I am fairly sure that Proving that a complex polynomial has no roots in the unit disk has been asked and answered before. My search with Approach0 finds only solutions on AoPS. – Any idea how to improve the search, or other ways to find a Q&A on MSE?

@MartinR It seems similar to this: Showing that the roots of a polynomial with descending positive coefficients lie in the unit disc.

@MartinSleziak That's it – it is the same question via the substitution q(z) = z^n p(1/z). Thanks!

Yes, the ordering is exactly opposite, substitution might solve it.

Probably among the linked questions you could find some questions with this formulation: math.stackexchange.com/questions/linked/188039

What I was trying was searching in the tag complex-analysis for keywords such as polynomial unit root "a_n" "a_0" or polynomial unit root real coefficients.

Maybe I should have simply tried complex-analysis+roots and complex-analysis+polynomials - and check the frequent tab in both cases.

Maybe it would be useful to add the (polynomials) tag to the question we found? (Just to make it easier to find in the future.)

@MartinSleziak Done.

Maybe I should try to remember that some people call it Eneström-Kakeya Theorem - in case I need search for something similar again. math.stackexchange.com/search?q=enestrom+kakeya