let-D-D-0-1-and-f-z-n-0-a-n-z-n-is-a-holomorphe-function-f-x-lt-1-1-z-prove-that-a-n-n-1-1-1-n-n-n-1-e-

Tinku Tara June 4, 2023 Relation and Functions 0 Comments

Question Number 37294 by math khazana by abdo last updated on 11/Jun/18

let D =D(0,1) and f(z) =Σ_(n=0) ^∞ a_n z^n is a holomorphe function / ∣f(x)∣< (1/(1−∣z∣)) prove that ∣a_n ∣≤ (n+1)(1+(1/n))^n ≤(n+1)e.

$${let}\:{D}\:={D}\left(\mathrm{0},\mathrm{1}\right)\:{and}\:{f}\left({z}\right)\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:{a}_{{n}} {z}^{{n}} \:{is}\:{a}\:{holomorphe} \\ $$$${function}\:/\:\:\mid{f}\left({x}\right)\mid<\:\:\frac{\mathrm{1}}{\mathrm{1}−\mid{z}\mid}\:\:{prove}\:{that} \\ $$$$\mid{a}_{{n}} \mid\leqslant\:\left({n}+\mathrm{1}\right)\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)^{{n}} \leqslant\left({n}+\mathrm{1}\right){e}. \\ $$

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