• D={z : ∣z∣<1} • H (A→B) denotes the set of holomorfic functions from A to B • We define: W={f∈H (D→R) : ∣∣f∣∣_W <∞ } where ∣∣ ∙ ∣∣_W : { (W,→,R_+ ),(f, ,(Σ_(n=0) ^∞ ((∣f^((n)) (0)∣)/(n!)))) :} Let f∈W Show that ∀g∈H ( f(D^� )), g○f∈W tip: show that ∣∣h∣∣_W ≤cste × Sup_(z∈D) {∣h(z)∣+∣h′′(z)∣} and that W is an algebra then, re−wright f=f_1 +f_2 with f_2 : z Σ_(n=N) ^∞ ((f^((n)) (0))/(n!))z^n with N great enough to make sure that Σ_(n=0) ^∞ ((g^((n)) (0))/(n!))f_2 ^( n) is well defined and converges over W. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙


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Question Number 178032 by TheHoneyCat last updated on 12/Oct/22
$• D={z : ∣z∣<1} • H (A→B) denotes the set of holomorfic functions from A to B • We define: W={f∈H (D→R) : ∣∣f∣∣_W <∞ } where ∣∣ ∙ ∣∣_W : { (W,→,R_+ ),(f, ,(Σ_(n=0) ^∞ ((∣f^((n)) (0)∣)/(n!)))) :} Let f∈W Show that ∀g∈H ( f(D^� )), g○f∈W tip: show that ∣∣h∣∣_W ≤cste × Sup_(z∈D) {∣h(z)∣+∣h′′(z)∣} and that W is an algebra then, re−wright f=f_1 +f_2 with f_2 : z Σ_(n=N) ^∞ ((f^((n)) (0))/(n!))z^n with N great enough to make sure that Σ_(n=0) ^∞ ((g^((n)) (0))/(n!))f_2 ^( n) is well defined and converges over W. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙$
$∙ D = {z : ∣ z ∣< 1}$ $∙ H (A \to B) denotes the set of holomorfic$ $functions from A to B$ $∙ We define :$ $W = {f \in H (D \to R) : ∣∣ f ∣ ∣_{W} < \infty}$ $where ∣∣ \cdot ∣ ∣_{W} : {\begin{cases} W & \to & R_{+} \\ f & \underset{n = 0}{\sum^{\infty}} \frac{∣ f^{(n)} (0) ∣}{n!} \end{cases}$ $Let f \in W$ $Show that \forall g \in H (f (\bar{D})), g \circ f \in W$ $t i p : s h o w t h a t$ $∣∣ h ∣ ∣_{W} ⩽ c s t e \times {Sup}_{z \in D} {∣ h (z) ∣ + ∣ h^{″} (z) ∣}$ $a n d t h a t W i s a n a l g e b r a$ $then, re - wright f = f_{1} + f_{2} with$ $f_{2} : z \underset{n = N}{\sum^{\infty}} \frac{f^{(n)} (0)}{n!} z^{n}$ $with N great enough to make sure that$ $\underset{n = 0}{\sum^{\infty}} \frac{g^{(n)} (0)}{n!} f_{2}^{n} is well defined and converges$ $over W .$ $\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$
Commented byJDamian last updated on 12/Oct/22
Besides I do not understand a half of your question, what damn is h? And cste?
Commented byTheHoneyCat last updated on 12/Oct/22
"h" is any function of W (so that you can take its norm). "cste" is any constant (so that it can be multiplied).
Commented byTheHoneyCat last updated on 12/Oct/22
sorry if it wasn't obvious from the sentence.
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