An example of the Bernstein-type inequalities we will discuss is the following: For a polynomial $p(z)=c_0+c_1z+/cdots c_n z^n$ of degree $n$, the following reverse-Holder inequality holds for Hardy-norms, $$||p||_{H^{/frac{3}{2}}(/mathbb{D})} /ge /frac{C}{/sqrt[6]{n}} ||p||_{H^{{2}}(/mathbb{D})},$$ where the power $/frac{1}{6}$ is optimal. This is a deterministic result, and we derive it with probabilistic methods during our investigation of random operator theory on the Bergman space.

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