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

Subject Areas : Statistics

Gholamreza  Talebi ¹ (Department of Mathematics Faculty of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Islamic Republic of Iran)
Ali  Ebrahimi Meymand ² (Department of Mathematical Sciences, Vali-e-Asr University of Rafsanjan, Rafsanjan, Islamic Republic of Iran.)

Received: 2021-04-13 Accepted : 2021-11-24 Published : 2022-06-22

Keywords: ماتریس هاسدورف, اندازه بورل احتمال, نامساوی هاردی, ماتریس نورلوند,

Abstract :

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.

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