Mean growth of the derivative of infinite blaschke products

Abstract

If f is an analytic function in the unit disc and 0<r<1, we set and we let n(r,f) denote the number of zeros of f; in the disc . We prove that if B is an interpolating Blaschke product with positive zeros then the quantities M ₁(r,B ¹) and n(r,B) are comparable and we use this result to prove that for any positive continuous function φ defined in [0,1) with φ(r)∞, as r←1, there exists an infinite Blaschke product B such that as r←1. This latter result was previously known but our proof is simpler than the original one. We also obtain simplified proofs of some other related results.

Keywords:

• Blaschke product
• Hardy spaces
• Theorem of Privalov
• Normal function

AMS Mathematics Subject Classifications 1991:

• 30D50
• 30D55

^*This research has been supported in part by a grant from “El Ministerio de Educacióon y Cultura, Spain” (PB97–1081) and by a grant from “La Junta de Andalucia” (FQM–210).

^*This research has been supported in part by a grant from “El Ministerio de Educacióon y Cultura, Spain” (PB97–1081) and by a grant from “La Junta de Andalucia” (FQM–210).

Notes

^*This research has been supported in part by a grant from “El Ministerio de Educacióon y Cultura, Spain” (PB97–1081) and by a grant from “La Junta de Andalucia” (FQM–210).