ABSTRACT
T A N G E N T V A R I A T I O N P R I N C I P L E : S A T E L L I T E P R I N C I P L E S
1.1. Modifications of Length-Area Principle: Connection with Various Classes of Functions
1.1.1. The Ahlfors' classical length-area principle (Ahlfors [1], 1930). It is one of the remarkable relations of the theory of functions, unique in its generality and clearness. According to this principle, for any function w(z) regular in a domain D the following inequality is true:
where L(D,T(R)) is the sum of the lengths of the curves in D on which \w(z)\ — R (which is the same as the curves w~l(T(R))), where T(R) is the circle {w: \w\ = R} and
p(R) = — / n(D,Re?*)d'd, 2?r JO
where n (D, Re119) is the number of roots of the equation w(z) = Re1^ in D, according to their multiplicities, and S(D) is the area of D.
