Sains Malaysiana 46(9)(2017): 1355–1359

http://dx.doi.org/10.17576/jsm-2017-4609-01

Hardy's Inequality for Functions of Several Complex Variables

(Ketidaksamaan Hardy untuk Fungsi Beberapa Pemboleh Ubah Kompleks)

VANSAK SAM & KAMTHORN CHAILUEK*

Applied Analysis Research Unit, Department of Mathematics and Statistics

Prince of Songkla University, Hatyai, Songkhla, 90110, Thailand

Received: 31 August 2016/Accepted: 18 January 2017

ABSTRACT

We obtain a generalization of Hardy’s inequality for functions in the Hardy space H1 (Bd), where Bd is the unit ball {z = (z1, …, zd) ∈ In particular, we construct a function φ on the set of d –dimensional multi-indices {n = (n1, …, nd) | ni ∈  {0}} and prove that if f(z) = Σ anzn is a function in H1 (Bd), then ≤ Moreover, our proof shows that this inequality is also valid for functions in Hardy space on the polydisk H1 (Bd).

Keywords: Hardy’s inequality; Hardy space and Hilbert’s inequality

ABSTRAK

Kami memperoleh generalisasi ketidakseimbangan Hardy’s untuk fungsi dalam ruang Hardy H1 (Bd), dengan Bd adalah unit bola {z = (z1, …, zd) ∈ Secara khususnya, kami membina fungsi φ pada set indeks pelbagai d –dimensi {n = (n1, …, nd) | ni ∈  {0}} dan membuktikan bahawa jika f(z) = Σ anzn adalah fungsi di H1 (Bd), kemudian ≤ Selain itu, bukti kami menunjukkan bahawa ketidakseimbangan ini juga adalah sah untuk fungsi dalam ruang Hardy ke atas polidisk H1 (Bd).

Kata kunci: Ketidaksamaan Hardy; Ruang Hardy dan ketidaksamaan Hilbert

REFERENCES

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Paulsen, V.I. & Singh, D. 2015. Extension of the inequalities of Hardy and Hilbert, arXive: http://arxiv.org/pdf/1502.05909. pdf.

Sababheh, M. 2008a. Hardy-type inequality on the real lines. J. of Ineq. in Pure and Applied Math. 9(3): No. 72.

Sababheh, M. 2008b. On an argument of Korner and Hardy's inequality. Analysis Mathematica 34: 51-57.

Zhu, K. 2004. Translating inequalities between Hardy and Bergman spaces. Amer. Math. Monthly. 111(6): 520-525.

*Corresponding author; email: kamthorn.c@psu.ac.th