Comparison of modular spaces of analytic functions in the half-plane (Q1072062)

The author considers some kinds of modular function spaces of analytic functions. An increasing and convex function \(\psi\) (u) for \(u\geq 0\) is called an N-function, if it satisfies the following conditions \[ (1)\quad \lim_{u\to 0+}\psi (u)/u=0,\quad (2)\quad \lim_{u\to \infty}\psi (u)/u=\infty. \] Let \(\psi_ 1\) and \(\psi_ 2\) be two N- functions. We say that \(\psi_ 2\) is not weaker than \(\psi_ 1\) and write \(\psi_ 1<\psi_ 2\) if there exist constants \(a,b>0\) such that the inequality holds \(\psi_ 1(u)\leq a\psi_ 2(bu)\) for \(u\geq 0\). If \(\psi_ 1<\psi_ 2\) and \(\psi_ 2<\psi_ 1\) simultaneously, we write \(\psi_ 1\sim \psi_ 2.\) An N-function \(\psi\) is said to satisfy condition \(\Delta_ 2\), if for some constant \(\alpha >1\) the inequality is satisfied: \(\psi\) (2u)\(\leq \alpha \psi (u)\) for \(u\geq 0\). Let us denote by \(\Omega\) the right half- plane; and by A(\(\Omega)\) the space of analytic functions in \(\Omega\). If \(\psi\) is an N-function and f is any complex function defined and measurable in the interval \((-\infty,+\infty)\), we define \(\rho_{\psi}(f)=\int^{\infty}_{-\infty}\psi (| f(t)|)dt\) and \(\rho_{\psi}(F)=\sup \{\int^{\infty}_{-\infty}\psi (F(x+iy)|)dy\); \(x>0\}\) where F is analytic in \(\Omega\). The class of functions \(F\in A(\Omega)\) for which \(\rho_{\psi}(F)<\infty\) is called the Hardy-Orlicz class in the half- plane \(\Omega\) and denoted \(H^{\psi}\); the class of functions \(F\in A(\Omega)\) for which there exists a number \(k>0\) such that \(kF\in H^{\psi}\) is called Hardy-Orlicz space and denoted \(H^{*\psi}\); \(H^{\circ \psi}\) is the totallity of functions \(F\in A(\Omega)\) such that kF\(\in H\) for every number \(k>0\). Obviousely \(H^{\circ \psi}\subset H^{\psi}\subset H^{*\psi}\). The typical theorem is in the following as Orlicz considered: Theorem. The following four conditions are mutually equivalent: 1) \(H^{\circ \psi}=H^{*\psi}\), 2) \(H^{\circ \psi}=H^{\psi}\), 3) \(H^{\psi}=H^{*\psi}\), 4) \(\psi\) satisfies condition \((\Delta_ 2)\).