## Hokkaido Mathematical Journal

### Paley's inequality of integral transform type

#### Abstract

Let $\{n_k\}_{k=1}^\infty$ be a sequence of positive integers with Hadamard gap. For an analytic function $F(z) = \sum_{n=0}^{\infty}a_n z^n$ in the unit disc satisfying $\sup_{0 \lt r \lt 1}$ $\int_0^{2\pi} |F(re^{i\theta})|\, d\theta \lt \infty$, the inequality $( \sum_{k=1}^{\infty}|a_{n_k}|^2 )^{1/2} \lt \infty$ holds, which is familiar as Paley's inequality. In this paper, an integral transform version of this inequality is established.

#### Article information

Source
Hokkaido Math. J., Volume 38, Number 2 (2009), 233-247.

Dates
First available in Project Euclid: 21 July 2009

https://projecteuclid.org/euclid.hokmj/1248190076

Digital Object Identifier
doi:10.14492/hokmj/1248190076

Mathematical Reviews number (MathSciNet)
MR2522913

Zentralblatt MATH identifier
1179.42006

#### Citation

KANJIN, Yuichi; SATO, Kunio. Paley's inequality of integral transform type. Hokkaido Math. J. 38 (2009), no. 2, 233--247. doi:10.14492/hokmj/1248190076. https://projecteuclid.org/euclid.hokmj/1248190076