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I have been working on controlling the behavior of an exponential function within a large positive interval of real numbers and I would like to identify a constant in the estimate I am using.

Let $r>1$ and let $ A = [u^{\frac{1}{r}}, u^{\frac{1}{r}} +1 ] $ for $u>>1$ (that is $u$ is a very large positive number). Let $y \in A$ then I am interested in the behavior of $y^2$ but we know:

$$ u^{\frac{2}{r}} \leq y^2 \leq ( u^{\frac{1}{r}} +1 )^2$$

Now since $u>>1$ we can get for some constant $K>0$ the following upper bound for $y^2$:

$y^2 \leq K u^{\frac{2}{r}}$
What is a simple way to prove this estimate and how do we determine $K$?

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1 Answer 1

up vote 1 down vote accepted

$(u^{1/7}+1)^2 \le K u^{2/7}$ if $u^{1/7}+1 \le \sqrt{K} u^{1/7}$, and this is true for any $K > 1$ as long as $u^{1/7} \ge \frac{1}{\sqrt{K}-1}$

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