The build-up method is a powerful class of propagation rules that generate self-dual codes over finite fields and unitary rings. Recently, it was extended to non-unitary rings of order 4, to generate quasi self-dual codes. In the present paper, we introduce three such propagation rules to generate self-orthogonal, self-dual and quasi self-dual codes over a special non-unitary ring of order 9. As an application, we classify the three categories of codes completely in length at most 3, and partially in lengths 4 and 5, up to monomial equivalence.

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.

In this paper, new generalizations of Newton-type inequalities for the class of strongly convex functions by utilizing Riemann–Liouville fractional integrals are established. New estimates are obtained for functions that are strongly convex. The established inequalities are further improved. Examples, along with graphs, are provided to demonstrate the validity of the newly established inequalities and comparisons with existing results. It is expected that 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.

This paper establishes an extended theoretical framework centered on the duality of codes constructed over a special class of non-unital, commutative, local rings of order $p^2$, where p is a prime satisfying $p\equiv 1mod4$ or $p\equiv 3mod4$. The work expands the traditional scope of coding theory by developing and adapting a generalized recursive approach to produce quasi-self-dual and self-dual codes within this algebraic setting. While the method for code generation is rooted in the classical build-up technique, the primary focus is on the duality properties of the resulting codes—especially how these properties manifest under different congruence conditions on p. Computational examples are provided to illustrate the effectiveness of the proposed methods.

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.

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.