# Find a closed form for a parameter in an inequality

Doing some math I came up to this expression, where $d,\epsilon>0$ $$\left(1+\frac{d}{2}+{\epsilon}d\right)^{20} <\left(1+\frac{d}{2}-{\epsilon}d\right)^{21}$$ Since for $\epsilon\rightarrow 0$ it holds, for every fixed $d>0$ one can find an enough small $\epsilon$ such that this inequality holds.

I'm curious if there is a closed form expressing $\epsilon$ as a function of $d$ for which the inequality holds, in other words a function $\epsilon(d)$ such that $$\left(1+\frac{d}{2}+d\cdot{\epsilon(d)}\right)^{20} <\left(1+\frac{d}{2}-d\cdot{\epsilon(d)}\right)^{21}$$ holds for every $d>0$.

I guess it could be something like $$\epsilon(d)= \begin{cases} \frac{1}{d^c+c}& \text{if }d\geqslant 1\\ d^c&\text{if }d< 1 \end{cases}$$ for $c$ big enough, and with the aid of a mathematical software it could be determined explicitly, but I have no experience in using them.

Maybe, am I missing some straightforward manipulations?

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Tried few calculations in Maple and seems it works for all $\epsilon \leq \frac{1}{82}$, independently on $d$. But I don't see why. – Sil Apr 5 at 20:29