Let-f-C-R-R-n-N-M-n-f-n-and-u-n-2-n-1-M-n-M-n-1-for-n-1-Show-that-if-M-1-lt-2M-0-M-2-then-u-n-lt-u-n-1-for-n-1-

Question Number 127631 by snipers237 last updated on 31/Dec/20

Let f∈C^∞ (R,R) , ∀ n∈N M_n =∣∣f^((n)) ∣∣_∞ and u_n =((2^(n−1) M_n )/M_(n−1) ) for n≥1 Show that if M_1 <(√(2M_0 M_2 )) then u_n <u_(n+1) for n≥1

$${Let}\:{f}\in{C}^{\infty} \left(\mathbb{R},\mathbb{R}\right)\:,\:\forall\:{n}\in\mathbb{N}\:\:\:{M}_{{n}} =\mid\mid{f}^{\left({n}\right)} \mid\mid_{\infty} \:\: \\ $$$${and}\:\:{u}_{{n}} =\frac{\mathrm{2}^{{n}−\mathrm{1}} {M}_{{n}} }{{M}_{{n}−\mathrm{1}} }\:\:\:{for}\:{n}\geqslant\mathrm{1}\: \\ $$$${Show}\:{that}\:{if}\:\:\:{M}_{\mathrm{1}} <\sqrt{\mathrm{2}{M}_{\mathrm{0}} {M}_{\mathrm{2}} }\:{then}\:{u}_{{n}} <{u}_{{n}+\mathrm{1}} \:{for}\:{n}\geqslant\mathrm{1} \\ $$

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Let-f-C-R-R-n-N-M-n-f-n-and-u-n-2-n-1-M-n-M-n-1-for-n-1-Show-that-if-M-1-lt-2M-0-M-2-then-u-n-lt-u-n-1-for-n-1-

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