On integral inequalities for trigonometric polynomials and their derivatives

Let $\Phi^+$ be the set of nondecreasing functions $\varphi$ defined on $(0,\infty)$ which admit a representation $\varphi(u)=\psi(\ln u)$, where the function $\psi$ is convex (below) on $(-\infty,\infty)$. To the class $\Phi^+$ belong, for example, the functions $\ln u$, $\ln^+u$, $u^p$ when $p>0$, and also any function $\varphi$ which is convex on $(0,\infty)$. In this paper it is shown, in particular, that if $\varphi\in\Phi^+$, then for any trigonometric polynomial $T_n$ of order $n$ the following inequality holds for all natural numbers $r$:
$$ \int_0^{2\pi}\varphi\bigl(\bigl|T_n^{(r)}(t)|\bigr)\,dt\leqslant\int_0^{2\pi}\varphi\bigl(n^r\bigl|T_n(t)\bigr|\bigr)\,dt. $$
This inequality may be considered a generalization of the inequalities of S. N. Bernstein and A. Zygmund.
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