Banach Journal of Mathematical Analysis

On Hardy-type inequalities for weighted means

Zsolt Páles and Paweł Pasteczka

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Abstract

Our aim in this article is to establish weighted Hardy-type inequalities in a broad family of means. In other words, for a fixed vector of weights (λn)n=1 and a weighted mean M, we search for the smallest extended real number C such that

n=1λnM((x1,,xn),(λ1,,λn))Cn=1λnxnfor allx1(λ). The main results provide a complete answer in the case when M is monotone and satisfies the weighted counterpart of the Kedlaya inequality. In particular, this is the case if M is symmetric, concave, and the sequence (λnλ1++λn)n=1 is nonincreasing. In addition, we prove that if M is a symmetric and monotone mean, then the biggest possible weighted Hardy constant is achieved if λ is the constant vector.

Article information

Source
Banach J. Math. Anal., Volume 13, Number 1 (2019), 217-233.

Dates
Received: 3 June 2018
Accepted: 23 July 2018
First available in Project Euclid: 16 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1542358830

Digital Object Identifier
doi:10.1215/17358787-2018-0023

Mathematical Reviews number (MathSciNet)
MR3892700

Zentralblatt MATH identifier
07002039

Subjects
Primary: 26D15: Inequalities for sums, series and integrals
Secondary: 39B62: Functional inequalities, including subadditivity, convexity, etc. [See also 26A51, 26B25, 26Dxx]

Keywords
weighted mean concave mean Hardy inequality Kedlaya inequality

Citation

Páles, Zsolt; Pasteczka, Paweł. On Hardy-type inequalities for weighted means. Banach J. Math. Anal. 13 (2019), no. 1, 217--233. doi:10.1215/17358787-2018-0023. https://projecteuclid.org/euclid.bjma/1542358830


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