62 



Mr Hardy, On a theorem of Mr 0. Polya 



= 2 log 2cr - 2 log (2o- + 1) + ^ = ^ - 2 log (l + ^M > 0. 



3. When 6 increases from towards -|-7r, or decreases towards 

 — hir, (T increases from 1 towards oo . Also 



^^ = 21og(2c.-l)-21og2^ + ^=2 1og(l-^^)+-^<0, 



da- 



Thus <I> steadily decreases and ^ steadily increases. Moreover 



^ (0) = 0, ^ (0) = 4 log 2 - 3 log 3 ; 



and it is easily verified that both <J» and ^ tend to the limit 



log 2 - 1 

 when 6 tends to ^tt. 



We thus obtain, in the first place, 



i,.=of-'.<--vnj:;y(,^)</«}=o(i). 



Secondly, we observe that, if S is any positive number, we have 



^{d)<^{h) = -7^<0 



for h^e^^ir, -^ir^eK-h. 



Hence we may replace the limits in Kn by - 8 and h, the re- 

 mainder of the integral being of the form 



4. All that remains, then, is to prove that 

 /„ = ,U2»f n^^)w(* = 0(l); 



j_5 1 (2/lcr) 



The function ^{6) may now be expanded in powers of ^. We 

 find without difficulty that 



where A = log 2 - ^ > 0. 



It follows that 



and we have 



h = 0bn\ ^e-^»^HO(»^*)^^ 

 = 0U/n r e--^^''^'de\^0{l). 



