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 gλ,n(z) = λ z (bz−1)n , λ ∈ R\{0}, z ∈ C\{0}, n ∈ N\{1}, b > 0, b 6= 1 in the present paper which obtains from generating function of the unified generalized Apostol-type polynomials. The real fixed points of gλ,n(x), x ∈ R \ {0} with their stability are found for n odd and n even. It is shown that gλ,n(z) has infinite number of singular values. Further, it is seen that some critical values of gλ,n(z) lie in the closure of the disk and other lie in the exterior of the disk with center at the origin.