# 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.

J. Hadamard [1] and M.A. Lavrent'ev [2] were the first to compute and to give applications of first-order variations. To obtain variations in some specific class of univalent functions often is a complicated task by itself, on account of the non-linear nature of families of such functions. This task was solved for certain classes of functions in simply-connected and multiply-connected domains.

#### References

 [1] J. Hadamard, "Leçons sur le calcul des variations" , 1 , Hermann (1910) [2] M.A. Lavrent'ev, "On some properties of univalent functions" Mat. Sb. , 4(46) : 3 (1938) pp. 391–458 (In Russian) (French abstract) [3] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian) [4] K.I. Babenko, "The theory of extremal problems for univalent functions of class " Proc. Steklov Inst. Math. , 101 (1975) Trudy Mat. Inst. Steklov. , 101 (1972)