Growth estimates for logarithmic derivatives of Blaschke products and of functions in the Nevanlinna class

Abstract

We prove growth estimates for logarithmic derivatives of functions in the Nevanlinna class. Blaschke products with radially restricted zero sequences will be of particular interest. Our results are sharper in a certain sense than the corresponding estimates in [2] obtained for meromorphic functions in the unit disc. E 1 dr 1 À r < y; ð1:1Þ such that for all z A D satisfying jzj B E 1 , we have 263 2000 Mathematics Subject Classification: Primary 30D50; Secondary 30A10.

Sharp Logarithmic Derivative Estimates With Applications to Ordinary Differential Equations in the Unit Disc

“…Sharp growth estimates for the maximum modulus of the generalized logarithmic derivative f (k) / f ( j) , where f is meromorphic in D and k and j are integers satisfying k > j ≥ 0, are obtained in [3,8]. The special cases where f either belongs to the Nevanlinna class or is a Blaschke product are further discussed in [9]. In addition, growth estimates for integral means of generalized logarithmic derivatives of meromorphic functions are obtained in [4,10].…”

“…The special cases where f either belongs to the Nevanlinna class or is a Blaschke product are further discussed in [9]. In addition, growth estimates for integral means of generalized logarithmic derivatives of meromorphic functions are obtained in [4,10].…”

Entire Functions of Bounded L-Index: Its Zeros and Behavior of Partial Logarithmic Derivatives

“…The first problem is the establishment of sharp estimates for the logarithmic derivatives of the functions in the unit disc outside some exceptional set. Chyzhykov et al [13][14][15][16] considered various formulations of the problem. The obtained estimates were used to study properties of holomorphic solutions of differential equations.…”