1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 389

is bounded in D. In 1925, H. Looman [80] showed that the boundedness

condition (6.23-1) can be replaced by the assumption that the limit in

(6.23-1) is finite at every point of D, and Besicovitch [16] refined the last

condition by requiring it to hold in D except at most on a subset that is

the union of a sequence of sets of finite measure.

P. Montel in [87] proved that if f = u +iv is continuous, the first partial

derivatives of u and v exist and are bounded, and if the Cauchy-Riemann

equations are satisfied everywhere in D, then f is analytic in D. In 1910,

Ch. de la Vallee-Poussin [122] showed that the boundedness of the partial
derivatives may be replaced by their finiteness and summability (in the
Lebesgue sense); then it is sufficient that the Cauchy-Riemann equations
be satisfied almost everywhere.
In 1913, Montel [88] stated without proof the following theorem, a proof
of which was supplied by Looman [79] in 1923. Looman's proof was found
to be faulty. A satisfactory one was finally provided by D. Menchoff in his
monograph Les conditions de monogeneite, Paris, 1936 [83].

Theorem of Looman-Menchoff. If the functions u(x,y) and v(x,y) are

continuous in an open set D, have first partial derivatives with respect to
x and with respect to y nearly everywhere in D (i.e, except possibly on a
countable subset of D), and if u., = Vy, Uy = -v., holds almost everywhere

in D, then f = u +iv is analytic in D. A proof of the Looman-Menchoff

theorem can be found in S. Saks [107], p. 199; see also J. D. Gray and
A. S. Morris [54].
The monogeneity of a function at a point z E D is equivalent to the
existence of the ordinary limits

(1) Az--+0 lim I ~w LJ.Z I= p and (2) Az--+0 lim Arg ~w L..l.Z = 'I/;

provided that f'(z) -:/= 0 in the second case; or in geometric terms, the
monogeneity off is equivalent to the constancy of the magnification ratio
together with the preservation of angles (in size as well as orientation) at
that point. Further attempts at characterizing analyticity in D have been
based on the idea of isolating one or the other of these properties, with the
addition of some supplementary conditions.
The following theorem of H. Bohr [17] represents a first result in this

direction: If f is continuous and univalent in D (i.e., if no two different

points have the same image), and if at nearly every point of D the limit p
exists, with 0 < p < +oo, then either for J is analytic in D. Rademacher
in [98] improved Bohr's theorem by removing the condition p -:/= 0.
A similar theorem based on the preservation of angles was given by

Menchoff [82], as follows: If f is continuous and univalent in D, and if at

nearly every point of D the limit 1/J exists, then f is analytic everywhere