# Variation of a univalent function

A concept in the theory of univalent functions (cf. Univalent function). In a given domain $D$ of the complex plane let there be given a function $f( z)$ and a family $F( z, \lambda )$ of functions depending on a real parameter $\lambda$, $0 \leq \lambda \leq \Lambda$, $\Lambda > 0$, which are univalent in $D$ for all $\lambda \in [ 0, \Lambda ]$. Suppose that $F( z, 0) = f( z)$. One forms the difference $F( z, \lambda ) - f( z) \equiv \Phi ( z, \lambda )$. The $n$- th order variation, or the $n$- th variation, $n = 1, 2 \dots$ of the function $f( z)$( along the family $F( z, \lambda )$) is the coefficient ${q _ {n} } ( z)$ of $\lambda ^ {n}$ in the expansion of $\Phi ( z, \lambda )$ with respect to the parameter $\lambda$, on the condition that the residual term
$$\phi _ {n} ( z , \lambda ) = \ \Phi ( z , \lambda ) - q _ {1} ( z) \lambda - \dots - q _ {n} ( z) \lambda ^ {n}$$
has a higher order of smallness than $\lambda ^ {n}$, uniformly with respect to $z$ in $D$, on compact sets in $D$ or in the closure of $D$. The selection of one of these additional conditions is usually determined by the nature of the problem whose solution involves variational methods connected with the variation of a univalent function.