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

The principal result of this paper is the establishment of the essentially sharp Markov-type inequality

for every $P \in \text{span}\{x^{\lambda _0}, x^{\lambda _1}, \ldots , x^{\lambda _n}\}$ 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

for every $P \in \text{\rm span}\{x^{\lambda _0}, x^{\lambda _1}, \ldots , x^{\lambda _n}\}$ with distinct real exponents $\lambda _j$ greater than $-1/p$ and for every $p \in [1, \infty ]$. Some related results are also discussed.