The subject of this paper is to find an \(n\)-th degree polynomial which maximizes \[ R(f)=\int_{I_a}|f(x)|^2dx/\int_{I_b}|f(x)|^2dx \] where \(f\) is a complex valued function and \(I_a\), \(I_b\) are real intervals. Special attention is paid to the asymptotic behaviour of this maximum for \(n\to \infty\) for the so-called interior problem \((I_a\subset I_b)\) and the adjacent exterior problem (\(I_a \cap I_b =\mathrm{right endpoint of}\, I_b = \mathrm{left endpoint of}\, I_a\)); the formulae given are rather intricate. The problem at hand is equivalent to finding the largest eigenvalue of a matrix eigenvalue problem of the type \(Ax=\lambda Bx\) with \(A,B\) real symmetric positive definite \((n+ 1) \times (n + 1)\) matrices. The solution of this problem leads to \((n + 1)\) real eigenvalues \(\lambda_0^{(n)}\geq \lambda_1^{(n)} \geq \ldots \geq \lambda_n^{(n)}>0\) (of which \(\lambda_o^{(n)}\) is the maximum value of \(R(f)\) sought for) with real eigenvectors \(f_0^{(n)},\ldots ,f_n^{(n)}\) satisfying the double orthogonality property \[ (f_j^{(n)})^tBf_k^{(n)}=\delta_{jk},\, (f_j^{(n)})^tAf_k^{(n)}=\lambda_j^{(n)}\delta_j^{(n)}\, (j,k=0,1,\ldots ,n). \] Also the maximum problem is considered for two other kinds of concentrations, leading solutions using Tchebycheff polynomials (from \(|f(y)|^2/ \max \; |f(x)|^2;\;y\in I_a,\; S\subset I_b\)) and Legendre polynomials (from \(|f(y)|^2/\int_S|f(x)|^2dx;\; y\in I_a,\; S\in I_b\)). Furthermore an equivalent differential equation eigenvalue problem is derived in the case of the centered interior problem \((I_a=[-a,a]\), \(I_b=[-1,1];\; 01)\) including an asymptotic investigation for \(n\to \infty\) of these equations. Finally it must be said that the style of the paper is in tune with its title, very concentrated.