Wedad Saleh

In this paper, we present a novel identity for twice partially differentiable mappings. Based on this identity, new fractional Bullen-type inequalities for differentiable functions of two variables, which are convex on the coordinate via Riemann–Liouville fractional integral operators are derived. Other results are obtained by applying integral inequalities, including the Hölder, the improved Hölder, and the power mean inequalities. We apply these findings to special means. A numerical example with graphical illustrations is presented to demonstrate the validity and effectiveness of our theoretical findings.

The goal of this study is to establish a new type of Milne-type inequality in the scope of fractional calculus with the aid of proportional Caputo-hybrid operators. We will focus on two different scopes of regularity, which contain functions whose first and second derivatives are convex, and functions whose first and second derivatives are Lipschitz continuous. We will base these estimates on a new integral identity of proportional Caputo-hybrid integrals. We will show that the smoothness of the derivative influences the shape of the bounds. Convexity will cause symmetry. Lipschitz continuity will contain bounds on the modulus of continuity. To show that our results are accurate and easy to obtain, we included a full numerical example with graphics and applications to quadrature error estimation.

In this work, we introduce a new class of functions, referred to as multiplicative beta-convex functions, which extends several existing convexity concepts in the context of multiplicative calculus. Based on this notion, we establish a Hermite–Hadamard inequality involving multiplicative Katugampola fractional integrals. We also derive a new integral identity associated with these operators, which allows us to construct a unified family of parameterized three-point Newton–Cotes-type inequalities. These inequalities apply to functions whose first-order [ ]-derivatives, in [ ]-absolute value, are multiplicative beta-convex. By suitably selecting the parameters, our general framework yields various well-known inequalities as special cases, including midpoint, trapezoidal, Simpson-, and Bullen-type inequalities. These arise naturally under specific subclasses of beta-convexity, such as multiplicative s-, P-, and [ ]-convexity. To corroborate our theoretical results, we present a numerical example together with graphical illustrations that demonstrate the validity and sharpness of the obtained bounds. Our contributions advance the development of fractional multiplicative analysis and offer new perspectives for applications in numerical approximation and the theory of inequalities.

This paper introduces the first theory of tempered fractional integrals within the framework of [ ]-calculus, a multiplicative non-Newtonian system for positive-valued functions with positive arguments. We begin by formulating the multiplicative Riemann–Liouville integral in its pure multiplicative form and extend it to include an exponential tempering parameter. A new multiplicative [ ]-incomplete Gamma function is defined to characterize these operators. Furthermore, we introduce and analyze multiplicative convexity in [ ]-calculus, along with novel multiplicative formulations of the classical midpoint and trapezoidal quadrature rules. We then establish the Hermite–Hadamard inequalities for [ ]-convex functions and derive two novel multiplicative integral identities, leading to midpoint- and trapezium-type bounds. Numerical examples with graphical illustrations, applications to quadrature rules, and connections to special means validate our results. The proposed framework fills a critical gap in non-Newtonian analysis and provides new tools for modeling scale-invariant phenomena in economics, biology, and signal processing.

In this paper, we establish a new integral identity and derive novel Radau-type integral inequalities for functions whose first derivatives are convex. These results extend and generalize some existing inequalities of Radau type in the literature. As an application, we provide bounds for certain mathematical means based on the obtained inequalities. We believe that the findings presented here will stimulate further research into integral inequalities and their applications in related fields.

This paper explores integral inequalities within the framework of fractional integrals having exponential kernels through a parameterized approach. A novel integral identity with a parameter is introduced, forming the basis for deriving a general inequality. From this, several classical inequalities, including midpoint, trapezium, Bullen, Simpson, and corrected Simpson types, are obtained for differentiable convex functions. Numerical examples and graphical analyses are presented to verify the validity of the proposed results. The findings are further demonstrated through applications to special functions, showcasing the versatility of the developed framework.

In this paper, we introduce a new class of stochastic fractional integrals, namely, the stochastic mean‐square ‐Riemann–Liouville fractional integrals, by combining elements of fractional calculus and stochastic analysis. We first establish the necessary theoretical framework by recalling fundamental concepts from both domains. A novel integral identity involving these operators is then derived, which serves as a key tool for our main results. Using this identity, we prove several ‐fractional Maclaurin‐type inequalities for differentiable ‐convex stochastic processes. These inequalities extend classical deterministic results to the stochastic setting and generalize them via ‐fractional operators, offering enhanced flexibility in modeling uncertainty and memory effects. The obtained results contribute to the growing theory of stochastic fractional analysis and provide new tools for the study of probabilistic bounds and convexity in random environments.

This paper presents a novel framework for Katugampola fractional multiplicative integrals, advancing recent breakthroughs in fractional calculus through a synergistic integration of multiplicative analysis. Motivated by the growing interest in fractional calculus and its applications, we address the gap in generalized inequalities for multiplicative s-convex functions by deriving a Hermite–Hadamard-type inequality tailored to Katugampola fractional multiplicative integrals. A cornerstone of our work involves the derivation of two groundbreaking identities, which serve as the foundation for midpoint- and trapezoid-type inequalities designed explicitly for mappings whose multiplicative derivatives are multiplicative s-convex. These results extend classical integral inequalities to the multiplicative fractional calculus setting, offering enhanced precision in approximating nonlinear phenomena.