This paper presents a new general subfamily ${\mathcal{N}}_{{\Sigma }_m}^{u,v}(\eta ,\mu ,\gamma ,\ell )$ of the family ${\Sigma }_m$ that contains holomorphic normalized m-fold symmetric bi-univalent functions in the open unit disk $\mathbb{D}$ associated with the Ruscheweyh derivative operator. For functions belonging to the family introduced here, we find estimates of the Taylor–Maclaurin coefficients $\left|a_{m+1}\right|$ and $\left|a_{2m+1}\right|$, and the consequences of the results are discussed. The current findings both extend and enhance certain recent studies in this field, and in specific scenarios, they also establish several connections with known results.

This paper presents two subclasses of analytic and bi-univalent functions associated with the Ruscheweyh derivative operator to investigate the bounds for |a2| and |a3|, where a2 and a3 are the initial Tayler-Maclaurin coefficients. The current results would generalize and improve some corresponding recent works. Additionally, in certain cases, our estimates correct some of the existing coefficient bounds.

Linear B-spline functions are essential mathematical tools for deriving approximate solutions for both linear and nonlinear functional equations of arbitrary order due to their simplicity and computational efficiency and ability to approximate functions accurately. In this paper, a system of multi-fractional order nonlinear integro-differential equations of Volterra-Hammerstein Type (NIFDEs-VH) with variable coefficients, in which the multi-fractional derivatives are described in the Caputo sense, is solved using a first-degree B-spline curve. The proposed method utilizes linear B-splines functions and Gauss-Legendre quadrature for numerical integration to convert the system of NIFDEs-VH into a system of nonlinear algebraic equations containing unknown control points. Newton’s method is selected for its efficiency in iteratively solving nonlinear algebraic systems, specifically aiming to determine the control points and obtain linear approximate functions. Four algorithms are proposed and implemented in MatLab to operationalize these methods, offering general computational frameworks. The effectiveness and accuracy of the proposed approach are demonstrated through various illustrative examples. Each example includes graphical representations of both exact and approximate solutions. In addition, the results are presented in tabular form.

Starlike and convex functions have gained increased prominence in both academic literature and practical applications over the past decade. Concurrently, logarithmic coefficients play a pivotal role in estimating diverse properties within the realm of analytic functions, whether they are univalent or nonunivalent. In this paper, we rigorously derive bounds for specific Toeplitz determinants involving logarithmic coefficients pertaining to classes of convex and starlike functions concerning symmetric points. Furthermore, we present illustrative examples showcasing the sharpness of these established bounds. Our findings represent a substantial contribution to the advancement of our understanding of logarithmic coefficients and their profound implications across diverse mathematical contexts.

The primary objective of this article is to develop new methods based on quadratic B-spline functions to numerically solve the general nonlinear system of Volterra integro-fractional differential equations of Hammerstein type (NSVIFDE-H) described in the Caputo sense. Using the Gauss quadrature method for numerical integration and quadratic B-spline functions, we propose four novel strategies for solving the considered equations. A key advantage of the proposed schemes is their ability to transform the NSVIFDE-H into a nonlinear system of ordinary algebraic equations, which can be effectively solved using appropriate techniques. By deriving the general form of the Caputo fractional derivative of the quadratic B-spline curve, we convert the system into a set of nonlinear ordinary algebraic equations involving unknown control points. Newton’s method is then applied to solve the resulting system and determine the control points. Numerical examples, along with corresponding graphs and tables, are presented to demonstrate the effectiveness and efficiency of the new algorithms. These results are compared with exact solutions, and most of the general programs were developed using MATLAB software 9.2 to implement the solutions for the test examples. Furthermore, these mathematical methods can be applied to solve complex physical problems, improve system performance, and address physics-related challenges such as optimizing optical systems, enhancing signal processing for antennas, and refining image analysis.

In the present paper, we are merging two interesting and well-known classes, namely those of Bazilevič and close-to-convex functions associated with a new derivative operator. We derive coefficient estimates for this broad category of analytic, univalent and bi-univalent functions and draw attention to the Fekete–Szegö inequalities relevant to functions defined within the open unit disk. Additionally, we identify several specific special cases of our results by specializing the parameters.