Geodesic
Derivatives and Length-Preserving Maps
Canadian mathematical bulletin, Tome 30 (1987) no. 3, pp. 379-384

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DOI

Let a be a constant, |a| = 1. We shall prove meromorphic (M) and bounded-holomorphic (BH) versions of the following prototype: (P) Let f and g be holomorphic in a domain D. Then, |f'| = |g'| in D if and only if there exist constant a, b with f = ag + b in D. (M) Let f and g be meromorphic in D. Then, |f'|/(1 + |f|2) = |g'|/(1 + |g|2) in D if and only if there exist a, b with |b| ≦ ∞ such that f = a(g - b)/(\ + g). (BH) Let f and g be holomorphic and bounded, |f| < 1, |g| < 1, in D. Then, |f'|/ (1 - |f|2) = |g'|/(1 - |g|2) in D if and only if there exist a, b with |b| < 1, such that f = a(g - b)/(1 - g).
DOI : 10.4153/CMB-1987-056-x
Mots-clés : Primary, 30A99, Secondary, 30C45 
Yamashita, Shinji. Derivatives and Length-Preserving Maps. Canadian mathematical bulletin, Tome 30 (1987) no. 3, pp. 379-384. doi: 10.4153/CMB-1987-056-x
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     title = {Derivatives and {Length-Preserving} {Maps}},
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     doi = {10.4153/CMB-1987-056-x},
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