Real and Complex Analysis (Rudin)

1S8 REAL AND COMPLEX ANALYSIS
for all Z E Q and all € > O. If we fix Z E Q and then let € -
desired result I I(z) I :$; 1.
We now turn to the general case. Put
g(z) = M(a)(b-Z)/(b-a)M(b)(z-a)/(b-a),
0, we obtain the
(7)
where, for M > 0 and w complex, M W is defined by
M W = exp (w log M),
and log M is real. Then g is entire, g has no zero, 1/g is bounded in n,
I g(a + iy) I = M(a),
I g(b + iy) I = M(b),
and hence Ilg satisfies our previous assumptions. Thus I fig I :$; 1 in Q, and
this gives (3). (See Exercise 7.)
IIII
12.9 Theorem Suppose
Q={X+iY:IYI 0 so that IX < {J < 1. For € > 0, define
For ZEn,
h,(z) = exp { _€(ePZ + e- PZ)}.
Re [ePZ + e- PZ] = (e PX + e- PX) cos {Jy ~ b(eP x + e-P1 (5)
where b = cos ({JnI2) > 0, since I {J I < 1. Hence
I h,(z) I :$; exp { -€b(ePX + e- PX)} < 1 (z En). (6)
It follows that I fh,1 :$; 1 on aQ and that
I I(z)h,(z) I :$; exp {Ae,*1 - €b(ePX + e-P1} (z E. n). (7)
(4)