  Let $X$ be a quasi-Banach space, $Y$ a $\gamma$-Banach space $(0<\gamma \leq
1)$ and $T$ a bounded linear operator from $X$ into $Y$. In this paper, we
prove that the first outer entropy number of $T$ lies between
$2^{1-1/\gamma}\|T\|$ and $\|T\|$; more precisely, $2^{1-1/\gamma}\|T\| \leq
e_1(T) \leq \|T\|,$ and the constant $2^{1-1/\gamma}$ is sharp. Moreover, we
show that there exist a Banach space $X_0$, a $\gamma$-Banach space $Y_0$ and a
bounded linear operator $T_0:X_0 \rightarrow Y_0$ such that $0 \neq e_k(T_0) =
2^{1-1/\gamma}\|T_0\| $ for all positive integers $k.$ Finally, the paper also
provides two-sided estimates for entropy numbers of embeddings between finite
dimensional symmetric $\gamma$-Banach spaces.
