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Konstantin M. Dyakonov

Publications and source records attributed to Konstantin M. Dyakonov.

31 records · Page 2Linked to original sources
arXiv2017

Boundary Gauss--Lucas type theorems on the disk

The classical Gauss--Lucas theorem describes the location of the critical points of a polynomial. There is also a hyperbolic version, due to Walsh, in which the role of polynomials is played by finite Blaschke products on the unit disk. We consider similar phenomena for generic inner functions, as well as for certain "locally inner" self-maps of the disk. More precisely, we look at a unit-norm function $f\in H^\infty$ that has an angular derivative on a set of positive measure (on the boundary) and we assume that its inner factor, $I$, is nontrivial. Under certain conditions to be discussed, it follows that $f'$ must also have a nontrivial inner factor, say $J$, and we study the relationship between the boundary singularities of $I$ and $J$. Examples are furnished to show that our sufficient conditions cannot be substantially relaxed.

math.CV↗
arXiv2016

Functions in Bloch-type spaces and their moduli

Given a suitably regular nonnegative function $ω$ on $(0,1]$, let $\mathcal B_ω$ denote the space of all holomorphic functions $f$ on the unit ball $\mathbb B_n$ of $\mathbb C^n$ that satisfy $$|\nabla f(z)|\le C\frac{ω(1-|z|)}{1-|z|},\qquad z\in\mathbb B_n,$$ with some fixed $C=C_f>0$. We obtain a new characterization of $\mathcal B_ω$ functions in terms of their moduli.

math.CV↗
arXiv2014

A characterization of Möbius transformations

We prove that the derivative $θ'$ of an inner function $θ$ is outer if and only if $θ$ is a Möbius transformation. An alternative characterization involving a reverse Schwarz--Pick type estimate is also given.

math.CV↗
arXiv2014

Two problems on coinvariant subspaces of the shift operator

Two problems are posed that involve the star-invariant subspace $K^p_θ$ (in the Hardy space $H^p$) associated with an inner function $θ$. One of these asks for a characterization of the extreme points of the unit ball in $K^\infty_θ$, while the other concerns the Fermat equation $f^n+g^n=h^n$ in $K^p_θ$.

math.FA↗
arXiv2013

A reverse Schwarz--Pick inequality

We prove a kind of "reverse Schwarz--Pick lemma" for holomorphic self-maps of the disk. The result becomes especially clear-cut for inner functions and casts new light on their derivatives.

math.CV↗
arXiv2012

Wronskians and deep zeros of holomorphic functions

Given linearly independent holomorphic functions $f_0,...,f_n$ on a planar domain $Ω$, let $\mathcal E$ be the set of those points $z\inΩ$ where a nontrivial linear combination $\sum_{j=0}^nλ_jf_j$ may have a zero of multiplicity greater than $n$, once the coefficients $λ_j=λ_j(z)$ are chosen appropriately. An elementary argument involving the Wronskian $W$ of the $f_j$'s shows that $\mathcal E$ is a discrete subset of $Ω$ (and is actually the zero set of $W$); thus "deep" zeros are rare. We elaborate on this by studying similar phenomena in various function spaces on the unit disk, with more sophisticated boundary smallness conditions playing the role of deep zeros.

math.CV↗
arXiv2012

Zeros of analytic functions, with or without multiplicities

The classical Mason-Stothers theorem deals with nontrivial polynomial solutions to the equation $a+b=c$. It provides a lower bound on the number of distinct zeros of the polynomial $abc$ in terms of the degrees of $a$, $b$ and $c$. We extend this to general analytic functions living on a reasonable bounded domain $Ω\subset\mathbb C$, rather than on the whole of $\mathbb C$. The estimates obtained are sharp, for any $Ω$, and a generalization of the original result on polynomials can be recovered from them by a limiting argument.

math.CV↗
arXiv2012

ABC-type estimates via Garsia-type norms

We are concerned with extensions of the Mason--Stothers $abc$ theorem from polynomials to analytic functions on the unit disk $\mathbb D$. The new feature is that the number of zeros of a function $f$ in $\mathbb D$ gets replaced by the norm of the associated Blaschke product $B_f$ in a suitable smoothness space $X$. Such extensions are shown to exist, and the appropriate $abc$-type estimates are exhibited, provided that $X$ admits a "Garsia-type norm", i.e., a norm sharing certain properties with the classical Garsia norm on BMO. Special emphasis is placed on analytic Lipschitz spaces.

math.CV↗
arXiv2012

Factoring derivatives of functions in the Nevanlinna and Smirnov classes

We prove that, given a function $f$ in the Nevanlinna class $N$ and a positive integer $n$, there exist $g\in N$ and $h\in BMOA$ such that $f^{(n)}=gh^{(n)}$. We may choose $g$ to be zero-free, so it follows that the zero sets for the class $N^{(n)}:=\{f^{(n)}: f\in N\}$ are the same as those for $BMOA^{(n)}$. Furthermore, while the set of all products $gh^{(n)}$ (with $g$ and $h$ as above) is strictly larger than $N^{(n)}$, we show that the gap is not too large, at least when $n=1$. Precisely speaking, the class $\{gh': g\in N, h\in BMOA\}$ turns out to be the smallest ideal space containing $\{f': f\in N\}$, where "ideal" means invariant under multiplication by $H^\infty$ functions. Similar results are established for the Smirnov class $N^+$.

math.CV↗
arXiv2010

Local ABC theorems for analytic functions

The classical $abc$ theorem for polynomials (often called Mason's theorem) deals with nontrivial polynomial solutions to the equation $a+b=c$. It provides a lower bound for the number of distinct zeros of the polynomial $abc$ in terms of $°{a}$, $°{b}$, and $°{c}$. We prove some "local" $abc$-type theorems for general analytic functions living on a reasonable bounded domain $Ω\subset\mathbb C$, rather than on the whole of $\mathbb C$. The estimates obtained are sharp, for any $Ω$, and they imply (a generalization of) the original "global" $abc$ theorem by a limiting argument.

math.CV↗
arXiv2010

Blaschke products and nonideal ideals in higher order Lipschitz algebras

We investigate certain ideals (associated with Blaschke products) of the analytic Lipschitz algebra $A^α$, with $α>1$, that fail to be "ideal spaces". The latter means that the ideals in question are not describable by any size condition on the function's modulus. In the case where $α=n$ is an integer, we study this phenomenon for the algebra $H^\infty_n=\{f:f^{(n)}\in H^\infty\}$ rather than for its more manageable Zygmund-type version. This part is based on a new theorem concerning the canonical factorization in $H^\infty_n$.

math.CV↗