Hardy’s inequalities in finite dimensional Hilbert spaces

Authors:
Dimitar K. Dimitrov, Ivan Gadjev, Geno Nikolov and Rumen Uluchev

Journal:
Proc. Amer. Math. Soc. **149** (2021), 2515-2529

MSC (2020):
Primary 26D10, 26D15; Secondary 33C45, 15A42

DOI:
https://doi.org/10.1090/proc/15467

Published electronically:
March 26, 2021

MathSciNet review:
4246802

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Abstract | References | Similar Articles | Additional Information

Abstract: We study the behaviour of the smallest possible constants $d_n$ and $c_n$ in Hardy’s inequalities \begin{equation*} \sum _{k=1}^{n}\Big (\frac {1}{k}\sum _{j=1}^{k}a_j\Big )^2\leq d_n \sum _{k=1}^{n}a_k^2, \qquad (a_1,\ldots ,a_n) \in \mathbb {R}^n \end{equation*} and \begin{equation*} \int _{0}^{\infty }\Bigg (\frac {1}{x}\int _{0}^{x}f(t) dt\Bigg )^2 dx \leq c_n \int _{0}^{\infty }f^2(x) dx,\qquad f\in \mathcal {H}_n, \end{equation*} for the finite dimensional spaces $\mathbb {R} ^n$ and $\mathcal {H}_n\colonequals \{f : \int _0^x f(t) dt =e^{-x/2} p(x)\ :\ p\in \mathcal {P}_n, p(0)=0\}$, where $\mathcal {P}_n$ is the set of real-valued algebraic polynomials of degree not exceeding $n$. The constants $d_n$ and $c_n$ are identified to be expressed in terms of the smallest zeros of the so-called continuous dual Hahn polynomials and the two-sided estimates for $d_n$ and $c_n$ of the form \begin{equation*} 4-\frac {c}{\ln n}< d_n, c_n<4-\frac {c}{\ln ^2 n} ,\qquad c>0 \end{equation*} are established.

- G. H. Hardy,
*Notes on some points in the integral calculus, LI. On Hilbert’s double-series theorem, and some connected theorems concerning the convergence of infinite series and integrals*, Messenger Math.**48**(1919), 107–112. - G. H. Hardy,
*Note on a theorem of Hilbert*, Math. Z.**6**(1920), no. 3-4, 314–317. MR**1544414**, DOI https://doi.org/10.1007/BF01199965 - G. H. Hardy,
*Notes on some points in the integral calculus, LX. An inequality between integral*, Messenger Math.**54**(1925), 150–156. - G. H. Hardy, J. E. Littlewood, and G. Pólya,
*Inequalities*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988. Reprint of the 1952 edition. MR**944909** - Mourad E. H. Ismail,
*The variation of zeros of certain orthogonal polynomials*, Adv. in Appl. Math.**8**(1987), no. 1, 111–118. MR**876957**, DOI https://doi.org/10.1016/0196-8858%2887%2990009-1 - Mourad E. H. Ismail,
*Monotonicity of zeros of orthogonal polynomials*, $q$-series and partitions (Minneapolis, MN, 1988) IMA Vol. Math. Appl., vol. 18, Springer, New York, 1989, pp. 177–190. MR**1019851**, DOI https://doi.org/10.1007/978-1-4684-0637-5_14 - Mourad E. H. Ismail,
*Classical and quantum orthogonal polynomials in one variable*, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2005. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey. MR**2191786** - R. Koekoek and R. F. Swarttouw,
*The Askey-scheme of hypergeometric orthogonal polynomials and its $q$-analogue*, Report 98-17, Delft University of Technology, 1998, http://homepage.tudelft.nl/11r49/documents/as98.pdf. - Alois Kufner, Lars-Erik Persson, and Natasha Samko,
*Weighted inequalities of Hardy type*, 2nd ed., World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. MR**3676556** - Alois Kufner, Lech Maligranda, and Lars-Erik Persson,
*The prehistory of the Hardy inequality*, Amer. Math. Monthly**113**(2006), no. 8, 715–732. MR**2256532**, DOI https://doi.org/10.2307/27642033 - Alois Kufner, Lech Maligranda, and Lars-Erik Persson,
*The Hardy inequality*, Vydavatelský Servis, Plzeň, 2007. About its history and some related results. MR**2351524** - E. Landau, I. Schur, and G. H. Hardy,
*A Note on a Theorem Concerning Series of Positive Terms: Extract from a Letter*, J. London Math. Soc.**1**(1926), no. 1, 38–39. MR**1575105**, DOI https://doi.org/10.1112/jlms/s1-1.1.38 - Gábor Szegő,
*Orthogonal polynomials*, 4th ed., American Mathematical Society, Providence, R.I., 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII. MR**0372517**

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Additional Information

**Dimitar K. Dimitrov**

Affiliation:
Departamento de Matemática, IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP, Brazil

MR Author ID:
308699

Email:
d_k_dimitrov@yahoo.com

**Ivan Gadjev**

Affiliation:
Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria

MR Author ID:
1090555

ORCID:
0000-0002-4444-9921

Email:
gadjev@fmi.uni-sofia.bg

**Geno Nikolov**

Affiliation:
Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria

MR Author ID:
131505

ORCID:
0000-0001-5608-2488

Email:
geno@fmi.uni-sofia.bg

**Rumen Uluchev**

Affiliation:
Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria

MR Author ID:
175915

ORCID:
0000-0002-9122-7088

Email:
rumenu@fmi.uni-sofia.bg

Received by editor(s):
July 20, 2020

Received by editor(s) in revised form:
September 14, 2020, and September 14, 2020

Published electronically:
March 26, 2021

Additional Notes:
Research supported by the Brazilian Science Foundations FAPESP under Grants 2016/09906-0 and 2016/10357-1 and CNPq under Grant 306136/2017-1 and the Bulgarian National Research Fund through Contract DN 02/14.

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American Mathematical Society