Talk:Univalent function

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Redundant but traditional

This article supports a long tradition of research in complex analysis of schlicht functions. In 1985 a long-standing conjecture was solved by de Branges's theorem. Currently MathSciNet turns up 279 articles on schlicht functions and 67 on univalent functions.

Nevertheless, a function (mathematics) is a special type of binary relation said to be univalent. Thus all functions are univalent and the term "univalent function" is redundant. This terminology in binary relations has been adopted by Gunther Schmidt in his books on relations, which have been well-received and widely used. In some situations the univalent property ( (xRy and xRz) implies y = z) holds, but the domain of the relation falls short of what is expected, leaving a partial function.

In summary, the analytic and injective functions that are said to be "univalent" represent a special and peculiar use of the term in complex analysis. Young students having studied binary relations, and the description of functions as univalent relations, may later be surprised by this special and peculiar use of the term. — Rgdboer (talk) 23:15, 25 July 2018 (UTC)[reply]