Carl's inequality for quasi-Banach spaces

We prove that for any two quasi-Banach spaces X and Y   and any α>0α>0 there exists a constant γα>0γα>0 such thatsup1≤k≤n⁡kαek(T)≤γαsup1≤k≤n⁡kαck(T) holds for all linear and bounded operators T:X→YT:X→Y. Here ek(T)ek(T) is the k-th entropy number of T   and ck(T)ck(T) is the k-th Gelfand number of T. For Banach spaces X and Y this inequality is widely used and well-known as Carl's inequality. For general quasi-Banach spaces it is a new result.