Georgia Irina Oros

In the present research, necessary and sufficient conditions for the univalence of the fractional integral of the Gaussian hypergeometric function are obtained by employing specific approaches from differential subordination theory. The first theorem provides the best dominant for a particular differential subordination, which is proved to be a Carathéodory function. These results facilitate the establishment of several corollaries that give the necessary and sufficient conditions for this fractional operator to be a function with a positive real part of the first derivative. The univalence properties obtained through this research enhance the potential for new applications in geometric function theory and related research domains.

In the present work, we introduce and study a new two-parameter generalization of Legendre-based Appell polynomials, defined through an explicit representation that unifies classical Legendre structures with the Appell polynomial framework. Starting from a generating function, we derive a three-term recurrence relation, a degree-lowering operator, an integro-partial degree-raising operator, and a corresponding integro-partial differential equation satisfied by the new family. A determinant representation is established via Cramer’s rule applied to the Cauchy-product expansion of the generating function. Several subfamilies of independent interest arise naturally as special cases, namely, Legendre-based Hermite–Frobenius–Euler polynomials, Legendre-based Miller–Lee polynomials, and both the probabilist’s and physicist’s variants of Legendre-based bi-variate Hermite polynomials. For each subfamily we record the corresponding recurrence relations, shift operators, differential equations, and determinant forms, and we illustrate the behavior of selected members through three-dimensional surface plots and real-root distribution diagrams. The framework presented here extends several constructions available in the recent literature and points to natural directions for future work, including connections with q-series, combinatorial identities, and symbolic-computation methods, which are outlined in the concluding section.

The findings of this study are connected with geometric function theory and were acquired by using linear operator is applied to investigate specific classes of meromorphic functions defined using the subordination. The first class introduced and investigated here, , is a generalized class of meromorphic functions. It is also shown that for particular choice of parameters for the new generalized class, the class of close-to-convex meromorphic functions emerges. Using the properties of the convolution and subordination, certain characterization properties of this class are proved involving combinations of the functions from the class. Further, three more classes, , , and , are defined in connection to this first class, developing new applications of this operator. The connections between the classes are given in the definition or in associated remarks and characterization properties are also proved, including combinations of functions belonging to those classes and inclusion relations.

This paper introduces a new family of multivariate Hermite-based Appell polynomials with a one-parameter extension, defined through an appropriate generating function. Explicit series representations, operational formulas, and a fractional integral representation are established, revealing their underlying algebraic structure. A determinant representation is derived, providing an effective characterization of the proposed polynomials. Moreover, several families of ordinary, integrodifferential, and partial differential equations satisfied by these polynomials are obtained via operational and factorization techniques. Various summation identities are also presented, enhancing their analytical and computational applicability. As notable special cases, the corresponding results for multivariate Hermite-based Bernoulli, Euler, and Genocchi polynomials are derived, demonstrating that the proposed framework unifies and generalizes several classical Appell-type polynomial families.

This paper demonstrates applications of q-calculus operators in geometric function theory with the use of differential subordination properties on non-Bazilevič functions with respect to k-symmetric in this study. The method of the differential subordination theory is applied in developing subordination results involving the new class of univalent functions introduced and studied in the work. Several geometric properties of the class of functions are established in the study. The results generalize many known results in literature. Such previously proved results are presented as corollaries. Different examples are also provided based on the results.

Fractional calculus has evolved as an effective mathematical tool for simulating complex dynamical systems in science and engineering. This study develops an enhanced fractional iterative technique for solving nonlinear equations that incorporates fractional calculus to improve accuracy, stability and computational efficiency. Traditional approaches frequently struggle with high computing costs and slow convergence, but the suggested method effectively balances accuracy and efficiency by adding Caputo fractional-order derivatives. Analysis of convergence reveals that the order of convergence of the suggested family of approaches is [ ]. Fractal analysis of the suggested numerical methods for solving nonlinear equations reveals that they outperform existing classical approaches in terms of convergence and stability. A comparison with existing methods shows that the newly developed schemes perform better in terms of residual error reduction, CPU time and convergence rate. Since increasing the fractional parameter [ ] from 0 to 1 greatly increases the method’s efficiency — near-optimal performance is seen around [ ] — it plays a critical role. The novel strategy generates fractals with greater elapsed time and consistency than existing strategies, according to numerical studies on engineering applications.

This study introduces a new linear operator constructed from the well-known Dziok–Srivastava linear hypergeometric operator, the Bessel function of the first kind, and convolution. Applying the new operator, a class of functions with positive real part is defined and investigated concerning inclusion relations, the necessary and sufficient conditions for functions to belong to this class, as well as starlikeness and convexity conditions. The properties of the new class to be convex of a certain order, close-to-convex and starlike of a certain order are highlighted and examples are also included to prove that the new outcome is applicable. The methods employed for the investigation are specific to the prolific theory of differential subordination initiated by S.S. Miller and P.T. Mocanu.