Area principle

The area of the complement to the image of a domain under a mapping by a function regular in it is non-negative. The area principle was first used in 1914 by T.H. Gronwall [1], who in this way proved the so-called exterior area theorem for functions of class $ \Sigma $— functions

$$ F (z) = z + \alpha _ {0} + \frac{\alpha _ {1} }{z} + \dots , $$

that are regular and univalent in the annulus $ \Delta ^ \prime = \{ {z } : {1 < | z | < \infty } \} $( cf. Univalent function). The area $ \sigma ( C F ( \Delta ^ \prime ) ) $ of the complement of the image $ F ( \Delta ^ \prime ) $ of $ \Delta ^ \prime $ under a mapping $ w = F (z) \in \Sigma $ can be determined by the formula

$$ \sigma ( C F ( \Delta ^ \prime ) ) = \pi \left ( 1 - \sum _ { k=1 } ^ \infty k | \alpha _ {k} | ^ {2} \right ) \geq 0 $$

and consequently

$$ \tag{1 } \sum _ { k=1 } ^ \infty k | \alpha _ {k} | ^ {2} \leq 1 . $$

By means of (1) the first results were obtained for functions of the classes $ \Sigma $ and $ S $, where $ S $ is the class of functions $ f (z) = z + \sum _ {k=2} ^ \infty a _ {k} z ^ {k} $ that are regular and univalent in the disc $ \Delta = \{ {z } : {| z | < 1 } \} $( cf. Bieberbach conjecture; Distortion theorems). A more general area theorem has been proved [2]. G.M. Goluzin [3] extended the area theorem to $ p $- valent functions in the disc (cf. Multivalent function).

The following area theorem has been proved [4]: Let $ F \in \Sigma $, $ \overline{B}\; = C F ( \Delta ^ \prime ) $, and let $ Q (w) $ be a regular function in $ \overline{B}\; $. If

$$ Q ( F (z) ) = \ \sum _ {k = - \infty } ^ { {+ } \infty } \alpha _ {k} z ^ {-k} \not\equiv \textrm{ const } ; $$

then

$$ \tag{2 } \sum _ {k = - \infty } ^ { {+ } \infty } k | \alpha _ {k} | ^ {2} \leq 0, $$

and equality holds only if the area $ \sigma ( \overline{B}\; ) $ of $ \overline{B}\; $ is zero.

By the area theorem for a certain class of univalent functions $ f (z) $, $ z \in B $, with $ B $ a domain, one usually understands any inequality having the property that equality holds if and only if the area of the complement $ \overline{G}\; $ of $ f (B) $ is zero, and the same applies for a class of systems of univalent functions $ \{ {f _ {k} (z) } : {z \in B _ {k} } \} _ {k=0} ^ {n} , n = 1 , 2 \dots $ where $ B _ {k} $ is a domain and $ \overline{G}\; $ is the complement of $ \cup _ {k=0} ^ {n} f _ {k} (B _ {f} ) $. Usually, such a theorem is proved by means of the area principle. That is, one considers any regular function $ Q (w) $, or more generally, one having a regular derivative, on $ \overline{G}\; $ and calculates the area $ \sigma ( Q ( \overline{G}\; ) ) $ of the image of $ \overline{G}\; $ under the mapping of the function $ Q $. Therefore, (2) is a certain extremely general area theorem in the class $ \Sigma $.

Let $ F \in \Sigma $ and let

$$ \mathop{\rm ln} \ \frac{F (t) - F (z) }{t - z } = \ \sum _ { p,q=1 } ^ \infty \omega _ {p,q} t ^ {-p} z ^ {-q} ,\ t , z \in \Delta ^ \prime . $$

One chooses a suitable function, regular in $ C F ( \Delta ^ \prime ) $, to be able to write (2) as

$$ \tag{3 } \sum _ { q=1 } ^ \infty q \left | \sum _ { p=1 } ^ \infty \omega _ {p,q} x _ {p} \right | ^ {2} \leq \ \sum _ { p=1 } ^ \infty \frac{1}{p} \ | x _ {p} | ^ {2} , $$

where the $ x _ {p} $ are any numbers not simultaneously equal to zero and such that

$$ \overline{\lim\limits}\; _ {p \rightarrow \infty } \ | x _ {p} | ^ {1/p} < 1 . $$

More general area theorems in $ \Sigma $ have also been obtained [5].

Area theorems have been proved for the following: the class $ \mathfrak M ( a _ {1} \dots a _ {n} ) $ of systems $ \{ {f _ {k} (z) } : {f _ {k} (0) = a _ {k} , z \in \Delta } \} _ {k=1} ^ {n} $ of functions $ f _ {k} $ that map the disc $ \Delta $ conformally and univalently onto domains without pairwise common points, i.e. onto non-adjacent domains [6]; the class $ \Sigma (B) $( $ \Sigma (B) $, $ B \ni \infty $, is the class of functions $ F $ that are regular and univalent in $ B \setminus \{ \infty \} $ and such that $ F ( \infty ) = \infty $, $ \lim\limits _ {z \rightarrow \infty } {F (z) } / z = 1 $), [7]; and non-overlapping multiple-connected domains (see [6] and also [8], [9]). All the area theorems for multiple-connected domains can be proved by the method of contour integration (cf. Contour integration, method of).

By the area method one understands methods of solving various problems in the theory of univalent functions by the use of area theorems.

For instance, from (3) one may obtain by means of the Cauchy inequality that

$$ \tag{4 } \left | \sum _ { p,q=1 } ^ \infty \omega _ {p,q} x _ {p} x _ {q} ^ \prime \right | ^ {2} \leq \ \sum _ { p=1 } ^ \infty \frac{1}{p} \ | x _ {p} | ^ {2} \sum _ { q=1 } ^ \infty \frac{1}{q} \ | x _ {q} ^ \prime | ^ {2} , $$

where $ x _ {p} $ and $ x _ {q} ^ \prime $ are such that the series on the right-hand side converge. If in (4), for example, $ x _ {p} = t ^ {-p} $, $ x _ {q} ^ \prime = z ^ {-q} $, $ | t | > 1 $, $ | z | > 1 $, one obtains the chord-distortion theorem

$$ \left | \mathop{\rm ln} \ \frac{F (t) - F (z) }{t - z } \right | ^ {2} \leq \mathop{\rm ln} \ \frac{| t | ^ {2} }{| t | ^ {2} -1 } \ \mathop{\rm ln} \ \frac{| z | ^ {2} }{| z | ^ {2} - 1 } . $$

Area theorems, for example in the class $ \mathfrak M ( a _ {1} \dots a _ {n} ) $, give necessary and sufficient conditions for a system $ \{ {f _ {k} (z) } : {f _ {k} (0) = a _ {k} , z \in \Delta } \} _ {k=1} ^ {n} $ of meromorphic functions $ f _ {k} $ to belong to the class $ \mathfrak M ( a _ {1} \dots a _ {n} ) $( see [6], p. 179).

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The inequalities (3) and (4) are a form of the Grunsky inequalities.

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