Geodesic
On the Modulii of Analytic Functions
Canadian mathematical bulletin, Tome 13 (1970) no. 3, pp. 325-327

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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 * 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. 
Sherman, Malcolm J. On the Modulii of Analytic Functions. Canadian mathematical bulletin, Tome 13 (1970) no. 3, pp. 325-327. doi: 10.4153/CMB-1970-062-4
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     title = {On the {Modulii} of {Analytic} {Functions}},
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     year = {1970},
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     doi = {10.4153/CMB-1970-062-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1970-062-4/}
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