The understanding of inequalities in convexity is crucial for studying local fractional calculus efficiency in many applied sciences. In the present work, we propose a new class of harmonically convex functions, namely, generalized harmonically --convex functions based on fractal set technique for establishing inequalities of Hermite-Hadamard type and certain related variants with respect to the Raina’s function. With the aid of an auxiliary identity correlated with Raina’s function, by generalized Hölder inequality and generalized power mean, generalized midpoint type, Ostrowski type, and trapezoid type inequalities via local fractional integral for generalized harmonically --convex functions are apprehended. The proposed technique provides the results by giving some special values for the parameters or imposing restrictive assumptions and is completely feasible for recapturing the existing results in the relative literature. To determine the computational efficiency of offered scheme, some numerical applications are discussed. The results of the scheme show that the approach is straightforward to apply and computationally very user-friendly and accurate.

中文翻译：

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一类新的凸函数的不等式及其在局部分数阶积分中的应用

对凸不等式的理解对于研究许多应用科学中的局部分数演算效率至关重要。在目前的工作中，我们提出了一类新的谐波凸函数，即广义谐波-基于分形集技术与对于雷纳的功能建立埃尔米特-哈德曼型和某些相关变量的不平等-凸函数。借助于与Raina函数相关的辅助恒等式，通过广义分数次积分，广义Hölder不等式和广义幂均值，广义中点型，Ostrowski型和梯形不等式--凸函数被理解。所提出的技术通过提供一些特殊的参数值或施加限制性假设来提供结果，并且对于重新获得相关文献中的现有结果是完全可行的。为了确定所提供方案的计算效率，讨论了一些数值应用。该方案的结果表明，该方法简单易用，并且在计算上非常人性化和准确。