Vol. 39, No. 1, 1971

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A note on the Löwner differential equations

Maurice Heins

Vol. 39 (1971), No. 1, 173–177

The object of the present note is to indicate a derivation of the Löwner differential equations [1] based on the derivation of an associated differential equation for Green’s function of the variable region relative to the defining parameter. Decisive in our treatment is the use of a certain normalized minimal positive harmonic function on the variable region. In fact, our starting point was the feeling that the Poisson kernel asserted its presence so strongly in the Löwner differential equations that the concomitant presence of a normalized minimal positive harmonic function on the variable region should appear naturally in the study of the question. We shall see that this is the case. A technical advantage of the present approach is that the “tip” lemmas of the classical proof are dispensed with.

Mathematical Subject Classification
Primary: 30A36
Received: 7 December 1970
Published: 1 October 1971
Maurice Heins