W.Rudin in *Real and Complex Analysis*(262) mentioned that 



> 
> **Theorem** Suppose $M$ is a vector space of continuous complex functions
>  on the closed unit disc $\bar U$,with
>  the following properites:
> 
> 
> (a) $1 \in M$
> 
> 
> (b)If $f \in M$,then also $jf \in
> > M$,where $j$ denote the identity
>  function :$j(z)=z$.
> 
> 
> (c)If $f \in M$,then
>  $||f||\_U=||f||\_T$,where T is the
>  boundary of $\bar U$.
> 
> 
> Then every $f\in M$ is holomorphic in
>  $U$.
> 
> 
> 


What I interested in is the origin and extension of this theorem.Where did it come form?
Is any references about this?



A converse of the maximum modulus Theorem
In the notes and comments section at the end of Real and Complex Analysis (page 397 in my copy), Rudin attributes this result "in a slightly different form" to:


W. Rudin, [Analyticity, and the maximum modulus principle](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1077465335). Duke Math. J. 20, (1953). 449–457.


