Sendov's conjecture for sufficiently high degree polynomials

Sendov’s conjecture asserts that if a complex polynomial $f$ of degree $n \geq 2$hasallofitszeroesinclosedunitdisk${ z: |z| \leq 1 }$, then for each such zero ${\lambda }_0$ there is a zero of the derivative $f^{\mathrm{\prime }}$ in the closed unit disk $\{z:\mathrm{\mid }z-{\lambda }_0\mathrm{\mid }\le 1\}$. This conjecture is known for $n<9$, but only partial results are available for higher $n$. We show that there exists a constant $n_0$ such that Sendov’s conjecture holds for $n \geq n_0$\mathrm{.}For$\lambda_0$ away from the origin and the unit circle we can appeal to the prior work of D'egot and Chalebgwa; for ${\lambda }_0$ near the unit circle we refine a previous argument of Miller (and also invoke results of Chijiwa when ${\lambda }_0$ is extremely close to the unit circle); and for ${\lambda }_0$ near the origin we introduce a new argument using compactness methods, balayage, and the argument principle.