The $L_p$ version of Newman’s Inequality for lacunary polynomials

Abstract:

The principal result of this paper is the establishment of the essentially sharp Markov-type inequality \[ \|xP’ (x)\|_{L_p[0,1]} \leq \left (1/p+12 \left ({\sum ^n_{j=0}}(\lambda _j + 1/p)\right )\right ) \|P\|_{L_p[0,1]}\] for every $P \in \operatorname {span}\{x^{\lambda _0}, x^{\lambda _1}, \ldots , x^{\lambda _n}\}$ with distinct real exponents $\lambda _j$ greater than $-1/p$ and for every $p \in [1, \infty ]$. A remarkable corollary of the above is the Nikolskii-type inequality \[ \|y^{1/p}P(y)\|_{L_\infty [0,1]} \leq 13 \left ({\sum ^n_{j=0}}(\lambda _j + 1/p)\right )^{1/p} \|P\|_{L_p[0,1]}\] for every $P \in \operatorname {span}\{x^{\lambda _0}, x^{\lambda _1}, \ldots , x^{\lambda _n}\}$ with distinct real exponents $\lambda _j$ greater than $-1/p$ and for every $p \in [1, \infty ]$. Some related results are also discussed.