a;b;c∈R, ∣ax^5 +bx^3 +cx∣≤1, ∀∣x∣≤1 proof: ∣a∣≤16, ∣b∣≤20, ∣c∣≤5


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Question Number 139644 by mathdanisur last updated on 30/Apr/21

$a; b; c \in R, ∣ {a x}^{5} + {b x}^{3} + c x ∣⩽ 1, \forall ∣ x ∣⩽ 1$ $p r o o f : ∣ a ∣⩽ 16, ∣ b ∣⩽ 20, ∣ c ∣⩽ 5$
	Answered by ajfour last updated on 30/Apr/21
	$let ∣x∣=r x=rcos θ+irsin θ ∣ar^5 e^(5iθ) +br^3 e^(3iθ) +cre^(iθ) ∣≤1 r^3 ∣ar^2 e^(2iθ) +b+(c/(r^2 e^(2iθ) ))∣≤1 r^3 ∣ar^2 (cos 2θ+isin 2θ)+b +(c/r^2 )(cos 2θ−isin 2θ)∣≤1 r^6 {[b+(ar^2 +(c/r^2 ))cos 2θ]^2 +(ar^2 −(c/r^2 ))^2 sin^2 2θ}≤1 r^6 {b^2 +2b(ar^2 +(c/r^2 ))cos 2θ +4accos^2 2θ+(ar^2 −(c/r^2 ))^2 }≤1 r^6 {b^2 +4ac[cos 2θ+(b/(4ac))(ar^2 +(c/r^2 ))]^2 −(b^2 /(4ac))(ar^2 +(c/r^2 ))^2 }≤1 ......$
	$l e t ∣ x ∣= r$ $x = r \cos θ + i r \sin θ$ $∣ {a r}^{5} e^{5 i θ} + {b r}^{3} e^{3 i θ} + {c r e}^{i θ} ∣⩽ 1$ $r^{3} ∣ {a r}^{2} e^{2 i θ} + b + \frac{c}{r^{2} e^{2 i θ}} ∣⩽ 1$ $r^{3} ∣ {a r}^{2} (\cos 2 θ + i \sin 2 θ) + b$ $+ \frac{c}{r^{2}} (\cos 2 θ - i \sin 2 θ) ∣⩽ 1$ $r^{6} {{[b + ({a r}^{2} + \frac{c}{r^{2}}) \cos 2 θ]}^{2} + {({a r}^{2} - \frac{c}{r^{2}})}^{2} \sin^{2} 2 θ} ⩽ 1$ $r^{6} {b^{2} + 2 b ({a r}^{2} + \frac{c}{r^{2}}) \cos 2 θ$ $+ 4 a c \cos^{2} 2 θ + {({a r}^{2} - \frac{c}{r^{2}})}^{2}} ⩽ 1$ $r^{6} {b^{2} + 4 a c {[\cos 2 θ + \frac{b}{4 a c} ({a r}^{2} + \frac{c}{r^{2}})]}^{2}$ $- \frac{b^{2}}{4 a c} {({a r}^{2} + \frac{c}{r^{2}})}^{2}} ⩽ 1$ $. . . . . .$
	Commented by mathdanisur last updated on 30/Apr/21

	$t h e n h o w s i r$
	Commented by mathdanisur last updated on 13/May/21

	$A j f o u r S i r c o n t i n u o p l e a s e . . .$
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