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The negation of your property 3. is called regular variation. Here is an example of a concave not regularly varying function (taken from this paper by Iksanov and Rösler, p.10) : Take $$f(x) = 2^{-k} x + 2^{k+1} - 3, \;\;\;\; x \in [4^k,4^{k+1})$$ Then for $x_n=4^n$, $y_n=3.4^n$, y_n=3\cdot4^n$, $$\lim_n \frac{f(2x_n)}{f(x_n)} = 2 \neq \frac{7}{5} = \lim_n \frac{f(2y_n)}{f(y_n)}.$$ |
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The negation of your property 3. is called regular variation. Here is an example of a concave not regularly varying function (taken from this paper by Iksanov and Rösler, p.10) : Take $$f(x) = 2^{-k} x + 2^{k+1} - 3, \;\;\;\; x \in [4^k,4^{k+1})$$ Then for $x_n=4^n$, $y_n=3.4^n$, $$\lim_n \frac{f(2x_n)}{f(x_n)} = 2 \neq \frac{7}{5} = \lim_n \frac{f(2y_n)}{f(y_n)}.$$ |
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