Coefficient estimation in nephroid domains and its use in fractional operators and neutrosophic Poisson distribution series

The relevance of nephroid domains and their relationships to fractional calculus operators have been brought to light by recent advances in the theory of analytic functions. Motivated by these developments, this study explores the geometric and analytic properties of a class of analytic functions, represented by ℳ ρ , λ ( q ). For this class, the Fekete–Szego inequality and associated coefficient bounds are obtained. The Erdelyi–Kober fractional integral operator is used to provide new connections between fractional operators and analytic function theory, which improves the results applicability. Furthermore, the class ℳ ρ , λ ( q ) is examined in connection with behavior of the functions ζ ϕ ( ζ ) and their inverse. To describe its structural characteristics, the convolution of ℳ ρ , λ ( q ) with normalized analytic functions is also covered. Finally, new insights into fractional and probabilitics extensions of analytic functions are revealed by obtaining specialized Fekete–Szego inequalities for subclass using the neutrosophic Poisson distribution series.