Journal of the
Korean Mathematical Society JKMS

We compute the holomorphic derivative of the harmonic measure associated to a $C^\infty$ bounded domain in the plane and show that the exact Bergman kernel function associated to a $C^\infty$ bounded domain in the plane relates the derivatives of the Ahlfors map and the Szeg\H{o} kernel in an $explicit$ way. We find several formulas for the exact Bergman kernel and the Szeg\H{o} kernel and the harmonic measure. Finally we survey some other properties of the holomorphic derivative of the harmonic measure.

Keywords: Bergman kernel, Szego kernel, Ahlfors map, harmonic measure

MSC numbers: 30C40, 32A36, 30C20