Antiderivative of holomorphic and bounded functionLet f : D ( 0 , 1 ) → C holomorphic and bounded. Do the antiderivatives belong in the disc algebra? Disc algebra = { f | f : holomorphic on D(0,1), continuous on the closed disc.

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Antiderivative of holomorphic and bounded functionLet   f  :  D  (  0  ,  1  )  →      C   holomorphic and bounded. Do the antiderivatives belong in the disc algebra? Disc algebra   =  {  f      |    f  : holomorphic on D(0,1), continuous on the closed disc.

Daniel Valdez · Accepted Answer

Step 1If       F    ′    =  f then       |    F  (  z  )  −  F  (  w  )      |    ≤  c      |    z  −  w      |  . Hence F is uniformly continuous in the open disk, and hence it extends continuously to the closed disk.Here&#039;s another argument that&#039;s maybe conceptually simpler, although it take a few more lines. If       F    ′    =  f in the open disk, define  F  (      e          i      t        )  =  F  (  0  )  +      ∫    0    1        ∂          ∂      r        F  (  r      e          i      t        )    d  r  =  F  (  0  )  +      ∫    0    1        e          i      t        f  (  r      e          i      t        )    d  r  ..Step 2It&#039;s easy to show that   F  (  r      e          i      t        )  →  F  (      e          i      t        )uniformly as   r  →  1.The second argument also shows that, for example, if   0  &amp;lt;  α  &amp;lt;  1 and       |    f  (  z  )      |    ≤  c  (  1  −      |    z      |        )          −      α       then F is in the disk algebra.

Antiderivative of holomorphic and bounded function Let f : D ( 0 , 1 )

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