Let f : D → D (unit disk) be a holomorphic function with f ( 0 ) = 0 , | f ′ ( 0 ) | &lt; 1. For f n = f ∘ ⋯ ∘ f, show that ∑ n = 1 ∞ f n ( z ) converges uniformly on compact subsets in D .I tried Schwarz lemma so that | f ( z ) | ≤ | z | , and I tried to use Weierstrass M test, but I don't know how f n is bounded. How to solve this problem?

Question

Let   f  :      D    →      D   (unit disk) be a holomorphic function with   f  (  0  )  =  0  ,      |        f    ′    (  0  )      |    &amp;lt;  1. For       f    n    =  f  ∘  ⋯  ∘  f, show that       ∑          n      =      1        ∞        f    n    (  z  ) converges uniformly on compact subsets in       D  .I tried Schwarz lemma so that       |    f  (  z  )      |    ≤      |    z      |  , and I tried to use Weierstrass   M test, but I don&#039;t know how       f    n   is bounded. How to solve this problem?

Nancy Phillips · Accepted Answer

We have to use effectively the assumptions, especially       |        f    ′    (  0  )      |    &amp;lt;  1.We first show the following:For any   r    (  0  &amp;lt;  r  &amp;lt;  1  ), there is a constant   c  &amp;lt;  1 (  c depends on   r) such that the inequality       |    f  (  z  )      |    ≤  c      |    z      |   holds for every   z    (      |    z      |    ≤  r  ).Proof.If there exists no such a constant   c, there is a sequence       |        z    n        |    ≤  r with |zn|≤r such that       |    f  (      z    n    )      |    &amp;gt;  (  1  −  1      /    n  )      |        z    n        |  ,   n  =  1  ,  2  ,  .  .  .. We may suppose without loss of generality that   {      z    n    } converges to a point       z    0      (      |        z    0        |    ≤  r  ). If       z    0    =  0, we have       |        f    ′    (  0  )      |    ≥  1, which contradicts the assumption       |        f    ′    (  0  )      |    ≥  1. If       z    0    ≠  0, we have       |    f  (      z    0    )      |    ≥      |        z    0        |  . Then Schwarz lemma leads to   f  (  z  )  =  ε  z with       |    ε      |    =  1, which contradicts       |        f    ′    (  0  )      |    &amp;lt;  1.Thus we have      |    f  (  z  )      |    &amp;lt;  c  r  ,        |        f    2    (  z  )      |    &amp;lt;  c      |    f  (  z  )      |    &amp;lt;      c    2    r  ,    .  .  .    ,        |        f    n    (  z  )      |    &amp;lt;      c    n    r  ,    .  .  .on compact subsets   {      |    z      |    ≤  r  }  ⊂      D   and we can use Weierstrass   M test.

ghulamu51 · Accepted Answer

Hint: there are   0  &amp;lt;      r    1    &amp;lt;      r    2    &amp;lt;  1 such that   f maps       r    2        D   into       r    1        D  . The Arzela-Ascoli theorem will also be useful.

Let f:D->D (unit disk) be a holomorphic function with f(0)=0,|f′(0)|<1. For f_n=f ∘⋯∘ f, show that ∑^∞_(n=1) f_n(z) converges uniformly on compact subsets in D.

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