  In this article, we improve on two explicit bounds of order $\log t$ for
$\sigma$ close to $1$: $|\zeta'(\sigma +it)/\zeta(\sigma +it)|$ and
$|1/\zeta(\sigma +it)|$. Our method provides marked improvements in the
constants, especially with regards to the latter. These savings come from
replacing a lemma of Borel and Carath\'{e}dory, with a Jensen-type one of
Heath-Brown's. Using an argument involving the trigonometric polynomial, we
additionally provide a slight asymptotic improvement: $1/\zeta(\sigma +it) \ll
(\log t)^{11/12}$.
