This paper investigates the third Hankel determinant, denoted $H_3\left(1\right)$, for functions within the subclass $\mathcal{R}{\mathcal{S}}_{\sum }^{\mathrm{*}}\left(\lambda \right)$ of bi-univalent functions associated with crescent-shaped regions ${\phi }_{⦅}\left(z\right)=z+\sqrt{1+z^2}$. The primary aim of this study is to establish upper bounds for $H_3\left(1\right)$. By analyzing functions within this specific geometric context, we derive precise constraints on the determinant, thereby enhancing our understanding of its behavior. Our results and examples provide valuable insights into the properties of bi-univalent functions in crescent-shaped domains and contribute to the broader theory of analytic functions.

We investigate some subclasses of regular and bi-univalent functions in the open unit disk that are associated with Lucas-Balancing polynomials in this work. For functions that belong to these subclasses, we obtain upper bounds on their initial coefficients. The Fekete–Szegö problem is also discussed. Along with presenting some new results, we also explore pertinent connections to earlier findings.

The aim of this paper is to introduce a new type of two-dimensional convexity by using total-order relations. In the first part of this paper, we examine the Hyers–Ulam stability of two-dimensional convex mappings by using the sandwich theorem. Our next step involves the development of Hermite–Hadamard inequality, including its weighted and product forms, by using a novel type of fractional operator having non-singular kernels. Moreover, we develop several nontrivial examples and remarks to demonstrate the validity of our main results. Finally, we examine approximate convex mappings and have left an open problem regarding the best optimal constants for two-dimensional approximate convexity.

The aim of this study is to investigate a certain sufficiency criterion for uniform convexity, strong starlikeness, and strong convexity of Rabtonov fractional exponential functions. We also study the starlikeness and convexity of order $\gamma$. Moreover, we find conditions so that the Rabotnov functions belong to the class of bounded analytic functions and Hardy spaces. Various consequences of these results are also presented.

In this paper, the antiplane state problem of an elliptic hole in a prestressed and prepolarized piezoelectric material loaded by constant, uniform remote shear stresses was performed. A compact and elementary form solution of the problem is obtained utilizing the conformal mapping technique and representation of the incremental stress and electrical fields by complex potentials. Using the boundary conditions, the coefficients of complex potentials developed as Laurent series, for the case of a piezoelectric material of class $$\bar{4}2m$$ 4 ¯ 2 m , and implicitly the components of incremental stress and electric displacement fields are obtained in a final compact and closed form.

In the last few years, numerous subfamilies of univalent functions, whether directly or indirectly associated with exponential functions, have been introduced and thoroughly investigated. Among these, the families ${\mathcal{S}}_e^{*}$, ${\mathcal{C}}_e$ and ${\mathcal{R}}_e$ defined by subordination to $e^z$ have been intensively investigated. While the coefficient problem on the class ${\mathcal{S}}_e^{*}$ and ${\mathcal{C}}_e$ has been solved in many cases, in this paper, we mainly intend to compute the sharp estimates of some initial coefficients, the Feketo–Szegö inequality, and the sharp bounds of second- and third-order Hankel determinants for functions belonging to the class ${\mathcal{R}}_e$. This work has the potential to significantly enrich and enhance the exploration of univalent functions in conjunction with exponential functions, making the field more comprehensive and robust.