Hermite-Padé approximants to exponential functions and an inequality of Mahler (Q1283814)

The main results of this paper are that the inequalities \[ |e^q-a|> q^{-19.183q} \quad\text{and}\quad |\log q-a|> q^{-19.183\log \log q} \] hold for every large positive integer \(q\) and \(a\). The proofs are based on using suitable approximation of Legendre-type polynomials. The method is similar to Hermite-Padé approximation.