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Talebi, Gholamreza, Ebrahimi Meymand, Ali. (1401). Extension of Carleman's inequality by means of infinite lower triangular matrices. , 8(36), 167-174.
Gholamreza Talebi; Ali Ebrahimi Meymand. "Extension of Carleman's inequality by means of infinite lower triangular matrices". , 8, 36, 1401, 167-174.
Talebi, Gholamreza, Ebrahimi Meymand, Ali. (1401). 'Extension of Carleman's inequality by means of infinite lower triangular matrices', , 8(36), pp. 167-174.
Talebi, Gholamreza, Ebrahimi Meymand, Ali. Extension of Carleman's inequality by means of infinite lower triangular matrices. , 1401; 8(36): 167-174.

Extension of Carleman's inequality by means of infinite lower triangular matrices

Journal of New Researches in Mathematics
مقاله 14، دوره 8، شماره 36، آذر و دی 2022، صفحه 167-174  XML اصل مقاله (261.42 K)
نوع مقاله: research paper
نویسندگان
Gholamreza Talebi email 1؛ Ali Ebrahimi Meymand2
1Department of Mathematics Faculty of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Islamic Republic of Iran
2Department of Mathematical Sciences, Vali-e-Asr University of Rafsanjan, Rafsanjan, Islamic Republic of Iran.
چکیده
Let H_μ=(h_(n,k) )_(n,k≥0) be the Hausdorff matrix associated with the probability measure . Graham Bennett in 1996 established the following extension of Carleman's inequality



[sumlimits_{n = 0}^infty {prodlimits_{k = 0}^n {{{left| {{x_k}} right|}^{{h_{n,k}}}}} } le {e^{int_0^1 {|log theta |dmu (theta )} }}sumlimits_{n = 0}^infty {left| {{x_n}} right|} .,,,,,,,(1)]



In this paper we show that the Hausdorff matrix in (1) can be replaced by any lower triangular matrix

[A = {left( {{a_{n,k}}} right)_{n,k ge 0}}]

for which the sum of each rows is one, provided that the constant in the right hand side, be replaced by

[left( {mathop {inf }limits_{p > 1} left| A right|_p^p} right)]. . . . . . . . .

As a consequence, we apply our results to Norlund matrices and weighted mean matrices to establish some new inequalities. Further, we show that being equal to 1 is an essential condition for the rows sum of A.
کلیدواژه‌ها
Borel probability measure؛ Hausdorff matrix؛ Norlund matrix؛ Hardy&rsquo؛ s inequality
مراجع

 

[1] F. Hausdorff. Summationsmethoden und Momentfolgen I. Mathematische Zeitschrift 9: 75-109 (1921).

 

[2] T. Carleman. Sur les fonctions quasi-analytiques. Comptes rendus, du  Congres des Mathematiciens Scandinaves, Helsingfors 181 – 196 (1922).

 

[3] G. Bennett. Factorizing the classical inequalities. Memoirs of the American Mathematical Society: 120 (576) 1-130 (1996).

 

[4] G. B. Folland.  Real Analysis: Modern techniques and their applications. A Wiley Series of Texts, Monographs and Tracts, 2nd edition, Kindle Edition (2013).

 

[5] J. Boos. Classical and Modern Methods in Summability. Oxford University Press, New York (2000).

 

[6] G. H. Hardy. An Inequality for Hausdorff mean.  Journal of the London Mathematical Society 18:  46 – 50 (1943).

 

[7] G. H. Hardy, J. E. Littlewood, G. Polya. Inequalities. 2nd edition, Cambridge University Press (1967).

 

[8] R. Lashkaripour, D. Forotannia. Extension of Hardy Inequality on Weighted Sequence Spaces. Journal of Sciences, Islamic Republic of Iran 20(2): 159-166 (2009).

 

[9] K. Endl. Untersuchungen über Moment Probleme bei Verfahren vom Hausdorffschen Typus. Mathematische Annalen 139: 403- 432 (1960).

 

[10] A. Jakimovski. The product of summability methods. New classes of transformations and their properties. Part 2 Tech. note, Contract No. AF 61: (1959) (052)187.

 

[11] A. Jakimovski, B. E. Rhoades, J. Tzimbalario. Hausdorff matrices as bounded operators over . Mathematische Zeitschrift 138: 173 -181 (1974).

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