Call a real number $\lambda>1$ **special** if it is a root of a polynomial $f(x)$ such that

- $f(x)$ is monic with integer coefficients,
- all roots of $f(x)$ are distinct,
- for all $z\in\mathbb{C}$, if $f(z)=0$ and $z\neq \lambda$ then $|z|<1$.

For example, the Golden Ratio $\frac{1+\sqrt{5}}{2}$ is special.

**Question**: Are there special numbers arbitrarily close to 1?

*Some Remarks*:

After messing around on Wolfram Alpha, I was able to beat the Golden Ratio using the unique real root of $x^3-x-1$, which is approximately 1.3247.

I have found lots of work on polynomials such that

*all*roots are on the unit disc, and some things where all but*two*roots are on the unit disc. But nothing close to what I want: exactly one real root not on the unit disc, and the rest strictly inside the unit disc.