Real fixed points and singular values of family of functions arising from generating function of unified generalized Apostol-type polynomials

Abstract

Our main objective is to study the real fixed points and singular values of a two-parameter family of transcendental meromorphic functions g λ,n (z) = λ z (b z −1) n , λ ∈ R\{0}, z ∈ C\{0}, n ∈ N\{1}, b > 0, b = 1 in the present paper which obtains from generating function of the unified generalized Apostol-type polynomials. The real fixed points of g λ,n (x), x ∈ R \ {0} with their stability are found for n odd and n even. It is shown that g λ,n (z) has infinite number of singular values. Further, it is seen that some critical values of g λ,n (z) lie in the closure of the disk and other lie in the exterior of the disk with center at the origin.