On some generalizations of the transfinite diameter (Q1086700)

The author gives a version of the Goluzin inequalities for univalent functions in a multiply connected domain. It is used to solve certain extremal problems. Let \(\Delta\) be a domain which contains \(\infty\) and is bounded by N smooth curves \(\Gamma_ k\), \(k=1,...,N\). Let f be analytic and univalent in \(\Delta\), except for a simple pole at \(\infty\), and let \(f'(\infty)=1\). Define the functional (1) \(\phi (f)=\sum^{n}_{j,k=1}x_ jx_ k\log | \frac{f(\zeta_ j)- f(\zeta_ k)}{\zeta_ j-\zeta_ k}|\) for n given points \(\zeta_ j\in \Delta\) and n fixed real constants \(x_ j\). Using a variation of Schiffer's type the author shows that a Goluzin type inequality holds \[ (2)\quad \phi (f)\leq 2s\sum^{n}_{j=1}x_ jg(\zeta_ j)-s^ 2\rho - \sum^{n}_{j,k=1}x_ jx_ k(g(\zeta_ j,\zeta_ k)-\log \frac{1}{| \zeta_ j-\zeta_ k|}), \] where \(g(\zeta)=g(\zeta,\infty)=\log | \zeta | +\rho +0(\frac{1}{| \zeta |})\) and g(\(\rho\),\(\eta)\) is Green's function of \(\Delta\), and \(s=\sum^{n}_{j=1}x_ j\). The number \(\rho\) is called Robin constant. The author shows how (2) can be used to get the well-known relation \(\rho =\log 1/d\) between the Robin constant and the transfinite diameter d. As another application a relation between the modulus M(A,B) of two disjoint closed sets A and B to the Green's function is given.