# The $L_p$ Version of Newman's Inequality for Lacunary Polynomials

Borwein, Peter and Erdelyi, Tamas (1994) The $L_p$ Version of Newman's Inequality for Lacunary Polynomials. [Preprint]

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Postscript
The principal result of this paper is the establishment of the essentially sharp Markov-type inequality $$\|xP^{\prime}(x)\|_{L_p[0,1]} \leq \left(1/p+12 \left(F{\sum^n_{j=0}} (\lambda_j + 1/p)\right)\right) \|P\|_{L_p[0,1]}$$ 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 $$\|y^{1/p}P(y)\|_{L_\infty[0,1]} \leq 13 \left({\sum^n_{j=0}} (\lambda_j + 1/p)\right)^{1/p} \|P\|_{L_p[0,1]}$$ for every $P \in \text{\rm span}\{x^{\lambda_0}, x^{\lambda_1}, \ldots, x^{\lambda_n}\}$ with distinct real exponents $\lambda_j$ greater than $-1/p$ and for every $p \in [1, \infty]$. Some related results are also discussed.