Colloquium publications.

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

149

be the function which corresponds to r as associated radius. Let
rl < r2 < r3
be three points of the interval of definition of <p(r). Then
1 log rl log.p(rl)
(2) 1 log r2 log p(r2) 0.
1 log r3 log e(r3)
This is the relation designated by Hartogs as the Fundamental
Property. It was proven by Fabry by means of Lemaire's theorem
cited above. Hartogs gave several proofs, one of which is
based on his function Rx defined below. He has also thrown
Fabry's proof into exceedingly simple form.*
The theorem admits the following interpretation. Let
(3) = log r, y = log s.
Then
1 x yl
(4) 1 x Y2 < 0.
1 x3 Y3
Hence the curve that represents y as a function of x, the above
curve (3), or
(5) y = (x),
is always continuous t and concave downward.
Let the function
f(x, y) = ZCm. nXyn
be analytic in the points (x, 0), where x I < R. If the associated
radius corresponding to one single value of r, r = ri, in the interval 0 < r < R is infinite, then the associated radius is infinite for
every value of r in this interval. In this case, then, the function
Sp(r) does not exist.
* Jahresber. D. M.-V., 16 (1907), p. 232.
t This property was established by A. Meyer, Stockholm Ved.-Ak. Forh. Ofv.,
40 (1883), No. 9, p. 15, and Phragmen, ibid., No. 10, p. 17. Cf. the reference
to Weierstrass, p. 148.