Bernstein-Nikolskii Inequality: Optimality with Respect to the Smoothness Parameter

In this paper, we study the form of the constant $C$ in the Bernstein–Nikolskii inequalities $\mathrm{\parallel }{f}^{(s)}{\mathrm{\parallel }}_q\lesssim C(s,p,q){\parallel f\parallel }_p,0<p<q\le \mathrm{\infty }$, for trigonometric polynomials and entire functions of exponential type. We obtain the optimal behavior of the constant with respect to the smoothness parameter $s$.