Singular Values of One Parameter Family \(\lambda\frac{b^2-1}{z}\)


Authors

Mohammad Sajid - College of Engineering, Qassim University, Buraidah, Al-Qassim, Saudi Arabia


Abstract

The singular values of one parameter family of entire functions \(f_\lambda(z)=\lambda\frac{b^2-1}{z}\) and \(f_\lambda(0)=\lambda\ln b, \quad \lambda\in \mathbb{R}-\{0\}, z\in \mathbb{C}, b>0, b\neq 1\) are investigated. It is shown that all the critical values of \(f_\lambda(z)\) belong to the right half plane for \(0 < b <1\) and the left half plane for \(b >1\). It is described that the function \(f_\lambda(z)\) has infinitely many singular values. It is also found that all these singular values are bounded and lie inside the open disk centered at origin and having radius \(\mid\lambda\ln b\mid\).


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