SCHWEITZER 1972 Problem 7

Let $f(x,y,z)$ be a nonnegative harmonic function in the unit ball of $\mathbb{R}^3$ for which the inequality $f(x_0,0,0) \leq \varepsilon^2$ holds for some $0\leq x_0 \leq 1$ and $0<\varepsilon<(1-x_0)^2$ . Prove that $f(x,y,z) \leq \varepsilon$ in the ball with center at the origin an radius $(1-3\varepsilon^{1/4}).$

P. Turan