In this paper, new generalizations of Newton-type inequalities for the class of hyperbolic $p$-convex functions by utilizing Riemann-Liouville fractional integrals are established. By means of an auxiliary identity connected with the Riemann-Liouville fractional integrals, new estimates are obtained for functions that are hyperbolic $p$-convex. The established inequalities are further improved by the effective use of Hölder's inequality and the power-mean inequality. Examples along with graphs are provided to demonstrate the validity of the newly established inequalities and comparisons with existing results. The results of this paper will open up new avenues of research and may be generalized to other types of fractional operators and generalized convex functions.

A novel class of generalized strongly modified (GSM) $(p,h)$-convex functions (CFs) was presented in paper and its fundamental properties were established. Schur, Hermite-Hadamard (H-H), and Fejér inequalities were proved for this new notion of convexity. Several illustrations have been incorporated by selecting several GSM $(p,h)$-CFs to substantiate the existence and feasibility of Schur, H-H, and Fejér-type inequalities. These inequalities are valuable resources for analyzing the characteristics of newly defined GSM $(p,h)$-CFs. A comparison was given to show that the results of this study represent a significant improvement over those of earlier publications.

This paper examines a family of convex interval-valued (IVC) functions using the Riemann-Liouville (RL) integrals. Hermite-Hadamard (HH) and Hermite-Hadamard-Fejér (HHF) type inclusions are developed by employing the $s$-type convexity of interval-valued (Ⅳ) functions. Some inclusions for the product of $s$-type (IVC) functions are also established involving RL integrals. All main results are furthur refined into inclusions and inequalities for $s$-type IVC functions and $s$-type convex point-valued functions, respectively, involving the ordinary integral. In addition, several consequences of the primary results are explored demonstrating the connections between point-valued convex functions, IVC functions, and $s$-type IVC functions. Each key conclusion is validated numerically. The results of this paper might open the path for new avenues in modeling, optimization problems, interval differential equations, and fuzzy Ⅳ functions that involve both discrete and continuous variables simultaneously.

This paper addresses the constrained system of quaternion matrix equations incorporating anti-Hermitian properties, driven by the significance of symmetric solutions in diverse applications. Solvability conditions are determined via rank equalities and relationships derived from Moore–Penrose inverses and induced projectors. Explicit solution representations are obtained, utilizing the Moore–Penrose inverse and projections. The originality of the results is established through a novel technique based on quaternion row-column determinant theory, supported by a numerical validation. This approach retains its innovative character even when extended to complex matrix equations using conventional determinants.