1 results
On non-separated zero sequences of solutions of a linear differential equation
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 64 / Issue 2 / May 2021
- Published online by Cambridge University Press:
- 30 April 2021, pp. 247-261
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- Article
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Let $(z_k)$
be a sequence of distinct points in the unit disc $\mathbb {D}$
without limit points there. We are looking for a function $a(z)$
analytic in $\mathbb {D}$
and such that possesses a solution having zeros precisely at the points $z_k$
, and the resulting function $a(z)$
has ‘minimal’ growth. We focus on the case of non-separated sequences $(z_k)$
in terms of the pseudohyperbolic distance when the coefficient $a(z)$
is of zero order, but $\sup _{z\in {\mathbb D}}(1-|z|)^p|a(z)| = + \infty$
for any $p > 0$
. We established a new estimate for the maximum modulus of $a(z)$
in terms of the functions $n_z(t)=\sum \nolimits _{|z_k-z|\le t} 1$
and $N_z(r) = \int_0^r {{(n_z(t)-1)}^ + } /t{\rm d}t.$
The estimate is sharp in some sense. The main result relies on a new interpolation theorem.
