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For real quadratic fields, there is the bound $$h\leq \lfloor\sqrt{\Delta}/2\rfloor$$

Is there anything similar for imaginary quadratic fields? More generally, I'm interested in a bound for $h$ involving the discriminant of the number field.

Javier
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    For the imaginary quadratic field $\Bbb Q(\sqrt{-d})$, the class number formula implies that $h \ll \sqrt{d} L(1,\chi_{-d})$; and the estimate $L(1,\chi_{-d}) \ll \log d$ is a standard bound in analytic number theory. Both can be given with explicit constants with enough work. – Greg Martin Aug 09 '16 at 16:59

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I'm turning Greg Martins comment into a community wiki answer by making everything more explicit.
If $\chi$ is a primitive Dirichlet character of conductor $q$ then by Corollary 1 of [1] we have $|L(1,\chi)| \leq \frac 1 2 \log q + 5/2 - \log 6 $. Combining this with the class number formula one gets for $d < -4$ a negeative fundamental discriminant that

$$h(d) \leq \frac{ \sqrt{|d|}}{ \pi} L(1,\chi) \leq \frac{ \sqrt{|d|}}{ \pi}(\frac 1 2 \log |d| + 5/2 -\log 6).$$

The area of getting better bounds on $L(1,\chi)$ is an active a area of research which so if you are willing to dive deeper in the literature one can improve on the factor $1/2$ at least for $-d \gg 0 $. For example the results of [2] imply that one can get a bound of $\frac{ \sqrt{|d|}}{ \pi}\frac 9 {20} \log |d| $ for $-d \geq 5 \cdot 10^{50}$.

[1] Ramaré, Olivier, Approximate formulae for (L(1,\chi)), Acta Arith. 100, No. 3, 245-266 (2001). ZBL0985.11037.

[2] Johnston, D. R.; Ramaré, O.; Trudgian, T., An explicit upper bound for (L(1,\chi)) when (\chi) is quadratic, Res. Number Theory 9, No. 4, Paper No. 72, 20 p. (2023). ZBL1547.11099.

Maarten Derickx
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