SCHWARZ LEMMA FOR REAL HARMONIC FUNCTIONS ONTO SURFACES WITH NON-NEGATIVE GAUSSIAN CURVATURE

Assume that f is a real rho-harmonic function of the unit disk D onto the interval (-1, 1), where rho(u v) = R(u) is a metric defined in the infinite strip (-1, 1) x R. Then we prove that vertical bar f(z)vertical bar(1-vertical bar z vertical bar(2)) 4/pi (1-f(z)(2) for all z epsilon D, provided that rho has a non-negative Gaussian curvature. This extends several results in the field and answers to a conjecture proposed by the first author in 2014. Such an inequality is not true for negatively curved metrics.