BIRKHOFF. — THE GENERALIZED RIEMANN PROBLEM. 525 



Now modify the contour C of integration in (4) to Di. The integrand 

 is analytic in t over the continua enclosed by C and Di, so that the 

 value of the integral will not thereby be altered. (It must be remem- 

 bered that X lies without C in (4).) From this modified form of (4) 

 we obtain 



(9) \J-ix)\^~ ! f \rnt)dt\+ f ,f^ 



X 



I 



upon applying (8) and (1). 



But the two integrals which appear in the right hand member of 

 this inequality tend to zero as the unspecified integer p increases; in 

 fact we have | r (x) \ < 1 within C so that r^ (.r) tends uniformly to 

 zero in any closed continuum within C, as p becomes infinite. It is 

 to be observed that the quantity | t — x \ which appears in the second 

 integrand is never less than the minimum distance from C to Ci, 

 since x lies without C, and / is a point of Ci. 



These considerations demonstrate that for a given positive e, 

 however small, the integer p may be chosen so large that for every 

 regular inner function <7+(.r), we have 



(10) maximum of | /" (x) | ^ e { maximum of g'^ (x) } along C. 



§ 2. On a Second Ancdogous Type of Integral. 

 In the same way we may treat an integral 



27r V— 1 Jc t — X 



where g~{x) is a regular outer function, and 2> is zero or a positive 

 integer. As before we denote the value of the integral for .r within C 

 by f*{x), and for x without C hy f-{x). A discussion parallel to the 

 earlier one in § 1 shows that /+(.t) and /~(.v) as thus defined are 

 regular inner and outer functions respectively; in this case equation 

 (4) is replaced by 



(40 r w = — ^ f r-' (0 r « "-4^^ ,t 



27rV— iJc 



t 



+ 



TT V— 1 JCi t — X 



