Zero distribution and factorization of analytic functions of slow growth in the unit disc

Abstract

For a meromorphic function f f in the unit disc, let the ρ ∞ \rho _\infty -order of the growth be the limit of the orders of L p L_p -norms of log ⁡ | f ( r e i θ ) | \log |f(re^{i\theta })| over the circle. In the case when the order of the maximum modulus function is smaller than 1, we describe zero distribution of canonical products and derive a new factorization theorem and logarithmic derivative estimates.