On a conjecture of Z. Jianzhong

Abstract

Let ϕ be a nonnegative, nondecreasing and nonconstant function defined on [0, ∞) such that Φ(t) = ϕ(et) is a convex function on (-∞, ∞). The Hardy-Orlicz space H (ϕ) is defined to be the class of all those functions f holomorphic in the open unit disc of the complex plane C satisfying The subclass H(ϕ)+ of H(ϕ) is defined to be the class of all those functions f ∈ H(ϕ) satisfying for almost all points eit of the unit circle. In 1990, Z. Jianzhong conjectured that H(ϕ)+ = H(ψ)+ if and only if H(ϕ) = H(ψ). In the present paper we prove that it is true not only on the unit disc of C but also on the unit ball of Cn.