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Orthogonal polynomials have been widely employed by renowned authors within the context of geometric function theory. This study is driven by prior research and aims to address the —Fekete-Szegö problem. Additionally, we provide bound estimates for the coefficients and an upper bound estimate for the second Hankel determinant for functions belonging to the category of analytical and bi-univalent functions. This investigation incorporates the utilization of Euler polynomials. [ABSTRACT FROM AUTHOR]

This work presents a novel investigation that utilizes the integral operator I p , λ n in the field of geometric function theory, with a specific focus on sandwich theorems. We obtained findings about the differential subordination and superordination of a novel formula for a generalized integral operator. Additionally, certain sandwich theorems were discovered. [ABSTRACT FROM AUTHOR]

This study presents a subclass S (β) of bi-univalent functions within the open unit disk region D . The objective of this class is to determine the bounds of the Hankel determinant of order 3, ( Ⱨ 3 (1) ). In this study, new constraints for the estimates of the third Hankel determinant for the class S (β) are presented, which are of considerable interest in various fields of mathematics, including complex analysis and geometric function theory. Here, we define these bi-univalent functions as S (β) and impose constraints on the coefficients │ a n │ . Our investigation provides the upper bounds for the bi-univalent functions in this newly developed subclass, specifically for n = 2, 3, 4, and 5. We then derive the third Hankel determinant for this particular class, which reveals several intriguing scenarios. These findings contribute to the broader understanding of bi-univalent functions and their potential applications in diverse mathematical contexts. Notably, the results obtained may serve as a foundation for future investigations into the properties and applications of bi-univalent functions and their subclasses. [ABSTRACT FROM AUTHOR]

The field of geometric function theory has thoroughly investigated starlike functions concerning symmetric points. The main objective of this work is to derive certain geometric properties, such as the starlikeness of order δ , convexity of order δ , k-starlikeness, k-uniform convexity, lemniscate starlikeness and convexity, exponential starlikeness and convexity, and pre-starlikeness for the Galué type Struve function (GTSF). Furthermore, the conditions for GTSF belonging to the Hardy space are also derived. The results obtained in this work generalize several results available in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we discuss and introduce a new study using an integral operator w k , μ m in geometric function theory, especially sandwich theorems. We obtained some conclusions for differential subordination and superordination for a new formula generalized integral operator. In addition, certain sandwich theorems were found. The differential subordination theory's features and outcomes are symmetric to those derived using the differential subordination theory. [ABSTRACT FROM AUTHOR]

The topics studied in the geometric function theory of one variable functions are connected with the concept of Symmetry because for some special cases the analytic functions map the open unit disk onto a symmetric domain. Thus, if all the coefficients of the Taylor expansion at the origin are real numbers, then the image of the open unit disk is a symmetric domain with respect to the real axis. In this paper, we formulate the q-differential operator associated with the q-Raina function using quantum calculus, that is the so-called Jacksons' calculus. We establish a new subclass of analytic functions in the unit disk by using this newly developed operator. The theory of differential subordination inspired our approach; therefore, we geometrically explore the most popular properties of this new operator: subordination properties, coefficient bounds, and the Fekete-Szegő problem. As special cases, we highlight certain well-known corollaries of our primary findings. [ABSTRACT FROM AUTHOR]

The symmetric differential operator is a generalization operating of the well-known ordinary derivative. These operators have advantages in boundary value problems, statistical studies and spectral theory. In this effort, we introduce a new symmetric differential operator (SDO) and its integral in the open unit disk. This operator is a generalization of the Sàlàgean differential operator. Our study is based on geometric function theory and its applications in the open unit disk. We formulate new classes of analytic functions using SDO depending on the symmetry properties. Moreover, we define a linear combination operator containing SDO and the Ruscheweyh derivative. We illustrate some inclusion properties and other inequalities involving SDO and its integral. [ABSTRACT FROM AUTHOR]