Let $\mathsf{\Omega }$ denote the class of functions $f\left(z\right)=z+a_2{z}^2+a_3{z}^3+\cdots$ belonging to the normalized analytic function class $\mathcal{A}$ in the open unit disk $\mathbb{U}=\left\{z:\left|z\right|<1\right\}$ , which are bi-univalent in $\mathbb{U}$ , that is, both the function f and its inverse $f^{-1}$ are univalent in $\mathbb{U}$ . In this paper, we introduce and investigate two new subclasses of the function class $\mathsf{\Omega }$ of bi-univalent functions defined in the open unit disc $\mathbb{U}$ , which are associated with a new differential operator of analytic functions involving binomial series. Furthermore, we find estimates on the Taylor–Maclaurin coefficients $|a_2|$ and $|a_3|$ for functions in these new subclasses. Several (known or new) consequences of the results are also pointed out.

In this work, we numerically investigate the fractional clannish random walker’s parabolic equations (FCRWPEs) and the nonlinear fractional Cahn–Allen (NFCA) equation using the Hybrid Decomposition Method (HDM). The analysis uses the Atangana–Baleanu fractional derivative (ABFD) in the Caputo sense, which has a nonsingular and nonlocal Mittag–Leffler kernel (MLk) and provides a more accurate depiction of memory and heredity effects, to examine the dynamic behavior of the models. Using nonlinear analysis, the uniqueness of the suggested models is investigated, and distinct wave profiles are created for various fractional orders. The accuracy and effectiveness of the suggested approach are validated by a number of example cases, which also support the approximate solutions of the nonlinear FCRWPEs. This work provides significant insights into the modeling of anomalous diffusion and complex dynamic processes in fields such as phase transitions, biological transport, and population dynamics. The inclusion of the ABFD enhances the model’s ability to capture nonlocal effects and long-range temporal correlations, making it a powerful tool for simulating real-world systems where classical derivatives may be inadequate.

The logarithmic functions have been used in a verity of areas of mathematics and other sciences. As far as we know, no one has used the coefficients of logarithmic functions to determine the bounds for the third Hankel determinant. In our present investigation, we first study some well-known classes of starlike functions and then determine the third Hankel determinant bound for the logarithmic coefficients of certain subclasses of starlike functions that also involve the sine functions. We also obtain a number of coefficient estimates. Some of our results are shown to be sharp.

In the current work, we introduce a special family of the function family of analytic and m-fold symmetric bi-univalent functions and obtain estimates of the Taylor–Maclaurin coefficients $|d_{m+1}|$ and $|d_{2m+1}|$ for functions in the special family. For $\delta$ a real number, Fekete–Szegö functional $|d_{2m+1}-\delta d_{m+1}^2|$ for functions in the special family is also estimated. We indicate several cases of the defined family and connections to existing results are also discussed.