In the framework of tempered fractional integrals, we obtain a fundamental identity for differentiable convex functions. By employing this identity, we derive several modifications of fractional Milne inequalities, providing novel extensions to the domain of tempered fractional integrals. The research comprehensively examines significant functional classes, including convex functions, bounded functions, Lipschitzian functions, and functions of bounded variation.

中文翻译：

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带调和分数阶积分的 Milne 型不等式的综合研究

在调和分数积分的框架中，我们获得了可微凸函数的基本恒等式。通过利用这个恒等式，我们得出了分数米尔恩不等式的几种修改，为调和分数积分领域提供了新的扩展。该研究全面研究了重要的函数类别，包括凸函数、有界函数、Lipschitzian 函数和有界变分函数。