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This paper introduces a new identity involving fractal–fractional integrals, which allow us to derive several new Bullen-type inequalities via generalized convexity. This study provides a significant advancement in the area of fractal–fractional inequalities, presenting a range of results not only for fractional integrals and fractal calculus, but also offering a refinement of the well-known Bullen-type inequality. We further explore the connections between generalized convexity and fractal–fractional integrals, showing how the concept of generalized convexity enables the establishment of error bounds for fractal–fractional integrals involving lower-order derivatives, with an emphasis on their applications in various fields. The findings expand the current understanding of fractal–fractional inequalities and offer new insights into the use of local fractional derivatives for analyzing functions with fractional-order properties. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel fractal–fractional identity, from which we derive new Simpson-type inequalities for functions whose first-order local fractional derivative exhibits generalized s-convexity in the second sense. This work introduces an approach that uses the first-order local fractional derivative, enabling the treatment of functions with lower regularity requirements compared to earlier studies. Additionally, we present two distinct methodological frameworks, one of which achieves greater precision by refining classical outcomes in the existing literature. The paper concludes with several practical applications that demonstrate the utility of our results. [ABSTRACT FROM AUTHOR]

This study aims to construct some new Milne-type integral inequalities for functions whose modulus of the local fractional derivatives is convex on the fractal set. To that end, we develop a novel generalized integral identity involving first-order generalized derivatives. Finally, as applications, some error estimates for the Milne-type quadrature formula and new inequalities for the generalized arithmetic and p-Logarithmic means are derived. This paper's findings represent a significant improvement over previously published results. The paper's ideas and formidable tools may inspire and motivate further research in this worthy and fascinating field. [ABSTRACT FROM AUTHOR]

In this article, we define and investigate the concept of generalizedγ$$ \gamma $$‐preinvex function on the Yang's fractal set ℝξ(0<ξ≤1)$$ {\mathbb{R}}&amp;#x0005E;{\xi}\kern3.0235pt \left(0&lt;\xi \le 1\right) $$. Based on the auxiliary definitions and involving local fractional integrals, we established several generalizations of Hermite‐Hadamard type inequalities under certain conditions. Additionally, we discuss some examples to test our outcomes and some applications in the form of bounds for generalized rth$$ {r}&amp;#x0005E;{th} $$ moment of a continuous random variable. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the notion of p -convex functions on fractal sets ℝ α ̂ (0 < α ̂ ≤ 1). Based on these novel ideas, we derived an auxiliary result depend on a three-step quadratic kernel by employing generalized p -convexity. Take into account the local fractal identity, we established novel Newton's type variants for the local differentiable functions. Several special cases are apprehended in the light of generalized convex functions and generalized harmonically convex functions. This novel strategy captures several existing results in the relative literature. Application is obtained in cumulative distribution function and generalized special weighted means to confirm the relevance and computational effectiveness of the considered method. Finally, we supposed that the consequences of this paper can stimulate those who are interested in fractal analysis. [ABSTRACT FROM AUTHOR]

In this paper, the definition of generalized s-preinvex function on Yang's fractal sets ℝ γ (0 < γ ≤ 1) is proposed, and the generalized Hermite–Hadamard's inequality for this class of functions is established. By using this convexity, some generalized Hermite–Hadamard type integral inequalities with parameters are established. For these inequalities, the absolute values of twice local fractional order derivative of the functions are generalized s-preinvex functions. Some special integral inequalities can be obtained by assigning special values to the obtained inequalities, and two examples are given to illustrate our results. Finally, we propose the applications of the results in numerical integration and error estimation. [ABSTRACT FROM AUTHOR]

In this paper, a new identity with parameters involving local fractional integrals is derived. Using this identity, some general local fractional integral inequalities for generalized preinvex functions are established. A parallel development is deduced for generalized preconcave functions. Taking special values for the parameters, some generalized midpoint inequalities, trapezoidal inequalities and Simpson inequalities are obtained. Finally, as some applications, error estimates of numerical integration for local fractional integrals are given. [ABSTRACT FROM AUTHOR]