Some supplements to S. B. Stechkin's inequalities in direct and inverse theorems on the approximation of continuous periodic functions

We give some supplements and comments to inequalities between elements of the sequence of best approximations $\{E_{n-1}(f)\}_{n=1}^{\infty}$ and the $k$th-order moduli of smoothness $\omega_k(f^{(r)};\delta),$ $\delta\in [0,+\infty)$, of a function $f\in C^r(\mathbb{T})$, where $k\in \mathbb{N},$ $r\in \mathbb{Z}_+$, $f^{(0)}\equiv f,$ $C^0(\mathbb{T})\equiv C(\mathbb{T}),$ and $\mathbb{T}=(-\pi,\pi]$, which were published by S. B. Stechkin in 1951 in the study of direct and inverse theorems of approximation of $2\pi$-periodic continuous functions. In particular, we prove the following results:
$\mathrm{(a)}$ the direct theorem or the Jackson–Stechkin inequality: $E_{n-1}(f)\le C_1(k)\omega_k(f;\pi/n)$, $n\in \mathbb{N}$, can be strengthened as $E_{n-1}(f)\le \rho_{n}^{(k)}(f)\equiv n^{-k}\max\{\nu^k E_{\nu-1}(f)\colon 1\le \nu\le n\}\le 2^kC_1(k)\omega_k(f;\pi/n),\ n\in \mathbb{N}$. This inequality is order-sharp on the class of all functions $f\in C(\mathbb{T})$ with a given majorant or with a given decrease order of the modulus of smoothness $\omega_k(f;\delta)$; namely: for any $k\in \mathbb{N}$ and $\omega\in \Omega_k(0,\pi]$, there exists a function $f_0(\,{\cdot}\,;\omega)\in C(\mathbb{T})$ ($f_0$ is even for odd $k$ and is odd for even $k$) such that $\omega_k(f_0;\delta)\asymp C_2(k)\omega(\delta)$, $\delta\in (0,\pi]$. Moreover, order equalities hold: $E_{n-1}(f_0)\asymp C_3(k)\rho_n^{(k)}(f_0)\asymp C_4(k)\omega_k(f_0;\pi/n)\asymp C_5(k)\omega(\pi/n),\ n\in \mathbb{N}$, where $\Omega_k(0,\pi]$ is the class of functions $\omega=\omega(\delta)$ defined on $(0,\pi]$ and such that $0<\omega(\delta)\!\downarrow\!0$ $(\delta\downarrow\!0)$ and $\delta^{-k}\omega(\delta)\!\downarrow$ $(\delta \uparrow)$;
$\mathrm{(b)}$ a necessary and sufficient condition under which the inverse theorem (without the derivatives), or the Salem–Stechkin inequality $\omega_k(f;\pi/n)\le C_6(k)n^{-k}\sum_{\nu=1}^n\nu^{k-1}E_{\nu-1}(f)$, $n\in \mathbb{N}$, holds is Stechkin's inequality $\|T_n^{(k)}(f)\|\le C_7(k) \sum_{\nu=1}^{n}\nu^{k-1}E_{\nu-1}(f),\ n\in \mathbb{N}$, where $T_n(f)\equiv T_n(f;x)$ is a trigonometric polynomial of best $C(\mathbb{T})$-approximation to the function $f$ (i.e., $\|f-T_n(f)\|=E_n(f),\ n\in \mathbb{Z}_+$);
$\mathrm{(c)}$ the inverse theorem (with the derivatives), or the Vallée-Poussin–Stechkin inequality $\omega_k(f^{(r)};$ $\pi/n)\le C_8(k,r)\big\{ n^{-k}\sum_{\nu=1}^{n}\nu^{k+r-1}E_{\nu-1}(f)+\sum_{\nu=n+1}^{\infty}\nu^{r-1}E_{\nu-1}(f)\big\}$ for any $n\in \mathbb{N}$, as well as Stechkin's earlier inequality $E_{n-1}(f^{(r)})\le C_9(r)\big\{ n^r E_{n-1}(f)+\sum_{\nu=n+1}^{\infty}\nu^{r-1}E_{\nu-1}(f)\big\},\ n\in \mathbb{N}$, where $E(f;r)\equiv$ $ \sum_{n=1}^{\infty}n^{r-1}E_{n-1}(f)<\infty$ (by S. N. Bernstein's theorem, this inequality guarantees that $f$ lies in $C^r(\mathbb{T})$, where $r\in\mathbb{N}$) can be supplemented with the following key inequalities: $\|f^{(r)}\|\le C_{10}(r)E(f;r)$ and $\|T_n^{(r)}(f)\|\le C_{7}(r)\sum_{\nu=1}^n\nu^{r-1}E_{\nu-1}(f)$, $n\in\mathbb{N}$. Moreover, all the inequalities formulated in this paragraph are pairwise equivalent; i.e., any of these inequalities implies any other and, hence, all the inequalities.