It is still an open problem whether for every compact operator u, the entropy numbers \(e_ k(u)\) and \(e_ k(u^*)\) of u and its dual \(u^*\), are asymptotically of the same order. We prove here that for every \(\alpha >1/2\), there exists numerical constants C(\(\alpha)\) and C’(\(\alpha)\), such that for any compact operator u with values in a Hilbert space, one has: \[ \sup_{k}k^{\alpha}e_ k(u^*)\leq C(\alpha)\sup_{k}k^{\alpha}e_ k(u)\leq C'(\alpha)\sup_{k}k^{\alpha}(Log k)^{\alpha +1}e_ k(u^*). \] This result follows from he comparison between entropy numbers and Gelfand numbers for which similar inequalities are proved.