-vanishing real valued functions f and 1…

We derive an upper bound for the mean of the supremum of the empirical process indexed by a class of functions that are known to have variance bounded by a small constant $\delta$. The bound is expressed in the uniform entropy integral of…

In this paper, subordination results are studied for certain subclass of p-valent meromorphic functions in the punctured unit disc having a pole of order p at the origin. The subclass under investigation is defined by using certain new…

In this note, we construct an algorithm that, on input of a description of a structurally stable planar dynamical flow $f$ defined on the closed unit disk, outputs the exact number of the (hyperbolic) equilibrium points and their locations…

We obtain an upper bound for the derivative of a Blaschke product, whose zeros lie in a certain Stolz-type region. We show that the derivative belongs to the space of analytic functions in the unit disk, introduced recently in \cite{FG}. As…

Differential subordination and superordination preserving properties for univalent functions in the open unit disk with an operator involving generalized Bessel functions are derived. Some particular cases involving trigonometric functions…