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For functions of the form ϕ (ξ) = ξ + ∑ n = 2 ∞ c n ξ n , we identified two new subclasses of bi-starlike functions and bi-convex functions by using Mathieu-type series defined in the disc Δ = { ξ ∈ C : | ξ | < 1 } . We derived constraints for | c 2 | and | c 3 | , and the subclasses are connected to the shell-shaped area. The Fekete–Szegö functional properties for the aforementioned function subclasses were also investigated. Additionally, a number of related corollaries are shown. [ABSTRACT FROM AUTHOR]

This paper introduces two novel subclasses of the function class Σ for bi-univalent functions, leveraging generalized telephone numbers and Binomial series through convolution. The exploration is conducted within the domain of the open unit disk. We delve into the analysis of initial Taylor-Maclaurin coefficients | a 2 | and | a 3 | , deriving insights and findings for functions belonging to these new subclasses. Additionally, Fekete-Szegö inequalities are established for these functions. Furthermore, the study unveils a range of new subclasses of Σ , some of which are special cases, yet have not been previously explored in conjunction with telephone numbers. These subclasses emerge as a result of hybrid-type convolution operators. Concluding from our results, we present several corollaries, which stand as fresh contributions in the domain of involution numbers involving hybrid-type convolution operators. [ABSTRACT FROM AUTHOR]

In the present investigation, we introduce a new class of meromorphic functions defined in the punctured unit disk Δ * : = { ϑ ∈ C : 0 < | ϑ | < 1 } by making use of the Erdély–Kober operator I ς , ϱ τ , κ which unifies well-known classes of the meromorphic uniformly convex function with positive coefficients. Coefficient inequalities, growth and distortion inequalities, in addition to closure properties are acquired. We also set up a few outcomes concerning convolution and the partial sums of meromorphic functions in this new class. We additionally state some new subclasses and its characteristic houses through specializing the parameters that are new and no longer studied in association with the Erdély–Kober operator thus far. [ABSTRACT FROM AUTHOR]