The inequality of Nikol'skii-Stechkin-Boas type with fractional derivatives in \(L_p\), \(0<p<1\) (Q2901593)

The trigonometric polynomials \(T_n(x)\) in \(L_p\), \(0<p<1\), are considered. Let \(\beta >0\), \(n\in {\mathbb N}\), \(T^{(\beta)}_n(x)\) be a derivative of order \(\beta\) of the trigonometrical polynomial \(T_n(x)\) and \(\Delta_{\delta}^{\beta} T_n(x)=\sum\limits_{\nu=0}^{\infty}(-1)^{\nu}\left(\beta\atop \nu\right)T_n\left(x+(\beta-\nu)\delta\right)\). For every \(0<p<1,\) \(\beta>0,\) \(n\in {\mathbb N},\) \(0<h\) and \(\delta<\frac{\pi}{n},\) it is proved that a quasinorm of \(T^{(\beta)}_n\) in \(L_p[-\pi,\pi]\) has lower estimate \(C_1\cdot h^{\,-\beta}\cdot\|\Delta_{h}^\beta T_n\|_{L_p[-\pi,\pi]}\) and upper estimate \(C_2\cdot \delta^{\,-\beta}\cdot\|\Delta_{\delta}^\beta T_n\|_{L_p[-\pi,\pi]}\) with some constants \(C_1\) and \(C_2\) depending only on \(p\) and \(\beta\). As corollary,the correspondent upper estimate for a modulus of smoothness of \(T_n\) of the order \(\beta\) and of the step \(h\) in \(L_p\) is given .