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Article

A Half-Discrete Hardy–Mulholland-Type Inequality Involving One Multiple Upper Limit Function and One Partial Sum

1
Institute of Applied Mathematics, Longyan University, Longyan 364012, China
2
School of Mathematics, Guangdong University of Education, Guangzhou 510303, China
*
Author to whom correspondence should be addressed.
Mathematics 2025, 13(15), 2497; https://doi.org/10.3390/math13152497 (registering DOI)
Submission received: 5 July 2025 / Revised: 31 July 2025 / Accepted: 1 August 2025 / Published: 3 August 2025
(This article belongs to the Special Issue Advances in Convex Analysis and Inequalities)

Abstract

In this paper, by using the techniques of real analysis, with the help of the Euler–Maclaurin summation formula, Abel’s summation by parts formula, and the differentiation mid-value theorem, we establish a half-discrete Hardy–Mulholland-type inequality involving one multiple upper limit function and one partial sum. Based on the obtained inequality, we characterize the condition of the best possible constant factor related to several parameters. At the end of the paper, we illustrate that some new half-discrete Hardy–Mulholland-type inequalities can be deduced from the special values of the parameters. Our results enrich the current results in the study of half-discrete Hardy–Mulholland-type inequalities.
Keywords: half-discrete Hardy–Mulholland’s inequality; multiple upper limit function; partial sum; best possible constant factor half-discrete Hardy–Mulholland’s inequality; multiple upper limit function; partial sum; best possible constant factor

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MDPI and ACS Style

Yang, B.; Wu, S.; Liao, J. A Half-Discrete Hardy–Mulholland-Type Inequality Involving One Multiple Upper Limit Function and One Partial Sum. Mathematics 2025, 13, 2497. https://doi.org/10.3390/math13152497

AMA Style

Yang B, Wu S, Liao J. A Half-Discrete Hardy–Mulholland-Type Inequality Involving One Multiple Upper Limit Function and One Partial Sum. Mathematics. 2025; 13(15):2497. https://doi.org/10.3390/math13152497

Chicago/Turabian Style

Yang, Bicheng, Shanhe Wu, and Jianquan Liao. 2025. "A Half-Discrete Hardy–Mulholland-Type Inequality Involving One Multiple Upper Limit Function and One Partial Sum" Mathematics 13, no. 15: 2497. https://doi.org/10.3390/math13152497

APA Style

Yang, B., Wu, S., & Liao, J. (2025). A Half-Discrete Hardy–Mulholland-Type Inequality Involving One Multiple Upper Limit Function and One Partial Sum. Mathematics, 13(15), 2497. https://doi.org/10.3390/math13152497

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