In mathematics, the Loewner differential equation, or Loewner equation, is an ordinary differential equation discovered by Charles Loewner in 1923 in complex analysis and geometric function theory. Originally introduced for studying slit mappings (conformal mappings of the open disk onto the complex plane with a curve joining 0 to ∞ removed), Loewner's method was later developed in 1943 by the Russian mathematician Pavel Parfenevich Kufarev (1909–1968). Any family of domains in the complex plane that expands continuously in the sense of Carathéodory to the whole plane leads to a one parameter family of conformal mappings, called a Loewner chain, as well as a two parameter family of holomorphic univalent self-mappings of the unit disk, called a Loewner semigroup. This semigroup corresponds to a time dependent holomorphic vector field on the disk given by a one parameter family of holomorphic functions on the disk with positive real part. The Loewner semigroup generalizes the notion of a univalent semigroup. The Loewner differential equation has led to inequalities for univalent functions that played an important role in the solution of the Bieberbach conjecture by Louis de Branges in 1985. Loewner himself used his techniques in 1923 for proving the conjecture for the third coefficient. The Schramm–Loewner equation, a stochastic generalization of the Loewner differential equation discovered by Oded Schramm in the late 1990s, has been extensively developed in probability theory and conformal field theory.
Subordinate univalent functions Let f {\displaystyle f} and g {\displaystyle g} be holomorphic univalent functions on the unit disk D {\displaystyle D} , | z | < 1 {\displaystyle |z|<1} , with f ( 0 ) = 0 = g ( 0 ) {\displaystyle f(0)=0=g(0)} .
f {\displaystyle f} is said to be subordinate to g {\displaystyle g} if and only if there is a univalent mapping φ {\displaystyle \varphi } of D {\displaystyle D} into itself fixing 0 {\displaystyle 0} such that
f ( z ) = g ( φ ( z ) ) {\displaystyle \displaystyle {f(z)=g(\varphi (z))}}
for | z | < 1 {\displaystyle |z|<1} . A necessary and sufficient condition for the existence of such a mapping φ {\displaystyle \varphi } is that
f ( D ) ⊆ g ( D ) . {\displaystyle f(D)\subseteq g(D).}
Necessity is immediate. Conversely φ {\displaystyle \varphi } must be defined by
φ ( z ) = g − 1 ( f ( z ) ) . {\displaystyle \displaystyle {\varphi (z)=g^{-1}(f(z)).}}
By definition φ is a univalent holomorphic self-mapping of D {\displaystyle D} with φ ( 0 ) = 0 {\displaystyle \varphi (0)=0} . Since such a map satisfies 0 < | φ ′ ( 0 ) | ≤ 1 {\displaystyle 0<|\varphi '(0)|\leq 1} and takes each disk D r {\displaystyle D_{r}} , | z | < r {\displaystyle |z|<r} with 0 < r < 1 {\displaystyle 0<r<1} , into itself, it follows that
| f ′ ( 0 ) | ≤ | g ′ ( 0 ) | {\displaystyle \displaystyle {|f^{\prime }(0)|\leq |g^{\prime }(0)|}}
and
f ( D r ) ⊆ g ( D r ) . {\displaystyle \displaystyle {f(D_{r})\subseteq g(D_{r}).}}
Loewner chain For 0 ≤ t ≤ ∞ {\displaystyle 0\leq t\leq \infty } let U ( t ) {\displaystyle U(t)} be a family of open connected and simply connected subsets of C {\displaystyle \mathbb {C} } containing 0 {\displaystyle 0} , such that
U ( s ) ⊊ U ( t ) {\displaystyle U(s)\subsetneq U(t)}
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