A note on the phase retrieval of holomorphic functions

We prove that if f and g are holomorphic functions on an open connected domain, with the same moduli on two intersecting segments, then $f=g$ up to the multiplication of a unimodular constant, provided the segments make an angle that is an irrational multiple of $\pi $ . We also prove that if f and g are functions in the Nevanlinna class, and if $|f|=|g|$ on the unit circle and on a circle inside the unit disc, then $f=g$ up to the multiplication of a unimodular constant.

Converse growth estimates for ODEs with slowly growing solutions

Let $$f_1,f_2$$ f 1 , f 2 be linearly independent solutions of $$f''+Af=0$$ f ′ ′ + A f = 0 , where the coefficient A is an analytic function in the open unit disc $${\mathbb {D}}$$ D of the complex plane $${\mathbb {C}}$$ C . It is shown that many properties of this differential equation can be described in terms of the subharmonic auxiliary function $$u=-\log \, (f_1/f_2)^{\#}$$ u = - log ( f 1 / f 2 ) # . For example, the case when $$\sup _{z\in {\mathbb {D}}} |A(z)|(1-|z|^2)^2 < \infty $$ sup z ∈ D | A ( z ) | ( 1 - | z | 2 ) 2 < ∞ and $$f_1/f_2$$ f 1 / f 2 is normal, is characterized by the condition $$\sup _{z\in {\mathbb {D}}} |\nabla u(z)|(1-|z|^2) < \infty $$ sup z ∈ D | ∇ u ( z ) | ( 1 - | z | 2 ) < ∞ . Different types of Blaschke-oscillatory equations are also described in terms of harmonic majorants of u. Even if $$f_1,f_2$$ f 1 , f 2 are bounded linearly independent solutions of $$f''+Af=0$$ f ′ ′ + A f = 0 , it is possible that $$\sup _{z\in {\mathbb {D}}} |A(z)|(1-|z|^2)^2 = \infty $$ sup z ∈ D | A ( z ) | ( 1 - | z | 2 ) 2 = ∞ or $$f_1/f_2$$ f 1 / f 2 is non-normal. These results relate to sharpness discussion of recent results in the literature, and are succeeded by a detailed analysis of differential equations with bounded solutions. Analogues for the Nevanlinna class are also considered, by taking advantage of Nevanlinna interpolating sequences. It is shown that, instead of considering solutions with prescribed zeros, it is possible to construct a bounded solution of $$f''+Af=0$$ f ′ ′ + A f = 0 in such a way that it solves an interpolation problem natural to bounded analytic functions, while $$|A(z)|^2(1-|z|^2)^3\, dm(z)$$ | A ( z ) | 2 ( 1 - | z | 2 ) 3 d m ( z ) remains to be a Carleson measure.

From ℋ∞ to 𝒩. Pointwise properties and algebraic structure in the Nevanlinna class

This survey shows how, for the Nevanlinna class 𝒩 of the unit disc, one can define and often characterize the analogues of well-known objects and properties related to the algebra of bounded analytic functions ℋ∞: interpolating sequences, Corona theorem, sets of determination, stable rank, as well as the more recent notions of Weak Embedding Property and threshold of invertibility for quotient algebras. The general rule we observe is that a given result for ℋ∞ can be transposed to 𝒩 by replacing uniform bounds by a suitable control by positive harmonic functions. We show several instances where this rule applies, as well as some exceptions. We also briefly discuss the situation for the related Smirnov class.