Dyadic perspectives on the weighted Pólya–Knopp inequality

Mehmet Zeki Sarıkaya

Dyadic perspectives on the weighted Pólya–Knopp inequality

Abstract

In this paper, we present a direct proof of the weighted Pólya-Knopp inequality based on a dyadic decomposition approach. The method relies on basic properties of nonincreasing functions and standard weight conditions in the sense of Ariño-Muckenhoupt. Compared with existing approaches, the argument avoids the use of auxiliary technical lemmas and provides a more transparent route to the inequality.

Keywords

References

[1] Ariño, M. A., & Muckenhoupt, B. (1990). Maximal functions on classical Lorentz spaces and Hardy’s inequality with weights for nonincreasing functions. Transactions of the American Mathematical Society, 320, 727–735. https://doi.org/10.2307/2001699
[2] Carleson, L. (1954). A proof of an inequality of Carleman. Proceedings of the American Mathematical Society, 5, 932–933. https://doi.org/10.1090/S0002-9939-1954-0065601-3
[3] Stepanov, V. D. (1993). The weighted Hardy’s inequality for nonincreasing functions. Transactions of the American Mathematical Society, 338(1), 173–186. https://doi.org/10.2307/2154450
[4] Heinig, H. P., & Stepanov, V. D. (1993). Weighted Hardy inequalities for increasing functions. Canadian Journal of Mathematics, 45(1), 104–116. https://doi.org/10.4153/CJM-1993-006-3
[5] Andersen, K. F. (1991). Weighted generalized Hardy inequalities for nonincreasing functions. Canadian Journal of Mathematics, 43(6), 1121–1135. https://doi.org/10.4153/CJM-1991-065-9
[6] Sawyer, E. (1985). A weighted weak type inequality for the maximal function. Proceedings of the American Mathematical Society, 93(4), 610–614.
[7] Sawyer, E. (1986). Weighted inequalities for the one-sided Hardy-Littlewood maximal functions. Transactions of the American Mathematical Society, 297(1), 53–61. https://doi.org/10.2307/2000455
[8] Persson, L. -E., & Stepanov, V. D. (2002), Weighted integral inequalities with the geometric mean operator. Journal of Inequalities and Applications, 7, 727–746. https://doi.org/10.1155/S1025583402000371

Details

Primary Language

English

Subjects

Pure Mathematics (Other), Applied Mathematics (Other)

Journal Section

Research Article

Authors

Mehmet Zeki Sarıkaya ^*
0000-0002-6165-9242
Türkiye

Publication Date

June 5, 2026

Submission Date

April 30, 2026

Acceptance Date

May 30, 2026

Published in Issue

Year 2026 Volume: 2 Number: 1

IZ

https://izlik.org/JA78CM39SZ

Cite

RIS / Bibtex

APA

Sarıkaya, M. Z. (2026). Dyadic perspectives on the weighted Pólya–Knopp inequality. Düzce Mathematical Research, 2(1), 67-75. https://izlik.org/JA78CM39SZ

AMA

1.Sarıkaya MZ. Dyadic perspectives on the weighted Pólya–Knopp inequality. Düzce Mathematical Research. 2026;2(1):67-75. https://izlik.org/JA78CM39SZ

Chicago

Sarıkaya, Mehmet Zeki. 2026. “Dyadic Perspectives on the Weighted Pólya–Knopp Inequality”. Düzce Mathematical Research 2 (1): 67-75. https://izlik.org/JA78CM39SZ.

EndNote

Sarıkaya MZ (June 1, 2026) Dyadic perspectives on the weighted Pólya–Knopp inequality. Düzce Mathematical Research 2 1 67–75.

IEEE

[1]M. Z. Sarıkaya, “Dyadic perspectives on the weighted Pólya–Knopp inequality”, Düzce Mathematical Research, vol. 2, no. 1, pp. 67–75, June 2026, [Online]. Available: https://izlik.org/JA78CM39SZ

ISNAD

Sarıkaya, Mehmet Zeki. “Dyadic Perspectives on the Weighted Pólya–Knopp Inequality”. Düzce Mathematical Research 2/1 (June 1, 2026): 67-75. https://izlik.org/JA78CM39SZ.

JAMA

1.Sarıkaya MZ. Dyadic perspectives on the weighted Pólya–Knopp inequality. Düzce Mathematical Research. 2026;2:67–75.

MLA

Sarıkaya, Mehmet Zeki. “Dyadic Perspectives on the Weighted Pólya–Knopp Inequality”. Düzce Mathematical Research, vol. 2, no. 1, June 2026, pp. 67-75, https://izlik.org/JA78CM39SZ.

Vancouver

1.Mehmet Zeki Sarıkaya. Dyadic perspectives on the weighted Pólya–Knopp inequality. Düzce Mathematical Research [Internet]. 2026 Jun. 1;2(1):67-75. Available from: https://izlik.org/JA78CM39SZ