Computation of weighted Jensen and Lah-Ribarič inequalities for analogue of $(g, h;m)$-convexity

Matloob Anwar; Muna Elsadig; Ghulam Farid; Josip Pečarić; Jongsuk Ro; Matloob Anwar; Muna Elsadig; Ghulam Farid; Josip Pečarić; Jongsuk Ro

doi:10.3934/math.20261312

AIMS Mathematics

2026, Volume 11, Issue 10: 33543-33570. doi: 10.3934/math.20261312

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Research article

Computation of weighted Jensen and Lah-Ribarič inequalities for analogue of $(g, h;m)$-convexity

1.
School of Natural Sciences, NUST, Islamabad, Pakistan
2.
Department of Information Systems, College of Computer and Information Sciences, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
3.
Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan
4.
Croatian Academy of Sciences and Arts, Zagreb, Croatia
5.
School of Electrical and Electronics Engineering, Chung-Ang University, Dongjak-gu, Seoul 06974, Republic of Korea
6.
Department of Intelligent Energy and Industry, Chung-Ang University, Dongjak-gu, Seoul 06974, Republic of Korea

Received: 08 August 2026 Revised: 11 September 2026 Accepted: 29 September 2026 Published: 15 October 2026
MSC : 26D15, 26A51, 39B62

We formulate an $H$-analogue of $(g, h;m)$-convexity by treating the nonnegative coefficient $H$ as an independent structural variable. Two endpoint representations of a point in an interval are used to establish weighted Lah-Ribarič inequalities for the resulting convex and concave classes. Under concavity or convexity assumptions on $H$, weighted Jensen's inequality yields normalized endpoint refinements and corresponding reverse estimates. A further extension is obtained when $H$ is supermultiplicative; its natural concave counterpart follows under submultiplicativity. Finite-sum versions produce $H$-analogues of the Giaccardi, Popoviciu, and Petrović inequalities. We also derive an increasing-functional inequality and an estimate for the associated Jensen gap. The framework contains the $(g, h;m)$-convex setting and, through linear, power, constant, and product-generated choices of $H$, recovers or extends a broad collection of known Lah-Ribarič and Jensen-type inequalities. A complementary two-coefficient formulation is also established to cover the general $(g, h;\alpha-m)$-convex case without identifying $ h(1-t^\alpha) $ with $h((1-t)^\alpha)$.
- $(g, h;m)$-convexity,
- Lah-Ribaričinequality,
- Jensen inequality,
- supermultiplicative functions,
- integral inequalities
Citation: Matloob Anwar, Muna Elsadig, Ghulam Farid, Josip Pečarić, Jongsuk Ro. Computation of weighted Jensen and Lah-Ribarič inequalities for analogue of $(g, h;m)$-convexity[J]. AIMS Mathematics, 2026, 11(10): 33543-33570. doi: 10.3934/math.20261312

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Abstract

We formulate an $H$-analogue of $(g, h;m)$-convexity by treating the nonnegative coefficient $H$ as an independent structural variable. Two endpoint representations of a point in an interval are used to establish weighted Lah-Ribarič inequalities for the resulting convex and concave classes. Under concavity or convexity assumptions on $H$, weighted Jensen's inequality yields normalized endpoint refinements and corresponding reverse estimates. A further extension is obtained when $H$ is supermultiplicative; its natural concave counterpart follows under submultiplicativity. Finite-sum versions produce $H$-analogues of the Giaccardi, Popoviciu, and Petrović inequalities. We also derive an increasing-functional inequality and an estimate for the associated Jensen gap. The framework contains the $(g, h;m)$-convex setting and, through linear, power, constant, and product-generated choices of $H$, recovers or extends a broad collection of known Lah-Ribarič and Jensen-type inequalities. A complementary two-coefficient formulation is also established to cover the general $(g, h;\alpha-m)$-convex case without identifying $ h(1-t^\alpha) $ with $h((1-t)^\alpha)$.

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