In this paper, we introduce a new proportional Caputo-hybrid-conformable (PCHC) fractional integral operator that combines proportional Caputo-hybrid and conformable fractional structures within a unified framework. By employing this operator, a fundamental integral identity is established, and several Hermite-Hadamard-Mercer-type inequalities for convex functions are derived. The proposed framework establishes a boundary limiting connection with the proportional Caputo-hybrid operator. In particular, as $ \beta\to1 $ and $ \eta\to1^{-} $, and after setting $ m = a $ and $ n = b $, Theorem 2.1 reduces to Theorem 1.7 of [M. Z. Sarikaya, On Hermite-Hadamard type inequalities for proportional Caputo-hybrid operator, Konuralp J. Math., 11 (2023), 31-39]. Several limiting cases and numerical examples are provided to illustrate the effectiveness and applicability of the proposed results. These findings contribute to the study of fractional integral inequalities and convex analysis in generalized fractional settings.
Citation: Jen Chieh Lo. Hermite-Hadamard-Mercer-type inequalities via proportional Caputo-hybrid-conformable operator[J]. AIMS Mathematics, 2026, 11(9): 28055-28077. doi: 10.3934/math.20261119
In this paper, we introduce a new proportional Caputo-hybrid-conformable (PCHC) fractional integral operator that combines proportional Caputo-hybrid and conformable fractional structures within a unified framework. By employing this operator, a fundamental integral identity is established, and several Hermite-Hadamard-Mercer-type inequalities for convex functions are derived. The proposed framework establishes a boundary limiting connection with the proportional Caputo-hybrid operator. In particular, as $ \beta\to1 $ and $ \eta\to1^{-} $, and after setting $ m = a $ and $ n = b $, Theorem 2.1 reduces to Theorem 1.7 of [M. Z. Sarikaya, On Hermite-Hadamard type inequalities for proportional Caputo-hybrid operator, Konuralp J. Math., 11 (2023), 31-39]. Several limiting cases and numerical examples are provided to illustrate the effectiveness and applicability of the proposed results. These findings contribute to the study of fractional integral inequalities and convex analysis in generalized fractional settings.
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Jen Chieh Lo. Hermite-Hadamard-Mercer-type inequalities via proportional Caputo-hybrid-conformable operator[J]. AIMS Mathematics, 2026, 11(9): 28055-28077. doi: 10.3934/math.20261119
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