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This paper investigates the third Hankel determinant, denoted H3(1), for functions within the subclass RS∑*(λ) of bi-univalent functions associated with crescent-shaped regions φ⦅z=z+1+z2. The primary aim of this study is to establish upper bounds for H3(1). 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.

This study introduces the notion of complex t-neutrosophic graphs (CTNGs) as a powerful tool for understanding and displaying complex interactions that are sometimes difficult to understand. It demonstrates that CTNGs may accurately reflect complicated interactions involving several components or dimensions within a particular scenario. It also instructs the basic set operations of CTNGs and analyzes notions like homomorphism and isomorphism within this framework. Furthermore, the research describes a practical application of CTNGs. It illustrates their value in addressing biodiversity conservation by taking into account a variety of relevant factors. The paper uses this application to highlight the flexibility and effectiveness of CTNGs as a tool for decision-makers to visualize and prioritize activities targeted at improving biodiversity conservation.

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 γ. 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.

Geometric function theory, a subfield of complex analysis that examines the geometrical characteristics of analytic functions, has seen a sharp increase in research in recent years. In particular, by employing subordination notions, the contributions of different subclasses of analytic functions associated with innovative image domains are of significant interest and are extensively investigated. Since ℜ(1+sinh(z))≯0, it implies that the class Ssinh* introduced in reference third by Kumar et al. is not a subclass of starlike functions. Now, we have introduced a parameter λ with the restriction 0≤λ≤ln(1+2), and by doing that, ℜ(1+sinh(λz))>0. The present research intends to provide a novel subclass of starlike functions in the open unit disk U, denoted as Ssinhλ*, and investigate its geometric nature. For this newly defined subclass, we obtain sharp upper bounds of the coefficients an for n=2,3,4,5. Then, we prove a lemma, in which the largest disk contained in the image domain of q0(z)=1+sinh(λz) and the smallest disk containing q0(U) are investigated. This lemma has a central role in proving our radius problems. We discuss radius problems of various known classes, including S*(β) and K(β) of starlike functions of order β and convex functions of order β. Investigating Ssinhλ* radii for several geometrically known classes and some classes of functions defined as ratios of functions are also part of the present research. The methodology used for finding Ssinhλ* radii of different subclasses is the calculation of that value of the radius r<1 for which the image domain of any function belonging to a specified class is contained in the largest disk of this lemma. A new representation of functions in this class, but for a more restricted range of λ, is also obtained.

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 Se*, Ce and Re defined by subordination to ez have been intensively investigated. While the coefficient problem on the class Se* and Ce 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 Re. 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.

In this paper, evolution formulas for the first non-zero eigenvalue of the operator LT,f on a weighted closed Riemannian manifold along the Ricci flow as well as along the Yamabe flow are formulated. Some monotonic quantities are also derived for the normalized Ricci flow on Bianchi classes.

In this paper we presented some numerical methods for minimization problems as Variational Inequalities of Elliptic (IVE). We treated the numerical analysis, which are obtained based on abstract algorithms and concrete numerical algorithms, described by software (in Pascal). [ABSTRACT FROM AUTHOR]