Refined Conditions for the Inclusion Properties of Special Functions in Lemniscate and Nephroid Domains

For D: =z∈C: |z|<1, this paper derives refined conditions for the inclusion of special functions in lemniscate and nephroid domains focusing on solutions to the differential equations of the form znF″ (z) +a (z) zn−1F′ (z) +b (z) F (z) +d (z) =0, n∈1, 2, z∈D, with normalization F (0) =1, where a (z), b (z) and d (z) are analytic in D. Using advanced techniques from geometric function theory, we generalize and improve existing results, particularly for classes of functions defined by differential equations. Specific applications include generalized Bessel functions, regular Coulomb wave functions, and associated Laguerre polynomials, where we derive improved bounds for their inclusion in lemniscate domains. Additionally, we present open problems, supported by numerical experiments, to guide future research in this direction.