Explicit estimates for the logarithmic derivative and the reciprocal of the Riemann zeta-function

In this article, we improve on two explicit bounds of order log t for σ close to $1$: |ζ'(σ +it)/ζ(σ +it)| and |1/ζ(σ +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.