Question Number 178032 by TheHoneyCat last updated on 12/Oct/22
$$\bullet\:{D}=\left\{{z}\::\:\mid{z}\mid<\mathrm{1}\right\} \\ $$$$\bullet\:\mathscr{H}\:\left({A}\rightarrow{B}\right)\:\mathrm{denotes}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{holomorfic} \\ $$$$\mathrm{functions}\:\mathrm{from}\:{A}\:\mathrm{to}\:{B} \\ $$$$\bullet\:\mathrm{We}\:\mathrm{define}: \\ $$$${W}=\left\{{f}\in\mathscr{H}\:\left({D}\rightarrow\mathbb{R}\right)\::\:\mid\mid{f}\mid\mid_{{W}} <\infty\:\right\} \\ $$$$\mathrm{where}\:\:\mid\mid\:\centerdot\:\mid\mid_{{W}} \::\:\begin{cases}{{W}}&{\rightarrow}&{\mathbb{R}_{+} }\\{{f}}&{ }&{\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mid{f}^{\left({n}\right)} \left(\mathrm{0}\right)\mid}{{n}!}}\end{cases} \\ $$$$ \\ $$$$\mathrm{Let}\:{f}\in{W} \\ $$$$\mathrm{Show}\:\mathrm{that}\:\forall{g}\in\mathscr{H}\:\left(\:{f}\left(\bar {{D}}\right)\right),\:{g}\circ{f}\in{W} \\ $$$${tip}:\:{show}\:{that} \\ $$$$\:\mid\mid{h}\mid\mid_{{W}} \leqslant{cste}\:×\:\mathrm{Sup}_{{z}\in{D}} \left\{\mid{h}\left({z}\right)\mid+\mid{h}''\left({z}\right)\mid\right\} \\ $$$${and}\:{that}\:{W}\:{is}\:{an}\:{algebra} \\ $$$$ \\ $$$$\mathrm{then},\:\mathrm{re}−\mathrm{wright}\:{f}={f}_{\mathrm{1}} +{f}_{\mathrm{2}} \:\mathrm{with} \\ $$$${f}_{\mathrm{2}} :\:{z}\: \underset{{n}={N}} {\overset{\infty} {\sum}}\frac{{f}^{\left({n}\right)} \left(\mathrm{0}\right)}{{n}!}{z}^{{n}} \\ $$$$\mathrm{with}\:{N}\:\mathrm{great}\:\mathrm{enough}\:\mathrm{to}\:\mathrm{make}\:\mathrm{sure}\:\mathrm{that} \\ $$$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{g}^{\left({n}\right)} \left(\mathrm{0}\right)}{{n}!}{f}_{\mathrm{2}} ^{\:{n}} \:\mathrm{is}\:\mathrm{well}\:\mathrm{defined}\:\mathrm{and}\:\mathrm{converges} \\ $$$$\mathrm{over}\:{W}. \\ $$$$\:\:\:\:\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot \\ $$
Commented by JDamian last updated on 12/Oct/22
Besides I do not understand a half of your question, what damn is h? And cste?
Commented by TheHoneyCat 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 by TheHoneyCat last updated on 12/Oct/22
sorry if it wasn't obvious from the sentence.