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.

In the present paper we introduce a new class of analytic functions f in the open unit disk normalized by f (0) = f ^′ (0) − 1 = 0, associated with exponential functions. The aim of the present paper is to investigate the third-order Hankel determinant H ₃ (1) for this function class and obtain the upper bound of the determinant H ₃ (1).

Nanofluids, with their enhanced thermal properties, provide innovative solutions for improving heat transfer efficiency in renewable energy systems. This study investigates a numerical simulation of bioconvective flow and heat transfer in a Williamson nanofluid over a stretching wedge, incorporating the effects of chemical reactions and hydrogen diffusion. The system also includes motile microorganisms, which induce bioconvection, a phenomenon where microorganisms’ collective motion creates a convective flow that enhances mass and heat transport processes. This mechanism is crucial for improving the distribution of nanoparticles and maintaining the stability of the nanofluid. The unique rheological behavior of Williamson fluid, extensively utilized in hydrometallurgical and chemical processing industries, significantly influences thermal and mass transport characteristics. The governing nonlinear partial differential equations (PDEs), derived from conservation laws and boundary conditions, are converted into dimensionless ordinary differential equations (ODEs) using similarity transformations. MATLAB’s bvp4c solver is employed to numerically analyze these equations. The outcomes highlight the complex interplay between fluid parameters and flow characteristics. An increase in the Williamson nanofluid parameters leads to a reduction in fluid velocity, with solutions observed for the skin friction coefficient. Higher thermophoresis and Williamson nanofluid parameters elevate the fluid temperature, enhancing heat transfer efficiency. Conversely, a larger Schmidt number boosts fluid concentration, while stronger chemical reaction effects reduce it. These results are generated by fixing parametric values as $0.1<\varpi <1.5, 0.1<\mathit{Nr}<3.0, 0.2<\mathrm{Pr}<0.5, 0.1<\mathit{Sc}<0.4, \mathrm{and} 0.1<\mathit{Pe}<1.5.$ This work provides valuable insights into the dynamics of Williamson nanofluids and their potential for thermal management in renewable energy systems. The combined impact of bioconvection, chemical reactions, and advanced rheological properties underscores the suitability of these nanofluids for applications in solar thermal, geothermal, and other energy technologies requiring precise heat and mass transfer control. This paper is also focused on their applications in solar thermal collectors, geothermal systems, and thermal energy storage, highlighting advanced experimental and computational approaches to address key challenges in renewable energy technologies.

In this article, we demonstrated various Hermite–Hadamard and Fejér type inequalities for modified h-convex functions. We showed several inequalities for the products of two modified h-convex functions. New identities related to inequalities in various forms are also established for different values of the $h\left({\phi }_t\right)$ function. We believe that the approach presented in this paper will inspire more research in this area.

Bi-univalent functions associated with the leaf-like domain within the open unit disk are investigated and a new subclass is introduced and studied in the research presented here. This is achieved by applying the subordination principle for analytic functions in conjunction with $\mathit{q}$-calculus. The class is proved to be not empty. By proving its existence, generalizations can be given to other sets of functions. In addition, coefficient bounds are examined with a particular focus on $|{\alpha }_2|$ and $|{\alpha }_3|$ coefficients, and Fekete–Szegö inequalities are estimated for the functions in this new class. To support the conclusions, previous works are cited for confirmation.