Multivariate version of Turan’s type inequality and its applications to the estimation of uniform moduli of smoothness of periodic functions

The following results are proved in the paper.
$\bf{Theorem~1.}$ Let $m \ge 1,\ f\in L_1(\mathbb{T}^m),\ l,k\in \mathbb N,\ l> m,\ \rho=l-(k+m),$ and $\sum_{n=1}^{\infty}n^{m-1}\omega_{l}(f;d/n)_{1,m}<\infty$. Then $f$ is equivalent to some function $\psi\in C(\mathbb{T}^m)$ and
$(a)$  $\displaystyle \omega_{k}\Big(\psi;\frac{d}{n}\Big)_{\infty,m} \le C_{1}(k,l,m)\bigg\{\sum\limits_{\nu=n+1}^{\infty}\nu^{m-1}\omega_{l}\Big(f;\frac{d}{\nu}\Big)_{1,m}+\chi (\rho)n^{-k}\sum\limits_{\nu=1}^{n}\nu^{k+m-1}\omega_{l}\Big(f;\frac{d}{\nu}\Big)_{1,m}\bigg\},\quad n\in \mathbb N,$
where $\omega_{l}(f;\delta)_{1,m}$ is the $l$ th-order complete modulus of smoothness of $f$, $\omega_{k}(\psi;\delta)_{\infty,m}$ is the $k$ th-order complete modulus of smoothness of $\psi$, $\mathbb{T}^m=(-\pi,\pi]^{m}$, $d=\pi m^{1/2}$, $\chi(t)=0$ for $t\le 0$, and $\chi(t)=1$ for $t>0$.
In the case $l=k+m\ (\Rightarrow \chi(\rho)=0)$, the proof of estimate (a) relies substantially on the inequality
$(b)$  $\displaystyle n^{-k}\max\limits_{|\alpha|=k}\Big\|\frac{\partial^{|\alpha|}T_{n,\ldots,n;1}(f;x)}  {\partial x^{\alpha}}\Big\|_{\infty,m} \le C_{2}(k,m)n^{m}\omega_{k+m}\Big(f;\frac{d}{n+1}\Big)_{1,m},\quad n\in \mathbb N$,
 where $T_{n,\ldots,n;1}(f;x_{1},\ldots,x_{m})$ is a polynomial of best $L_{1}(\mathbb{T}^m)$-approximation to $f$ of order $n\in \mathbb N$ with respect to the variable $x_{i}$ $(i=\overline{1,m})$ and $\alpha=(\alpha_{1},\ldots,\alpha_{m})$, $\alpha_{j} \in \mathbb Z_{+}$ $(j=\overline{1,m})$, is a multiindex of length $|\alpha|=k$. Inequality (b) is proved by using a multivariate version of Turan's type inequality: for each trigonometric polynomial $t_{n_{1},\ldots,n_{m}}(x_{1},\ldots,x_{m})$ of order $n_{i} \in \mathbb N$ with respect to the variable $x_{i}$ $(i=\overline{1,m})$, we have the inequality
$(c)$   $\displaystyle \Big\|\frac{\partial^{k}t_{n_{1},\ldots,n_{m}}(x)}{\partial x^{\alpha}}\Big\|_{\infty,m} \le \Big(\frac{\pi}{2}\Big)^m \Big\|\frac{\partial^{k+m}t_{n_{1},\ldots,n_{m}}(x_{1},\ldots,x_{m})}{\partial x_{1}^{\alpha_{1}+1}\ldots\partial x_{m}^{\alpha_{m}+1}}\Big\|_{1,m},$
 which follows directly from a similar inequality (with $k=0$ in inequality $(c)$) but holds under the conditions $\frac{1}{2\pi}\displaystyle\int\nolimits_{0}^{2\pi}t_{n_{1},\ldots,n_{i},\ldots,n_{m}}(x_{1},\ldots,x_{i}-y_{i},\ldots,x_{m})\, dy_{i}=0,$ $i=\overline{1,m}.$
Estimate (a) is order-sharp in the class $H_{1,m}^l[\omega]=\{f\in L_1(\mathbb{T}^m):\ \omega_{l}(f;\delta)_{1,m} \le \omega (\delta)$, $\delta \in (0,d]\}$,  where $\omega \in \Omega_{l}(0,d]$ is the class of functions $\omega =\omega (\delta)$ defined on $(0,d]$ and satisfying the conditions $0<\omega (\delta)\downarrow 0\ (\delta \downarrow 0)$ and $\delta^{-l}\omega(\delta)\downarrow(\delta\uparrow)$.
$\bf{Theorem~2.}$  Let $m\ge 1,\ l,k\in \mathbb N,\ l>m,\ \rho =l-(k+m),\ \omega \in \Omega_{l}(0,d],$ and $\sum_{n=1}^{\infty}n^{m-1}\omega(d/n) <\infty$. Then
$$ \sup\Big\{ \omega_{k} \Big(\psi;\frac{d}{n}\Big)_{\infty,m}:\ f\in H_{1,m}^{l} [\omega]\Big\} \asymp \sum_{\nu=n+1}^{\infty}\nu^{m-1}\omega\Big(\frac{d}{\nu}\Big) +\chi(\rho) n^{-k}\sum_{\nu=1}^{n}\nu^{k+m-1}\omega\Big(\frac{d}{\nu}\Big),\quad n\in \mathbb N, $$
where $\psi$ is the corresponding function from the class $C(\mathbb{T}^m)$ equivalent to $f\in H_{1,m}^{l}[\omega]$.