Inequalities of Ono numbers and class numbers associated to imaginary quadratic fields (Q1030580)

\textit{T. Ono} [Arithmetic of algebraic groups and its applications, St. Paul's international exchange series occasional papers VI, St. Paul's Univ., Tokyo (1986)] originally conjectured that \(h(D) \leq 2^{p(D)}\), but this was disproved by \textit{F. Sairaiji} and \textit{K. Shimizu} [Proc. Japan Acad., Ser. A 77, No. 2, 29--31 (2001; Zbl 0988.11052)]. In this article, the author proves the existence of a constant \(c_0\) such that \(h(D) \leq c^{p(D)}\) for only finitely many \(D\) whenever \(c < c_0\), and that \(h(D) \leq c^{p(D)}\) for infinitely many \(D\) when \(c > c_0\). In addition, it is shown that \(\root 4\of {2} \leq c_0 \leq \sqrt{2}\).