any complex number for which Then we have for the logarithmic derivative In particular, if is a zero of and then or This may also be written as Taking their conjugates, we get that is a convex combination of the roots of . If then for some , and is still a convex combination of the roots of . See also Marden's theorem Bôcher's theorem Sendov's conjecture Routh–Hurwitz theorem Hurwitz's theorem (complex analysis) Descartes' rule of signs Rouché's theorem Properties of polynomial roots Cauchy interlacing theorem Notes References . Craig Smorynski: MVT: A Most Valuable Theorem. Springer, 2017, ISBN 978-3-319-52956-1, pp. 411–414 External links Lucas–Gauss Theorem by Bruce Torrence, the Wolfram Demonstrations Project. Gauss-Lucas theorem as interactive illustration Convex analysis Articles containing proofs Theorems in complex analysis Theorems about polynomials