Hp空间中函数的模和零点分布

ON MODULUS AND DISTRIBUTION OF ZEROS OF FUNCTION IN Hp SPACES

讨论了N类空间、B空间、N~*类空间和Hp空间中模和零点分布的关系;给出了最大模的一个精确估计式.

Let f(z) be analytic in |z|<1,f(0)■0,{z_n} be the sequence of zerosin |z|<l,|z_n|=p_n,n=1,2,…Denote M_f(p)=■|f(z)|,m_f(p)= ■|f(z)|Theorem 1 If f∈N,then for 0<p<1,exp{-(1+ρ)/(1-ρ)N(f)}·M_f(ρ)≤■(ρ+ρ_n)/(1+ρρ_n),(1)exp{(1+ρ)/(1-ρ)N(f)}·m_f(ρ)≥((丨f(0)丨)/(■ρ_n))^((1+ρ)/(1-ρ))■(丨ρ-ρ_n丨)/(丨1-ρρ_n丨),(2)where N(f)=■1/(2π)∫_0^(2π) log^+丨f(pe^(i■)丨dθ.Inequality(1)and(2)are exact.Theorem 2 If f∈Hp(p>0),then((1+ρ)/(1-ρ)Hp(f)^(1/ρ) M_f(ρ)≤(ρ+丨a丨)/(1+丨a丨ρ)·■(ρ+ρ_n)/(1+ρρ_n),(0<ρ<1).If f∈N,then exp(-(1+ρ)/(1-ρ)N(f)).M_f(ρ)≤(ρ+丨a丨)/(1+丨a丨ρ)■(ρ+ρ_n)/(1+ρρ_n)where a=(f(0))/(E(f)■ρ_n),E(f)=exp{1/(2π)∫_0^(2π)log丨f(e^(if))丨dt}andHp(f)=■1/(2π)∫_0^(2n)丨f(pe^(it)丨~p dt.