Localization of the eigenvalues of a pencil of positive definite matrices

Let $A$ and $B$ be real square positive definite matrices close to each other. A domain $S$ on the complex plane that contains all the eigenvalues $\lambda$ of the problem $Az=\lambda Bz$ is constructed analytically. The boundary $\partial S$ of $S$ is a curve known as the limacon of Pascal. Using the standard conformal mapping of the exterior of this curve (or of the exterior of an enveloping circular lune) onto the exterior of the unit disc, new analytical bounds are obtained for the convergence rate of the minimal residual method (GMRES) as applied to solving the linear system $Ax=b$ with the preconditioner $B$.