New version of this lecture: https://youtu.be/18zREgtbSHM
Harnack's inequality and Harnack's principle for harmonic functions. Online lectures for Complex Analysis I at Oklahoma State University. After uploading I noticed that there is an easier estimate in for the q, that is you can prove f(z_j) \le 3^2 f(z_{j+1}). It's an exercise to simplify this proof.
See https://www.jirka.org/ca/ for the (free online) book "Guide to Cultivating Complex Analysis" these lectures are based on.
See https://youtube.com/playlist?list=PLRfQb6m35rf5vZaOlSXVmt0hgJepfEqCw for a more complete course (work in progress)
