In the article, we describe the Grüss type inequality, provide some related inequalities by use of suitable fractional integral operators, address several variants by utilizing the generalized proportional Hadamard fractional (GPHF) integral operator. It is pointed out that our introduced new integral operators with nonlocal kernel have diversified applications. Our obtained results show the computed outcomes for an exceptional choice to the GPHF integral operator with parameter and the proportionality index. Additionally, we illustrate two examples that can numerically approximate these operators.

中文翻译：

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考虑广义比例Hadamard分数积分算子的新估计

在本文中，我们描述了Grüss型不等式，使用适当的分数积分算子提供了一些相关的不等式，通过利用广义比例Hadamard分数分数（GPHF）积分算子解决了几种变体。需要指出的是，我们引入的具有非本地内核的新积分运算符具有多种应用。我们获得的结果表明，对于具有参数和比例指数的GPHF积分算子来说，是一种特殊选择的计算结果。此外，我们说明了两个可以在数值上近似这些运算符的示例。