The inclusion of certain classes of functions

For any sequence $\{N_k\}$ with $\{N_k\}\downarrow0$ we find sharp theorems on the inclusion of the classes $\{f:f\in L(0,2\pi),\ E_k^{(1)}(f)=O(N_k)\}$, where $E_k^{(1)}(f)$ is the best approximation (in $L$) of $f$ by trigonometric polynomials of order no greater than $k$, in the class $L_\varphi(L)$ with slowly growing $\varphi$ and in the class $L^\nu$, $1<\nu<\infty$.