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Fractal Fract
Volume 9
Issue 8
10.3390/fractalfract9080519
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Wirtinger-Type Inequalities Involving Tempered Ψ-Fractional Derivatives with Applications

by
Qingzhe Wu
1,
Muming Zhang
2,*,
Jing Shao
1,
Muhammad Samraiz
3,*,
Humaira Javaid
3 and
Saima Naheed
3
1
College of Sciences, Shenyang University, Shenyang 110044, China
2
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
3
Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan
*
Authors to whom correspondence should be addressed.
Fractal Fract. 2025, 9(8), 519; https://doi.org/10.3390/fractalfract9080519
Submission received: 27 June 2025 / Revised: 4 August 2025 / Accepted: 6 August 2025 / Published: 8 August 2025
Versions Notes

Abstract

In this work, we explore Wirtinger-type inequalities involving tempered Ψ-Caputo fractional derivatives by utilizing Taylor’s formula. We establish more general inequalities for the same operator in Lp norms for p>1 by using Hölder’s inequality. Special cases are discussed in the form of remarks by highlighting their relationships with the existing literature. The derived results are also verified through illustrative examples, including tables and graphs. Moreover, applications of the obtained inequalities are discussed in the context of arithmetic and geometric means.
Keywords: Wirtinger inequality; tempered Ψ-Caputo fractional derivative; Taylor’s formula; Hölder’s inequality Wirtinger inequality; tempered Ψ-Caputo fractional derivative; Taylor’s formula; Hölder’s inequality

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MDPI and ACS Style

Wu, Q.; Zhang, M.; Shao, J.; Samraiz, M.; Javaid, H.; Naheed, S. Wirtinger-Type Inequalities Involving Tempered Ψ-Fractional Derivatives with Applications. Fractal Fract. 2025, 9, 519. https://doi.org/10.3390/fractalfract9080519

AMA Style

Wu Q, Zhang M, Shao J, Samraiz M, Javaid H, Naheed S. Wirtinger-Type Inequalities Involving Tempered Ψ-Fractional Derivatives with Applications. Fractal and Fractional. 2025; 9(8):519. https://doi.org/10.3390/fractalfract9080519

Chicago/Turabian Style

Wu, Qingzhe, Muming Zhang, Jing Shao, Muhammad Samraiz, Humaira Javaid, and Saima Naheed. 2025. "Wirtinger-Type Inequalities Involving Tempered Ψ-Fractional Derivatives with Applications" Fractal and Fractional 9, no. 8: 519. https://doi.org/10.3390/fractalfract9080519

APA Style

Wu, Q., Zhang, M., Shao, J., Samraiz, M., Javaid, H., & Naheed, S. (2025). Wirtinger-Type Inequalities Involving Tempered Ψ-Fractional Derivatives with Applications. Fractal and Fractional, 9(8), 519. https://doi.org/10.3390/fractalfract9080519

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