Simpson-Type Integral Inequalities for Differentiable $(m_1,m_2)$-Exponentially Convex Functions via Variable-Order Fractional Integrals
Abstract
In this study, new Simpson-type integral inequalities are obtained by combining the flexible structure provided by variable-order fractional computation with the geometric properties of $(m_1,m_2)$-exponentially convex functions. First, a new integral identity involving variable-order Riemann–Liouville fractional integral operators is established for differentiable functions. This identity is used as a fundamental tool in the development of Simpson-type error estimators and integral inequalities. Then, various Simpson-type upper bounds are derived under the assumption that the absolute value of derivative of the function is $(m_1,m_2)$-exponentially convex. For this purpose, different types of error estimators are obtained using the Hölder inequality, the power-averaging approach, and appropriate integral techniques. The presented results include many classical and fractional Simpson inequalities in the literature as special cases and provide important generalizations within the framework of variable-order fractional integral operators. In addition, appropriate special cases and results are given to demonstrate the applicability of the obtained findings. The obtained results enrich the theory of Simpson-type inequalities within the framework of variable-order fractional calculus and provide a unified approach for deriving new error estimates associated with exponentially convex structures.
Keywords
Simpson-type inequalities, Fractional integrals of variable order, Riemann–Liouville fractional integral operators, (m1, m2)-Exponentially convex functions
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