The present investigation focus on applying the theories of differential subordination, differential superordination and related sandwich-type results for the study of some subclasses of symmetric functions connected through a linear extended multiplier operator, which was previously defined by involving the $q$-Choi-Saigo-Srivastava operator. The aim of the paper is to define a new class of analytic functions using the aforementioned linear extended multiplier operator and to obtain sharp differential subordinations and superordinations using functions from the new class. Certain subclasses are highlighted by specializing the parameters involved in the class definition, and corollaries are obtained as implementations of those new results using particular values for the parameters of the new subclasses. In order to show how the results apply to the functions from the recently introduced subclasses, numerical examples are also provided.

The operational and algebraic framework offers a powerful and systematic approach for investigating the structural properties of hybrid special polynomial families. In this work, we introduce a new class of Gould–Hopper Sheffer-based Appell polynomials (GHSbAP) by combining the Gould–Hopper polynomial structure with the general theory of Sheffer and Appell sequences. The offered construction is implemented by means of exponential generating functions and operational methods based upon the principle of monomiality. Basic properties of the GHSbAP family are constructed including generating functions, series representations, operational identities, quasi-monomial behavior, and differential equations. Further, the computationally efficient characterization of determinant representation is derived through the relation between Sheffer sequences and generalized Riordan arrays. A number of illustrative cases such as Gould–Hopper-Sheffer based Bernoulli and Euler polynomials are demonstrated.

The roots of non-linear equations are a major challenge in many scientific and professional fields. This problem has been approached in a number of ways, including use of the sequential Newton's method and the traditional Weierstrass simultaneous iterative scheme. To approximate all of the roots of a given nonlinear equation, sequential iterative algorithms must use a deflation strategy because rounding errors can produce inaccurate results. This study aims to develop an efficient numerical simultaneous scheme for approximating all nonlinear equations' roots of convergence order 12. The numerical outcomes of the considered engineering problems show that, in terms of accuracy, validations, error, computational CPU time, and residual error, recently developed simultaneous methods perform better than existing methods in the literature.

In this scholarly article, we present novel generalized subclasses of $\nu$ uniformly starlike functions of order $\rho$, designated as $M(\tau ,\rho ,\nu )$, alongside $\nu$ uniformly convex functions of order $\rho$, referred to as $N(\tau ,\rho ,\nu )$. We provide comprehensive coefficient characterizations that delineate the conditions under which analytic functions are classified within the newly established subclasses of uniformly starlike and uniformly convex families, respectively. Furthermore, we conduct an analysis of the implications of the Bessel function and explore the consequences of the Gaussian hypergeometric function on these mathematical classes to substantiate an inclusion property for analytic functions that reside within these subclasses.

The petroleum industry has a complex, inflexible and challenging supply chain (SC) that impacts both the national economy as well as people’s daily lives with a range of services, including transportation, heating, electricity, lubricants, as well as chemicals and petrochemicals. In the petroleum industry, supply chain management presents several challenges, especially in the logistics sector, that are not found in other industries. In addition, logistical challenges contribute significantly to the cost of oil. Uncertainty regarding customer demand and supply significantly affects SC networks. Hence, SC flexibility can be maintained by addressing uncertainty. On the other hand, in the real world, decision-making challenges are often ambiguous or vague. In some cases, measurements are incorrect owing to measurement errors, instrument faults, etc., which lead to a pentagonal fuzzy number (PFN) which is the extension of a fuzzy number. Therefore, it is necessary to develop quantitative models to optimize logistics operations and supply chain networks. This study proposed a linear programming model under an uncertain environment. The model minimizes the cost along the refineries, depots, multimode transport and demand nodes. Further developed pentagonal fuzzy optimization, an alternative approach is developed to solve the downstream supply chain using the mixed-integer linear programming (MILP) model to obtain a feasible solution to the fuzzy transportation cost problem. In this model, the coefficient of the transportation costs and parameters is assumed to be a pentagonal fuzzy number. Furthermore, defuzzification is performed using an accuracy function. To validate the model and technique and feasibility solution, an illustrative example of the oil and gas SC is considered, providing improved results compared with existing techniques and demonstrating its ability to benefit petroleum companies is the objective of this study.