Singular Values of One Parameter Family \(λ ((e^{z}-1)/z)^{m}\) - Sajid, Mohammad (International Journal of Applied Mathematical Research, 2015)
In the present paper, the singular values of one parameter family of entire functions \(f_{λ}(z)=λ\bigg(\dfrac{e^{z}-1}{z}\bigg)^{m}\) and \(f_{λ}(0)=λ\), \(m\in \mathbb{N}\backslash \{0\}\), \(λ\in \mathbb{R} \backslash \{0\}\), \(z \in \mathbb{C}\) is investigated. It is shown that all the critical values of \(f_{λ}(z)\) lie in the left half plane. It is also found that the function \(f_{λ}(z)\) has infinitely many bounded singular values and lie inside the open disk centered at origin and having radius \(|λ|\).