Modulus of continuity of harmonic functions
Suppose that u(z) is a harmonic function in a plane domain G and that the modulus of continuity of u(z) is majorised by a nondecreasing function \(\mu\) (t), \(\mu\) (2t)\(\leq 2\mu (t)\), on the boundary \(\partial G\). What kind of upper bound can be obtained for \(| u(z_ 1)-u(z_ 2)|\) when \(z_ 1,z_ 2\in \bar G?\) Making use of various estimates of the harmonic measure the author gives quite explicit bounds such as \[ | u(z_ 1)-u(z_ 2)| \leq \mu (| z_ 1-z_ 2|)(K_ 1+K_ 2S(| z_ 1-z_ 2|)), \] for \(z_ 1,z_ 2\in \bar G\), where \(K_ 1\) and \(K_ 2\) are some constants (at least one of them must depend on G) and S(t) is some function.
scientific article; zbMATH DE number 4099646