Steiner symmetrization and the initial coefficients of univalent functions

Theorem 1. The functions f and $f^*$ defined above satisfy the inequality $$|a_1|^2 - \operatorname{Re} a_1 a_{-1} \geqslant |a_1^|^2 - \operatorname{Re} a_1^ a_{-1}^*.$$ (2) This inequality has the following interpretation in terms of capacity. Let a set E be symmetric with respect to the real axis and let $H \setminus E$ be a simply connected domain, where $H := \{w : \operatorname{Im} w > 0\}$ . We denote by g the function which takes the domain $H \setminus E$ conformally and univalently onto the half-plane H in such a way that $$\lim_{w \to \infty} [g(w) - w] = 0.$$ The limit $$\operatorname{hcap}(E\cap H) = \lim_{w\to\infty} w[g(w)-w]$$ is referred to as the half-plane capacity (from infinity) of the set $E \cap H$ ([2], p. 69). In view of the expansion of f, we conclude that g has the following form in a neighbourhood of the point at infinity: $$g(w) = w + \frac{a_1^2 - a_1 a_{-1}}{w} + \dots,$$ where $a_1$ and $a_{-1}$ are real numbers. Thus, $$hcap(E \cap H) = |a_1|^2 - \operatorname{Re} a_1 a_{-1}$$ and the inequality (2) can be written in the form $$hcap(E \cap H) \geqslant hcap(E^* \cap H).$$ Theorem 1 is supplemented by the following assertion.

Theorem 2. Let a function $\widetilde{f}(z) = \widetilde{a}_1 z + \widetilde{a}_0 + \widetilde{a}_{-1}/z + \dots$ map D conformally and univalently onto the exterior of a continuum $\widetilde{E} \subset E^*$ . Then $$|a_1^|^2 - \operatorname{Re} a_1^ a_{-1}^* \ge |\widetilde{a}_1|^2 - \operatorname{Re} \widetilde{a}_1 \widetilde{a}_{-1}.$$ (3) Proofs of Theorems 1 and 2 are given in the concluding part of the paper. To obtain the inequality (2), we need the symmetrization with respect to a circle [3], as described in § 2. The inequality (3) follows from a result of Schiffer which was established using Hadamard's formula for the variation of the Green function ([4], § 3). As applications of Theorems 1 and 2, we prove covering results for the well-known class $\Sigma$ of functions $f(z) = z + a_0 + a_{-1}/z + \ldots$ that are meromorphic and univalent in D [5].

Corollary 1. Let f be a function belonging to the class $\Sigma$ and let $w_0$ be an arbitrary point of the complement $E = \mathbb{C}_w \setminus f(D)$ . Then the inequality $$\frac{m_f^4(w_0, \varphi) + 16R_f^4(w_0)}{8m_f^2(w_0, \varphi)} \le 1 + \operatorname{Re} e^{-2i\varphi} a_{-1},$$ holds for any real number $\varphi$ , where $R_f(w_0) \geqslant 0$ stands for the radius of the largest disc centred at the point $w_0$ and belonging to the set E, and $m_f(w_0, \varphi)$ is the linear Lebesgue measure of the intersection of E with the line $\{w = w_0 + te^{i\varphi} : t \in \mathbb{R}\}$ . This inequality becomes equation for the functions $f(z) = w_0 + e^{i\varphi}\lambda^{-1}h^{-1}(\lambda h(e^{-i\varphi}z))$ with $h(\zeta) = \zeta + 1/\zeta$ and any $\lambda > 1$ . In particular, the following inequalities hold: $$\frac{1}{8}m_f^2\left(w_0, \frac{\arg a_{-1}}{2}\right) - 1 \leqslant |a_{-1}| \leqslant 1 - \frac{1}{8}m_f^2\left(w_0, \frac{\arg a_{-1} + \pi}{2}\right).$$ The right-hand inequality refines a well-known corollary to the area theorem: $|a_{-1}| \leq 1$ ([5], Ch. II, §4). Both inequalities supplement the classical bound $$m_f(w_0, \varphi) \leqslant 4 \ \forall \varphi,$$ which follows from (1). Namely, $$m_f\left(w_0, \frac{\arg a_{-1}}{2}\right) \leqslant \sqrt{8(1+|a_{-1}|)} \leqslant 4,$$ $$m_f\left(w_0, \frac{\arg a_{-1} + \pi}{2}\right) \leqslant \sqrt{8(1 - |a_{-1}|)}.$$ These inequalities become equalities when |a⁻1| = 1 and f(z) = z + w⁰ + e ²iϕ/z. It would be of interest to obtain sharp estimates for a fixed |a⁻1| 6= 1.

Corollary 2. Suppose that a function f(z) = z +a⁰ +a⁻1/z +. . . of class Σ satisfies the inequality $$\mu\left(\left(\mathbb{C}_w\setminus f(D)\right)\cap l(u)\right)\geqslant \alpha \qquad \forall u, \qquad \beta\leqslant u\leqslant \gamma.$$ for some α, β and γ. Then Re $$a_{-1} \le 1 - \frac{c^2}{2}(1 - k^2),$$ where the real constants c and k can be found from the condition $$c\int_{0}^{1} \sqrt{\frac{\zeta^{2} - k^{2}}{\zeta^{2} - 1}} d\zeta = \frac{\gamma - \beta}{2} - \frac{i\alpha}{2}, \ c > 0, \ 0 < k < 1.$$ This inequality becomes an equality for a function f of class Σ mapping D conformally and univalently onto the exterior of a rectangle with sides lying on the lines u = β, u = γ, γ − β < 4, and of an appropriate height α. Corollaries 1 and 2 are obtained by successively applying the inequalities (2) and (3). The list of assertions of this kind can readily be extended in the same way as the well-known applications of Steiner symmetrization to function theory ([3],[6]).

Lemma 1. If open sets $B_1$ and $B_2$ satisfy the conditions $\infty \in B_1$ and $\overline{B}_1 \subset B_2$ , then the inclusion relation $$S_v B_1 \subset \overline{\mathbb{C}}_w \setminus (\overline{\mathbb{C}}_w \setminus B_2)^ \qquad (R_v(\overline{\mathbb{C}}_w \setminus B_1) \supset (\overline{\mathbb{C}}_w \setminus B_2)^).$$ holds for all sufficiently large v > 0. The proof of Lemma 1 is clearly of a technical nature, and therefore we omit it. We only note the importance of the condition that $B_1$ is contained in a compact subset of $B_2$ . Then the closed set $\overline{\mathbb{C}}_w \setminus B_2$ is contained in $\overline{\mathbb{C}}_w \setminus B_1$ together with some neighbourhood U. Near the real axis, the rays passing through the point iv and intersecting the neighbourhood U tend to lines parallel to the imaginary axis as $v \to \infty$ . Here the 'logarithmic measure' in a neighbourhood of the circle |w - iv| = v tends to the Euclidean measure. Let $g_B(z, z_0)$ denote the Green function of the connected component of B which contains the point $z_0$ (with a pole at this point), where $g_B(z, z_0)$ is defined to be zero outside this connected component. Let $r(B, z_0)$ be the inner radius of the above component with respect to the point $z_0$ [3].

Lemma 2. If the connected components of the open set B have Green's functions and the points iv and $\infty$ belong to B, then $$\log[r(B,iv)r(B,\infty)] + 2g_B(iv,\infty) \leq \log[r(S_vB,iv)r(S_vB,\infty)] + 2g_{S_vB}(iv,\infty)$$ for any v > 0. Proof. This follows from [3], Theorem 1.7, Proposition 1.11 (see also [7], Theorem 1).