In this work, we prove a parameterized fractional integral identity involving differentiable functions. Then, we use the newly established identity to establish some new parameterized fractional Hermite–Hadamard–Mercer-type inequalities for differentiable function. The main benefit of the newly established inequalities is that these inequalities can be converted into some new Mercer inequalities of midpoint type, trapezoidal type, and Simpson’s type for differentiable functions. Finally, we show the validation of the results with the help of some mathematical examples and their graphs.

中文翻译：

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微分函数的分数阶 Hermite-Hadamard-MERCER 不等式的研究

在这项工作中，我们证明了涉及可微函数的参数化分数积分恒等式。然后，我们使用新建立的恒等式为可微函数建立一些新的参数化分数 Hermite-Hadamard-Mercer 型不等式。新建立的不等式的主要好处是，这些不等式可以转换为一些新的可微函数的中点型、梯形型和辛普森型的 Mercer 不等式。最后，我们借助一些数学示例及其图表来展示结果的验证。