Illinois Journal of Mathematics

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

Faruk F. Abi-Khuzam and Bassam Shayya

Full-text: Open access

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.

Article information

Source
Illinois J. Math., Volume 44, Issue 3 (2000), 551-555.

Dates
First available in Project Euclid: 20 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1256060415

Digital Object Identifier
doi:10.1215/ijm/1256060415

Mathematical Reviews number (MathSciNet)
MR1772428

Zentralblatt MATH identifier
0961.30021

Subjects
Primary: 30D20: Entire functions, general theory
Secondary: 30D35: Distribution of values, Nevanlinna theory 42A50: Conjugate functions, conjugate series, singular integrals

Citation

Abi-Khuzam, Faruk F.; Shayya, Bassam. On a singular integral estimate for the maximum modulus of a canonical product. Illinois J. Math. 44 (2000), no. 3, 551--555. doi:10.1215/ijm/1256060415. https://projecteuclid.org/euclid.ijm/1256060415


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