article · Mathematics
This research defines a new mathematical category known as (h1,h2)-Godunova-Levin preinvex functions, which generalises earlier concepts in mathematical analysis. Using this framework, the work establishes new formulations for Hermite-Hadamard, weighted Fejer, and trapezium-type inequalities. To validate these theoretical advances, non-trivial examples are provided to demonstrate that the newly derived mathematical relations hold true. The findings are further connected to practical mathematical operations, including the trapezoidal formula for numerical approximation, evaluations of probability density functions, special functions, and calculations of special means. Finally, the analysis considers the importance of order relations within these structures and outlines two open mathematical problems to guide subsequent research.
Mathematical inequalities are foundational to computational science, statistics, and engineering. By expanding the properties of preinvex functions, this study strengthens analytical tools used in numerical integration, probability estimation, and the evaluation of special mathematical functions, helping researchers establish more precise mathematical bounds and models.
This work represents early-stage fundamental mathematical research. Potential applications include improving algorithms for numerical integration and statistical analysis, which could interest software developers and quantitative analysts. However, the abstract indicates no direct commercial pathway or ready-to-market product, remaining at a purely theoretical stage.
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This note introduces a new class of preinvexity called (h1,h2)-Godunova-Levin preinvex functions that generalize earlier findings. Based on these notions, we developed Hermite-Hadamard, weighted Fejér, and trapezium type inequalities. Furthermore, we constructed some non-trivial examples in order to verify all the developed results. In addition, we discussed some applications related to the trapezoidal formula, probability density functions, special functions and special means. Lastly, we discussed the importance of order relations and left two open problems for future research. As an additional benefit, we believe that the present work can provide a strong catalyst for enhancing similar existing literature.
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DOI: 10.3390/math12030382
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