Caputo-Hadamard fractional Wirtinger-type inequalities via Taylor expansion with applications to classical means

Muhammad Samraiz; Humaira Javaid; Muath Awadalla; Hajer Zaway; Muhammad Samraiz; Humaira Javaid; Muath Awadalla; Hajer Zaway

doi:10.3934/math.2025730

AIMS Mathematics

2025, Volume 10, Issue 7: 16334-16354. doi: 10.3934/math.2025730

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Research article

Caputo-Hadamard fractional Wirtinger-type inequalities via Taylor expansion with applications to classical means

1.
Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan
2.
Department of Mathematics and Statistics, College of Science, King Faisal University, Al Ahsa 31982, Saudi Arabia

Received: 15 May 2025 Revised: 07 July 2025 Accepted: 14 July 2025 Published: 21 July 2025
MSC : 26A33, 26D10, 26D15, 41A58

In this paper, we explored Caputo-Hadamard fractional Wirtinger-type inequalities using Taylor's formula. The main findings were derived by utilizing Hölder's inequality to derive results for Caputo-Hadamard fractional derivatives in terms of $ \textbf{L}_{q} $ norms for $ q > 1 $. Through graphical interpretation, we confirmed the validity of the results. A flowchart summarizing the logical progression from lemma to theorem was added for clarity. Furthermore, inequalities were also derived for Hadamard fractional derivatives. Finally, we discussed the applications of Wirtinger-type inequalitie which incorporates arithmetic mean and geometric mean-type inequalities.
- Taylor's formula,
- Wirtinger inequality,
- Caputo-Hadamard fractional derivative,
- Hölder's inequality
Citation: Muhammad Samraiz, Humaira Javaid, Muath Awadalla, Hajer Zaway. Caputo-Hadamard fractional Wirtinger-type inequalities via Taylor expansion with applications to classical means[J]. AIMS Mathematics, 2025, 10(7): 16334-16354. doi: 10.3934/math.2025730

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Abstract

In this paper, we explored Caputo-Hadamard fractional Wirtinger-type inequalities using Taylor's formula. The main findings were derived by utilizing Hölder's inequality to derive results for Caputo-Hadamard fractional derivatives in terms of $ \textbf{L}_{q} $ norms for $ q > 1 $. Through graphical interpretation, we confirmed the validity of the results. A flowchart summarizing the logical progression from lemma to theorem was added for clarity. Furthermore, inequalities were also derived for Hadamard fractional derivatives. Finally, we discussed the applications of Wirtinger-type inequalitie which incorporates arithmetic mean and geometric mean-type inequalities.

References

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