UDC 517.54

MATHEMATICS

K. I. BABENKO

ON THE THEORY OF THE SECOND VARIATION OF FUNCTIONALS ON THE CLASS (S) OF UNIVALENT FUNCTIONS

(Presented by Academician A. N. Tikhonov on 24 II 1970)

By (S) we shall, as usual, denote the class of functions (f(z)) univalent in the disk (|z|<1) and normalized by the conditions (f(0)=0,\ f'(0)=1). We shall write the Taylor expansion of (f(z)\in S) in the form
[
f(z)=\sum_{0}^{\infty} a_k z^{k+1}, \qquad a_0=1.
]
The class (S) becomes bicompact if one introduces in it the metric
[
\rho(f,g)=\max_{|z|=1/2}|f(z)-g(z)|,\qquad f,g\in S.
]

For the construction of the theory of extremal problems on the class (S), it is important to be able, for a given (f(z)\in S), to find a function (g(z)\in S) lying in a prescribed (\varepsilon)-neighborhood of (f). This can be achieved, for example, by the following known device. A function (h(z)), regular in some annulus (K_{r_0}={z:\ r_0<|z|<1}), will be called admissible with respect to (f(z)) if (f(z)+h(z)) is regular and univalent in (K_{r_0}). Following G. M. Goluzin ((^1)), one can show that, by means of a change of variables, the singularities of the function (f+h) are removed and a function is obtained which is regular and univalent in the disk (|z|<1). Such a construction is always possible as soon as (h) satisfies a certain smallness condition. It is important to emphasize that the regularizing change of variables is made with a known degree of arbitrariness, and this circumstance is useful in various concrete cases.

As (h(z)) one can always take a function of the form
[
\varepsilon \int_{\mathscr C} \frac{f'(z)}{f(z)-f(t)}
\left(\frac{t f'(t)}{f(t)}\right)^2 \nu(dt),
\tag{1}
]
where (\nu(dt)) is a complex-valued measure with support (\mathscr C), lying in some disk (|z|\le r_1<1), and (\varepsilon) is a real constant. As a result of regularization and subsequent normalization we obtain a function (f(z,\varepsilon)\in S). For fixed (r_0<1), one can indicate an (\varepsilon_0>0) such that, for (|\varepsilon|<\varepsilon_0) and (|z|\le r_0), (f(z,\varepsilon)) is represented in the form of a convergent series
[
f(z,\varepsilon)=\sum_{0}^{\infty}\frac{\varepsilon^k}{k!}\,f_k(z),\qquad f_0(z)=f(z).
]
The function (f_k(z)), (k=1,2,\ldots), is regular for (|z|<1), and we shall call it the (k)-th variation of the function (f(z)). It is convenient to denote the first and second variations by (\delta f(z)), (\delta^2 f(z)), or by (\delta f(z;\nu)), (\delta^2 f(z;\nu)). For the first variation we have the well-known formula of M. Schiffer ((^2)),
[
\delta f(z)=
\int_{\mathscr C}
\left[
\frac{f'(z)}{f(z)-f(t)}
\left(\frac{D f(t)}{f(t)}\right)^2
+
\frac{D f(z)\,t}{t-z}
-
f(z)
\right]\nu(dt)
+
]
[
+
D f(z)\int_{\mathscr C}\frac{\bar t z}{1-\bar t z}\,\nu(dt)
+
iC_1\bigl(Df(z)-f(z)\bigr),
\tag{2}
]
where (D) is the differential operator (z\,d/dz), and (C_1) is an arbitrary real constant.

We shall not give the formula for (\delta^2 f(z)), in view of its cumbersomeness. The coefficients of the power series of the function (\delta f(z)) will be called the first variations of the coefficients (a_k,\ k=1,2,\ldots,) and will be denoted, respectively, by (\delta a_k,\ k=1,2,\ldots). Thus,

[
\delta f(z)=\sum_{1}^{\infty}\delta a_k z^{k+1}.
]

Assuming (w\ne 0), consider, for small (|z|), the expansion

[
\frac{f^2(z)}{f(z)-w}=\sum_{0}^{\infty} q_k(w) z^{k+1}.
\tag{3}
]

It is clear that (q_k(w)) is a polynomial in (1/w) of degree (k). We also introduce the functions

[
p_k(z)=\sum_{l=1}^{k}(k-l+1)a_{k-l}z^{-l},\quad l=1,2,\ldots
\tag{4}
]

From (2) we find

[
\delta a_k=\int_{\mathcal E}\mathfrak A_k(z)\,\nu(dz)+\int_{\mathcal E}\mathfrak B_k(\bar z)\,\overline{\nu(dz)}+iC_1ka_k,\quad k=1,2,\ldots,
\tag{5}
]

where

[
\mathfrak A_k(z)=q_k[f(z)]\left(Df(z)/f(z)\right)^2+p_k(z)+ka_k,
]

[
\mathfrak B_k(z)=p_k(1/z),\quad k=1,2,\ldots
\tag{6}
]

It is not hard to verify that (\mathfrak A_k(z), \mathfrak B_k(z),\ k=1,2,\ldots,) are regular functions in the disk (|z|<1), and moreover (\mathfrak A(0)=\mathfrak B(0)=0).

Taking admissible functions in the form (1), in a prescribed neighborhood (U) of the element (f,\ U={g:\rho(f,g)\le \varepsilon_0}), we obtain a fairly large set. Therefore, with the help of variations of this kind one can resolve the question of necessary conditions for a local extremum of a functional. If one considers the question of sufficient conditions, then it is necessary to use admissible functions of a more general form than (1).

We shall give the necessary conditions for a local extremum, restricting ourselves to functionals of the form (J(f)=F(a_1,\ldots,a_n)), where (F) is a twice continuously differentiable function of its arguments, defined in the ball

[
\sum_{1}^{n}|a_k|^2\le R^2
]

of sufficiently large radius. One of the necessary conditions for an extremum, namely the M. Schiffer equation, is well known. Assuming that the functional (J(f)) is stationary at the function (f), with the help of (5) we obtain, by virtue of the arbitrariness of the measure (\nu(dz)) and of the constant (C_1), the conditions

[
\sum_{1}^{n}\left[\bar\lambda_k\mathfrak A_k(z)+\lambda_k\overline{\mathfrak B_k(z)}\right]=0,
\tag{7}
]

[
\operatorname{Im}\sum_{1}^{n}\bar\lambda_kka_k=0,
\tag{8}
]

where (\lambda_k=2\partial F/\partial \bar a_k,\ k=1,\ldots,n). Applying (6), relation (7) can be written in the form

[
Q(f(z))\left(Df(z)/f(z)\right)^2+P(z)=0,
\tag{9}
]

where

[
Q(w)=\sum_{1}^{n}\bar\lambda_k q_k(w),\quad
P(z)=\sum_{1}^{n}\left[\bar\lambda_k\bigl(p_k(z)+ka_k\bigr)+\lambda_k\overline{p_k\left(\frac1z\right)}\right].
\tag{10}
]

Equation (9) is precisely the M. Schiffer equation. From (8) we obtain that (\operatorname{Im}P(z)=0), if (|z|=1).

If the functional (J) has a local maximum at the function (f), then the second variation of the functional must be nonpositive. For convenience of the computations, we first consider the case when (F) is a linear function

[
F=\operatorname{Re}\sum_{1}^{n}\overline{\lambda_k}a_k.
]

Then

[
\delta^2F=\operatorname{Re}\sum_{1}^{n}\overline{\lambda_k}\delta^2a_k.
]

Omitting all intermediate calculations, we give the final formula for (\delta^2F). We first introduce a number of notations. Put

[
\mathcal{K}(z,t)=\left[Df(t)/(f(z)-f(t))\right]^2-(Df(t)/f(t))^2-zt/(z-t)^2,
\tag{11}
]

and also

[
P^*(z)=\sum_{1}^{n}(k+1)\left[\overline{\lambda_k}p_k(z)-\lambda_k\overline{p_k\left(\frac{1}{z}\right)}\right],
\quad
R(z)=\sum_{1}^{n}\overline{\lambda_k}p_k(z).
\tag{12}
]

Next put

[
Q^(w,z)=\sum_{k=2}^{n}\overline{\lambda_k}\sum_{l=1}^{k-1}(k-l+1)z^{-l}q_{k-l}(w),
\quad
Q^(w)=\sum_{1}^{n}k\overline{\lambda_k}q_k(w).
\tag{13}
]

We now introduce two kernels: the symmetric kernel

[
\begin{aligned}
\mathfrak{A}(z,t)= {}& P(z)\mathcal{K}(z,t)+P(t)\mathcal{K}(t,z)+[zP^(t)-tP^(z)]/(z-t)+{}\
& +(Df(z)/f(z))^2[Q^(f(z),t)+Q^(f(z))]+(Df(t)/f(t))^2[Q^(f(t),z)+{}\
& \qquad +Q^(f(t))]-R(z)-R(t)+\sum_{1}^{n}\overline{\lambda_k}k^2a_k
\end{aligned}
\tag{14}
]

and the Hermitian kernel

[
\begin{aligned}
\mathfrak{B}(z,\overline{t})={}&
\frac{z\overline{t}}{(1-z\overline{t})^2}[P(z)+\overline{P(t)}]
-\frac{z\overline{t}}{1-z\overline{t}}[P^(z)+\overline{P^(t)}]
-\left(\frac{Df(z)}{f(z)}\right)^2 \times{}\
&\times Q^\left[f(z),\frac{1}{\overline{t}}\right]
-\overline{\left(\frac{Df(t)}{f(t)}\right)^2 Q^\left[f(t),\frac{1}{\overline{z}}\right]}
+R\left(\frac{1}{\overline{z}}\right)+R\left(\frac{1}{t}\right).
\end{aligned}
\tag{15}
]

Introduce one more function

[
\mathfrak{C}(z)=(Df(z)/f(z))^2Q^[f(z)]+P^(z)-R(z)+\overline{R(1/\overline{z})}
+\sum_{1}^{n}\overline{\lambda_k}k^2a_k.
\tag{16}
]

Then

[
\delta^2F=\operatorname{Re}\left{
\iint_{\mathcal{C}\mathcal{C}}\mathfrak{A}(z,t)\nu(dz)\nu(dt)
-\iint_{\mathcal{C}\mathcal{C}}\mathfrak{B}(z,\overline{t})\nu(dz)\overline{\nu(dt)}
+\right.
]

[
\left.
+\,2iC_1\int_{\mathcal{C}}\mathfrak{C}(z)\nu(dz)
-C_1^2\sum_{1}^{n}\overline{\lambda_k}k^2a_k
\right}.
\tag{17}
]

In order to obtain the second variation in the general case, one must add to expression (17) a quadratic form in (\delta a_k,\ k=1,2,\ldots), whose coefficients are computed from the second derivatives of the function (F).

Using the arbitrariness in the choice of the measure (\nu(dz)) and of the constant (C_1), it is easy to obtain from the condition (\delta^2F\leqslant 0) that

[
P(z)\geqslant 0 \quad \text{for } |z|=1.
\tag{18}
]

Definition 1. A univalent function (f(z)\in S) satisfying equation (9), in which for (P(z)) condition (19) is fulfilled, will be called an extremal univalent function.

By (S_n) we shall denote the set of extremal functions satisfying equation (9), in which (Q(w)) is a polynomial in (1/w) of degree not higher than (n). The term extremal can be justified by the fact that every such function gives a local extremum of some function-

nal; this is a rather deep fact, and we shall not need it. From equation (9) we obtain that on the circle (|z|=1) the function (f(z)) can have only a finite number of algebraic singular points. Thus, (f(z)) maps the disk (|z|<1) onto a domain whose boundary consists of a finite number of analytic arcs. On each such arc, by virtue of (9) and (18),

[
Q(f(z))(df(z)/f(z))^2 \geqslant 0,
]

i.e., these arcs belong to the trajectories of the differential (Q(w)(dw/w)^2). Let us take the union of the trajectories of this differential that have limiting end points at the zeros of (Q(w)). Adjoining to the set so constructed the point (w=0), we obtain a plane connected graph (\mathfrak G)—the graph of the differential (Q(w)(dw/w)^2). The structure of the graph of a quadratic differential has been studied in detail in the papers ((^3!-!^5)). It is important to note that every cycle of the graph (\mathfrak G) necessarily contains the point (w=0). It is not difficult to show that the image of the circle (|z|=1) under the mapping (w=f(z)) will be a tree (\mathfrak T \subset \mathfrak G). The point (w=\infty) belongs to the tree (\mathfrak T), and we shall regard it as the root of (\mathfrak T).

Below we shall study the second variation in the class (S_n). To each (f\in S_n) there corresponds equation (9), or, what is the same, equation (7), which is completely determined by the vector (\lambda=(\lambda_1,\ldots,\lambda_n)). We shall call these vectors associated with (f). The totality of all associated vectors will be denoted by (\Lambda_f), and the totality of those vectors for which (18) holds will be denoted by (K_e) or (K_e(f)); (K_e) is a cone. Note that (\dim K_e\geqslant 1). A vector (\lambda\in K_e) naturally determines the linear functional

[
L(f)=\operatorname{Re}\sum_{1}^{n}\overline{\lambda_k}a_k,
]

which plays an essential role in the class (S_n). We transform formula (17) for the second variation of the linear functional (L(f)). Let (D(\mathfrak T)) be the domain complementary to (\mathfrak T). Put

[
\xi(w)=\delta f(z),\quad z=f^{-1}(w);\qquad
\xi(w)=\sum_{1}^{\infty}\xi_k w^{k+1}.
]

The coefficients (\xi_k), (k=1,2,\ldots), are easily computed in terms of (\delta a_k), (k=1,2,\ldots). Let

[
Q(w)=\sum_{1}^{n} A_k w^{-k}.
]

Orient the tree (\mathfrak T) in an arbitrary way. Then on each edge of the tree the sides—the left and the right—will be determined. We shall denote the limiting values of the function (\varphi(w)) on (\mathfrak T) from the left and from the right, respectively, by (\varphi_+(w)), (\varphi_-(w)). Put (\xi^*(w)=\sqrt{Q(w)/w^2}\,\xi(w)), where some branch of the square root is taken. It turns out that the second variation is a quadratic functional of (\xi(w)), i.e., of the first variation. Namely:

[
\delta^2 L(f)=\operatorname{Re}\left{
\frac{1}{2\pi i}\int_{\mathfrak T}
\bigl[\xi_+^(w)d\overline{\xi_+^(w)}
-\xi_-^(w)d\overline{\xi_-^(w)}\bigr]
-
\frac{1}{2}\sum_{k=1}^{n}\xi_k
\sum_{l=1}^{n-k}(k+l+2)A_{k+l}\xi_l
\right},
\tag{19}
]

where the integral over (\mathfrak T) is understood as the sum of integrals over the oriented edges. Duren and Schiffer, in the paper ((^6)), assuming that the tree (\mathfrak T) consists of one edge, established a formula close to our formula (19).

Institute of Applied Mathematics
Academy of Sciences of the USSR
Moscow

Received
9 XII 1969

References

1. T. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Moscow, 1952.
2. M. Schiffer, Am. J. Math., 65, 341 (1943).
3. O. Teichmöller, Preuss. Akad. Wiss. Sitzungsber., 363 (1938).
4. A. C. Schaeffer, D. C. Spencer, Am. Math. Colloq. Publ., No. 35 (1950).
5. P. P. Dzhenkins, Univalent Functions and Conformal Mappings, Moscow, 1962.
6. P. Duren, M. Schiffer, J. d’Analyse Math., 10, Jérusalem (1962/63).