Singular values of two-parameter families λ((bz − 1)/z)μ and λ(z/(bz − 1))η

The singular values of two kinds of two-parameter families of functions (i) fλ,μ(z)=λ((bz−1)/z)μ and fλ,μ(0) = λ(ln b)μ, μ > 0, (ii) gλ,η(z)=λ(z/(bz−1))η and gλ,η(0) = λ/(ln b)η, η > 0; λ∈ℝ∖{0}, z∈ℂ, b > 0, b ≠ 1 are described. It is shown that all the critical values of fλ,μ(z) and gλ,η(z) lie interior and exterior of the disk centered at origin and having radii |λ(ln b)μ| and |λ/(ln b)η| respectively. Further, it is proved that both the functions fλ,μ(z) and gλ,η(z) have infinitely many singular values for all b > 0, b ≠ 1.