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 t for close to 1: |' (+it) / (+it) | and |1/ (+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\'edory, with a Jensen-type one of Heath-Brown's.