AbstractLet X be a quasi‐Banach space, Y be a γ‐Banach space and T be a bounded linear operator from X into Y. In this paper, we prove that the first outer entropy number of T lies between and ; more precisely, , and the constant is sharp. Moreover, we show that there exist a Banach space X0, a γ‐Banach space Y0 and a bounded linear operator such that for all positive integers k. Finally, the paper also provides two‐sided estimates for entropy numbers of embeddings between finite dimensional symmetric γ‐Banach spaces.