The problem to be considered in this note, in its most concrete form, is the determination of all quartets f 1, f 2, g 1, g 2 of functions analytic on some domain and satisfying

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where p > 0. When p = 2 the question can be reformulated in terms of finding a necessary and sufficient condition for (two-dimensional) Hilbert space valued analytic functions to have equal pointwise norms, and the answer (Theorem 1) justifies this point of view. If p ≠ 2, the problem is solved by reducing to the case p = 2, and the reformulation in terms of the norm equality of lp valued analytic functions gives no clue to the answer.