On the class numbers of algebraic number fields

Let $K$ be a number field of degree $n$ over $\mathbb Q$ and $d,h$, and $R$ be the absolute value of the discriminant, the class number, and the regulator, respectively, of $K$. It is known that if $K$ contains no quadratic subfield, then
$$ hR\gg\frac{d^{1/2}}{\log d}, $$
where the implied constant depends only on $n$. In Theorem 1 this lower estimate is improved for pure cubic fields.
Consider the family $\mathcal K_n$ where $K\in\mathcal K_n$ if $K$ is a totally real number field of degree $n$ whose normal closure has the symmetric group $S_n$ as its Galois group. Theorem 2: Fix $n\ge2$. There are infinitely many $K\in\mathcal K_n$ with
$$ h\gg d^{1/2}(\log\log d)^{n-1}/(\log d)^n, $$
where the implied constant depends only on $n$.
This is a somewhat greater improvement over W. Duke's analogous result with $h\gg d^{1/2}/(\log d)^n$ [MR 1966783 (2004g:11103)].