In this paper, the newly proposed concept of the generalized proportional fractional integral operator with respect to another function [Formula: see text] has been utilized to generate integral inequalities using convex function. This new concept will have the option to reduce self-similitudes in the fractional attractors under investigation. We discuss the implications and other consequences of the integral inequalities concerning the generalized proportional fractional integral operator with respect to another function [Formula: see text] are derived here and these outcomes permit us specifically to generalize some classical inequalities. Certain intriguing subsequent consequences of the fundamental hypotheses are also figured. It is to be supposed that this investigation will provide new directions in the quantum theory of capricious nature.

中文翻译：

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通过广义比例分数积分算子对另一个函数的积分不等式的新估计

在本文中，新提出的广义比例分数积分算子关于另一个函数的概念[公式：见正文]已被用于使用凸函数生成积分不等式。这个新概念将可以选择减少正在研究的分数吸引子的自相似性。我们讨论了关于广义比例分数积分算子相对于另一个函数的积分不等式的含义和其他后果 [公式：参见文本]，这些结果允许我们特别推广一些经典不等式。还计算了基本假设的某些有趣的后续后果。可以认为，这项研究将为反复无常的量子理论提供新的方向。