In an article I am reading, the author considers a class of functions:
$$ \{T \in C^1(\Bbb R^+)\text{ strictly increasing; } \ \ x\le T(x)\le xT'(x) \} $$
Can you give me a generic example of such a function?
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1 Answer
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Any monomial for $n\ge1$ satisfies $x(x^n)'=nx^n\ge x^n$, and the constraint $T(x)\le xT'(x)$ is linear (w.r.t positive linear combinations), so any polynomial $\sum_{k=1}^n a_kx^k$ will satisfy this condition as long as each $a_k\ge 0$. To satisfy $x\ge T(x)$, all we need is $a_1\ge 1$.
answered Apr 6, 2014 at 1:34
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$\begingroup$ thanks. Do you have an idea of a function with an at most polynomial increase? $\endgroup$– mookidApr 6, 2014 at 1:37
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$\begingroup$ @mookid Oops, I misread the OP as $xT(x)\le T'(x)$, instead of $T(x)\le xT'(x)$. Actually, most polynomials will satisfy your constraint; will edit. $\endgroup$ Apr 6, 2014 at 1:39
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$\begingroup$ @mookid the $x$ was moved across the inequality, so it actually completely changes the character of the differential equation. (You need exponentials to get a derivative which has one more $x$ factor in the derivative, but any polynomial can have one less $x$ factor in the derivative.) $\endgroup$ Apr 6, 2014 at 1:47
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$\begingroup$ exactly. I edited with only the answer, for clarity. thanks again. $\endgroup$– mookidApr 6, 2014 at 1:49