On a singular integral estimate for the maximum modulus of a canonical product

Abstract

If $f$ is a canonical product with only real negative zeros and non-integral order $\rho,n(t,0)$ is the zero counting function, and $B(r,f)=\mathrm{sup}_{0 \lt \theta \lt \pi}|\log f(re^{i \theta})|$, then $$r^{-q-1}B(r,f) \leq \pi\{M \varphi(r) + MH\varphi(r)\}+\int_{0}^{\infty}{\frac{\varphi(t)dt}{t+r}},$$ where $\varphi(t) = t^{-q-1} n(t,0)$, $H$ is the Hilbert transform operator and $M$ is the Hardy-Littlewood maximal operator.