On properties of associated quadratic differentials in some extremal problems

V. O. Kuznetsov

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 9, Tome 168 (1988), pp. 85-97

Résumé

One investigates properties of extremal configurations in the problem of the maximum of the $n$-th diameter $d_n(E)$ in the family of continua $E$ of unit capacity and in the problem of the maximum of the corresponding conformal invariant in a family of nonoverlapping domains. It is shown that for these problems the associated quadratic differentials do not have multiple zeros and that the Fekete points of the extremal continuum of the first of the mentioned problems are simple poles of the associated differential. It is also shown that the quadratic differential, associated with a support function of class $\Sigma$, does not have zeros of multiplicity $>2$. The paper continues the previous investigation of the author (MR 88c:30028).