Let f be analytic in the unit disc \(| z| <1\) with Re[f'(z)] the Poisson integral of a measure \(\mu\) on the unit circle, whose modulus of continuity is denoted by \(\omega_{\mu}\). It is shown that \(f\in BMOA\) and \(\lambda_ f(S_ h)\leq Ch{\tilde \omega}_{\mu}(h)^ 2\) for all Carleson sectors \(S_ h\), and \(\omega_*(f,h)\leq C{\tilde \omega}_{\mu}(h)\), where C is a constant independent of h and \(\mu\). In the above, \(\lambda_ f\) denotes the measure defined by \[ d\lambda_ f(z)=(1-| z|^ 2)| f'(z)|^ 2dA(z), \] \(\omega\) \({}_*(f,h)\) the oscillation of f, and \({\tilde \omega}{}_{\mu}(h)=h\int^{\infty}_{h}(\omega (s)/s^ 2)ds\).