The local modulus of continuity of an analytic function

Abstract

Let f(z) be an analytic function on the unit disk D and let ς ∈ ∂D. If f(ς) exists as a radial limit, then we define

$$ \omega (f,t\zeta ) = {\rm{sup}}\left\{ {\left| {f(z) - f(\zeta )} \right|:\left| {z - \zeta } \right| \le t,z, \in D} \right\}$$

and

$$\varpi (f,t,\zeta ) = ess\,{\rm{sup}}\left\{ {\left| {f({e^{i\Theta }}) - f(\zeta )\left| : \right|{e^{i\Theta }} - \zeta } \right| \le t} \right\}$$

.