Great post! I like your style and try to follow it closely.

Here are two examples of amplification, where the amplification trick gets rid of constants.

1. The maximum principle for holomorphic functions in a disk (which could be another domain). Let and be a holomorphic function. Then Cauchy’s integral formula gives you an inequality , with respect to the sup-norm over the boundary. Apply the inequality to , then take the -th root. The constant is changed into , which tends to as tends to infinity.

2. In matrix analysis, let be an algebra norm over , that is . Let denote the spectral radius of a matrix (which involves the modulus of complex eigenvalues too). Then . This would be obvious if the norm was subordinated to a complex norm (take an eigenvector associated with an eigenvalue of maximal modulus, bla-bla). If not, take any such subordinated norm , we thus have . By equivalence of norms, we obtain . Apply this to , use , then take the -th root and let tend to infinity.

All the best,

Denis

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