Novel notions of symmetric Hahn calculus and related inequalities

• Published in: Journal of inequalities and applications Vol. 2024; no. 1; pp. 147 - 24
• Main Authors: Butt, Saad Ihsan, Aftab, Muhammad Nasim, Fahad, Asfand, Wang, Yuanheng, Mohsin, Bandar Bin
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 20.11.2024; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Applications of Mathematics; Calculus; Hahn calculus; Hermite-Hadamard inequality; Hölder’s inequality; Mathematical functions; Mathematics; Mathematics and Statistics; Symmetric Hahn calculus; Symmetric Quantum calculus; Hölder’s inequality; 26A51; Hermite-Hadamard inequality; Symmetric Hahn calculus; 05A30; 39A13; 26D10; Symmetric Quantum calculus; 26D15; Hahn calculus
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-024-03228-9

In this manuscript, we demonstrate a graphical comparison analysis of the classical, quantum, and symmetric quantum derivatives for any continuous function, evaluated at z = 1 and q = 0.5 . We introduce the new notions of symmetric Hahn calculus and give the basic interpretations of derivative and integration of a function in Hahn symmetric quantum calculus. This enable us to to derive general Hölder’s inequality in symmetric Hahn calculus and as an application, we extract corresponding symmetric Hahn Minkowski’s inequality in general settings. Later, we construct an exciting Lemma in which we derive power rule of integration in symmetric Hahn calculus. An example is provided in which we give explicit analysis on four cases. Finally, these findings allow us to construct Hermite–Hadamard inequality and its related kinds in symmetric Hahn calculus. It is pertinent to mention that the particular case of our results recapture the results of symmetric quantum calculus.