An inequality relating the regulator and the discriminant of a number field

Let K be a number field of degree d with regulator R_K and absolute discriminant D_K. Let r(K) be the rank of the unit group in K, and let p(K) be the maximum of r(k) as k ranges over proper subfields of K. We prove

$\text{R}{}_{\mathrm{K}}\text{> c}{}_{\mathrm{d}}\text{(}\text{log}\text{γ}{}_{\mathrm{d}}\text{D}{}_{\mathrm{K}}\text{)}{}^{\mathrm{r(K)?\rho (K)}}$

for constants c_d, γ_d > 0 depending only on d.