Exploring a distinct group of analytical functions linked with Bernoulli's Lemniscate using the -derivative.

This research presents a new group of mathematical functions connected to Bernoulli's Lemniscate, using the -derivative. Expanding on previous studies, the research concentrates on determining coefficient approximations, the Fekete-Szego functional, Zalcman inequality, Krushkal inequality, along with the second and third Hankel determinants for this recently established collection of functions. Additionally, the study derives the Fekete-Szego inequality for the function and obtains the inverse function for this specific class.

Boundary values of Hankel and Toeplitz determinants for -convex functions.

The study of holomorphic functions has been recently extended through the application of diverse techniques, among which quantum calculus stands out due to its wide-ranging applications across various scientific disciplines. In this context, we introduce a novel q-differential operator defined via the generalized binomial series, which leads to the derivation of new classes of quantum-convex (q-convex) functions. Several specific instances within these classes were explored in detail.