208 Dr Biesz, Sur le principe de Phrcigmen-Lindeldf 



n 



Note by G.B.. Hardy. 



I know from my own experience that it is sometimes a little 

 difficult to pick out from Phragmen and Lindelof's classical memoir 

 the precise proposition of which one may have need; and Dr 

 Cramer's dissertation, referred to by Dr Riesz, is not easily acces- 

 sible to English readers. It may therefore be worth while to give 

 an explicit statement and proof of the particular theorems used 

 by Dr Riesz. 



1. Suppose that T is an angle of magnitude less than it, ivhose 

 vertex is at the origin, that C, C^ , and K are constants, and that 



(1) \f{^)\^c 



on the boundary of T, 



(2) \f{z)\^C\eJ^'- 

 throughout T. Then (1) holds throughout T. 



We may plainly suppose, without loss of generality, that the 

 boundaries of T are <^ = — a and ^ = a, where < a < \tt. Let 



IT 



§ be positive, \ <h < ^, and 

 so that 



1 / \ I -dr'' cos k(b ^ - dr'^ cos ka ^ -, 



\g{z)\=e "^ ^ e < 1 



throughout T. Finally, let 



f{z)g{z)^h{z). 

 Since \ g (z) \ ^ 1, we have \ h (z) \ ^ C on the boundary of T. 

 Also, since ^ > 1 and cos ka > 0, 



\ 7 / \ \ ^ n ^r - Sr'' cos ka, 

 \h{z)\^ C^e 



tends uniformly to zero when z tends to infinity in T. Hence 

 \ h (z) \ ^ — C if — a^(f>^a, r = R, and R is sufficiently large ; 

 and therefore at all points of the boundary of the region T {R) 

 defined by the inequalities just written; and therefore throughout 

 T{R). 



As R is arbitrarily large: 



\^{^)\ = \e-'''f{z)\^C 

 throughout T; and therefore, making S tend to zero, \f{z)\^C 

 throughout T. 



2. Suppose that T is the angle ^-^^ (f) ^ ^^, ivhere <^2~ (f>i < '^l 

 that 



(3) I / (re^"*') I ^ Ce"^"", 



(4) |/(re*"*^) |^Ce«^ 

 and that (2) is satisfied throughout T. Then 



