As titled. Intuitively I don't think there is any. But I wish to get a somehow rigorous argument, not even necessarily a proof. If anyone has any reference on this, it would also be much appreciated.
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1What about $e^{e^x}$ and so forth? Or $4^x$ – Vasili Jan 15 '21 at 16:59
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Ummm... $x^{x^{x^x}}$? – David G. Stork Jan 15 '21 at 17:00
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Everyone be missing the concave condition? – jlammy Jan 15 '21 at 17:01
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@jlammy: the question was changed to convex... – Jan 15 '21 at 17:03
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Well I'll guess we have to wait for OP to clarify -- the question wasn't edited by them. – jlammy Jan 15 '21 at 17:05
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1If the question is on concave functions, by definition of concavity any concave function $f$ is bounded above by $f(0)+(f(1)-f(0))x$ for all $x\ge1$. Therefore increasing concave functions are $O(x)$ as $x\to\infty$. – Jan 15 '21 at 17:13
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@jlammy pretty sure OP was actually talking about convex functions since he mentioned $e^x$ – 5201314 Jan 15 '21 at 17:18
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@5201314 That's provided the OP knows the difference between convex and concave, and gives a damn. But this is MSE, so... – Jan 15 '21 at 17:29
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If $f$ is convex, then $e^f$ is convex. See how to apply this to find infinitely many functions with faster and faster growth. – Jean-Claude Arbaut Jan 15 '21 at 18:27
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For some reason this question was edited from "concave" to "convex", it is now rolled back – uuddlrlrb4 Jan 15 '21 at 18:46
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@Gae.S. Could you please elaborate a little further on why this bound holds? – uuddlrlrb4 Jan 15 '21 at 18:58
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@Gae.S. I got it now. Your answer is basically what I was looking for. Thanks! – uuddlrlrb4 Jan 16 '21 at 13:21
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$\Gamma(x)$ (the factorial for non-integer values) is such a function. As is $2^{2^x}$ and $2$ to the power of the previous function and so on. Even $3^x$ grows faster than $e^x$ because its base is larger.
Parcly Taxel
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There are plenty of examples of convex functions that grow faster than exponential functions. Some examples are:
- Double exponential function https://en.wikipedia.org/wiki/Double_exponential_function
- Factorials / Gamma function https://en.wikipedia.org/wiki/Factorial
- Tetration https://en.wikipedia.org/wiki/Tetration
5201314
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