{"id":2252,"date":"2026-01-16T20:41:15","date_gmt":"2026-01-16T20:41:15","guid":{"rendered":"https:\/\/www.ethanepperly.com\/?p=2252"},"modified":"2026-02-19T20:44:52","modified_gmt":"2026-02-19T20:44:52","slug":"the-other-markovs-inequality","status":"publish","type":"post","link":"https:\/\/www.ethanepperly.com\/index.php\/2026\/01\/16\/the-other-markovs-inequality\/","title":{"rendered":"The Other Markov’s Inequality"},"content":{"rendered":"\n

If a polynomial function is trapped in a box, how much can it wiggle? This question is answered by Markov’s inequality<\/a>, which states that for a degree-\"n\" polynomial<\/a> \"p\" that maps \"[-1,1]\" into \"[-1,1]\", it holds that

(1) <\/span>   <\/span>\"\[\max_{x \in [-1,1]} |p'(x)| \le n^2. \]\"<\/p>That is, if a polynomial \"p\" is trapped within a square box \"[-1,1] \times [-1,1]\", the fastest it can wiggle\u2014as measured by its first derivative<\/a>\u2014is the square of its degree.<\/p>\n\n\n\n

How tight is this inequality? Do polynomials we know and love come close to saturating it, or is this bound very loose for them? A first polynomial which is natural to investigate is the power \"p(x) = x^n\". This function maps \"[-1,1]\" into \"[-1,1]\", and the maximum value of its derivative is

  <\/span>   <\/span>\"\[\max_{x \in [-1,1]} |p'(x)| = \max_{x \in [-1,1]} |nx^{n-1}| = n\ll n^2.\]\"<\/p>Markov’s inequality is quite loose for the power function.<\/p>\n\n\n\n

To saturate the inequality, we need something wigglier. The heavyweight champions<\/a> for polynomial wiggliness are the Chebyshev polynomials<\/a> \"T_n\", which are motivated and described at length in this previous post<\/a>. For our purposes, what’s important is that Chebyshev polynomials really know how to wiggle. Just look at how much more rapidly the degree-7 Chebyshev polynomial (blue solid line) moves around than the degree-7 power \"x^n\" (orange dashed line).<\/p>\n\n\n\n