This paper examines the fundamental system properties of pole mobility and stabilizability under bounded feedback for the case of single-input single-output (SISO) systems, where the constraints on the feedback are defined in terms of the L2 norm. It is shown that the bounded gain assumption implies a bounded norm condition on the coefficient vector associated with the closed-loop characteristic polynomial. Classical and new results on the root distribution of bounded coefficient polynomials are reviewed first. With these results, the closed-loop pole mobility of SISO systems under bounded state feedback is studied. It is shown that the assignable closed-loop poles are always within bounded regions, and alternative estimates for these regions are given. Exact boundaries are also established by utilizing computational methods. A common computational scheme is also developed to calculate the distances of stable polynomials from instability and the distance of unstable polynomials from stability. Finally, for open loop unstable systems, the problems of stabilization and delta-stabilization under the L2 norm restriction are considered. The minimal norm required for stabilizing or delta-stabilizing unstable systems can be found.