Functions in Bloch-type spaces and their moduli

Given a suitably regular nonnegative function $\omega$ on $(0,1]$, let $\mathcal B_\omega$ denote the space of all holomorphic functions $f$ on the unit ball $\mathbb B_n$ of $\mathbb C^n$ that satisfy $$|\nabla f(z)|\le C\frac{\omega(1-|z|)}{1-|z|},\qquad z\in\mathbb B_n,$$ with some fixed $C=C_f>0$. We obtain a new characterization of $\mathcal B_\omega$ functions in terms of their moduli.Comment: 9 pages; to appear in Ann. Acad. Sci. Fenn. Math. 41 (2016), No.

On the singular factor of a linear combination of holomorphic functions

We prove that the linear combinations of functions $f_0,...,f_n$ in $H^\infty$ have "few" singular inner factors, provided that the $f_j$'s are suitably smooth up to the boundary, while in general this is no longer true.Comment: 4 page

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.Comment: 9 page

Remembering Victor Petrovich Havin

These are reminiscences of V. P. Havin (1933--2015), founder of the modern St. Petersburg analysis school.Comment: 6 pages; to appear in the memorial volume "Tribute to Victor Havin: 50 years with Hardy spaces", Birkh\"auser Verlag, Base