-^Note by Mr Hardy 209 



(5) |/(re'"*) 1^ Ce'^W'- 



where h {(f>) is the function A cos (f> + B sin (f> which assumes the 



values «! and a^for <f> = (f>i and ^ = <^2- 



We may plainly suppose, without loss of generality, that 

 - ^1= ^2^^> where < ip < ^tt. 



Let g (z) = e-(-i-''^)« 



so that \ g (z) \ = e-'iW'- 



and f{z)9{z) = hiz). 



Plainly h (z) satisfies a condition of the type (2); and \ h (z) \ ^ C 

 for cf>= <f)^ and for <!> = <f>^. Hence \ h {z) \ ^ C throughout T, 

 which proves the theorem. 



It seems worth while also to fill up expHcitly the "petite lacune" 

 ni my proof in the Acta Mathematica signalised by Dr Riesz. It is 

 a question simply of proving that 



g (2) = fS± 

 '^ ^ ' sm TTZ 

 satisfies an inequality 



(6) \g{z)\< Ce^' 



for ^ ^ ^ Itt. Suppose that A is a positive constant, U the part 

 of the positive quadrant above, and F the part below, the line 

 y = X. Since | sin 772 | is greater than a constant throughout U, 

 (6) is satisfied in U, and we need only consider V. Also 



z lying in F and the contour of integration being the circle 

 \u — z\ =^ I. Hence/' (2) satisfies an inequality 



throughout F. 



Suppose now that z lies in F and that n is the integer nearest 

 to z (either, if there are two equidistant from z). Then 



f{z)=f{z)-f{n)=\^f'{u)du, 



J n 



the path of integration being rectilinear; and so 



\f{z)\^M\z-n\, 



where M is the maximum of |/' (w) | on the path of integration. 

 Hence 



F{z)\ = ^ l^ I ^ 6V^(- + 4 + ^) I t^^ 



sm TTZ\ ^ sm 772 



C'e^^^Ce^'-. 



which is the inequality required. This is substantially the argument 

 I used before, extended however to complex values of z. 



VOL. XX. PART I. 14 



