Structural properties and estimation of bounds for Jensen gaps and Hermite–Hadamard inequalities associated with $ \sigma $-convex functions

Yuanheng Wang; Muhammad Zakria Javed; Swaiza Akhlaq; Muhammad Uzair Awan; Awais Gul Khan; Hala Mostafa; Yuanheng Wang; Muhammad Zakria Javed; Swaiza Akhlaq; Muhammad Uzair Awan; Awais Gul Khan; Hala Mostafa

doi:10.3934/math.2026820

AIMS Mathematics

2026, Volume 11, Issue 7: 20195-20223. doi: 10.3934/math.2026820

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Research article

Structural properties and estimation of bounds for Jensen gaps and Hermite–Hadamard inequalities associated with $ \sigma $-convex functions

1.
Department of Mathematics, Zhejiang Guangsha Vocational and Technical University of Construction, Jinhua 322100, China
2.
Department of Mathematics, Government College University Faisalabad, Pakistan
3.
Department of Information Technology, College of Computer and Information Sciences, Princess Nourah bint Abdulrahman University, P. O. Box 84428, Riyadh 11671, Saudi Arabia

Received: 28 April 2026 Revised: 11 June 2026 Accepted: 25 June 2026 Published: 09 July 2026
MSC : 26A42, 26A48, 26A51, 26D10, 26D15

Generalized convexity plays a prolific role in the development and refinement of inequalities. This manuscript assesses some key regularity properties of $ \sigma $ convex functions and related generalized fundamental inequalities. Particularly, we proved the relation between $ \sigma $-convex functions and classical convexity. Based on this result, we discussed the first-order characterization of $ \sigma $-convex functions. Next, we derived some new refinements of Jensen's inequality, including an interpolating Jensen inequality and bounds of Jensen's functional. By taking the benefit of $ \sigma $-convexity, we constructed general a Hermite–Hadamard inequality and its weighted analog for symmetric functions as well as and several other inequalities. Our results are novel due to their unifying characteristics because several robust estimates and inequalities can be recaptured for different choices of the transformation function $ \sigma $. Finally, we addressed some useful applications to normed spaces, entropy measures, and the general mean inequality.
- Convex function,
- Jensen inequality,
- Hermite–Hadamard inequality,
- Quasi-mean,
- Hölder inequality,
- Shannon entropy
Citation: Yuanheng Wang, Muhammad Zakria Javed, Swaiza Akhlaq, Muhammad Uzair Awan, Awais Gul Khan, Hala Mostafa. Structural properties and estimation of bounds for Jensen gaps and Hermite–Hadamard inequalities associated with $ \sigma $-convex functions[J]. AIMS Mathematics, 2026, 11(7): 20195-20223. doi: 10.3934/math.2026820

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Abstract

Generalized convexity plays a prolific role in the development and refinement of inequalities. This manuscript assesses some key regularity properties of $ \sigma $ convex functions and related generalized fundamental inequalities. Particularly, we proved the relation between $ \sigma $-convex functions and classical convexity. Based on this result, we discussed the first-order characterization of $ \sigma $-convex functions. Next, we derived some new refinements of Jensen's inequality, including an interpolating Jensen inequality and bounds of Jensen's functional. By taking the benefit of $ \sigma $-convexity, we constructed general a Hermite–Hadamard inequality and its weighted analog for symmetric functions as well as and several other inequalities. Our results are novel due to their unifying characteristics because several robust estimates and inequalities can be recaptured for different choices of the transformation function $ \sigma $. Finally, we addressed some useful applications to normed spaces, entropy measures, and the general mean inequality.

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