Lobachevskii Journal of Mathematics Vol. 9, 2001, 37 – 46 \begin{matrix}c \\ ◯ \end{matrix}Andrei V. Kazantsev

Andrei V. Kazantsev

ON A PROBLEM OF POLYA AND SZEGÖ

(submitted by F. G. Avhadiev)

Abstract. We give a new proof of a theorem, which is originally due to Gehring and Pommerenke on the triviality of the extrema set M_{f} of the inner mapping radius |f′(ζ)|(1−|ζ|^{2}) over the unit disk in the plane, where the Riemann mapping function f satisfies the well-known Nehari univalence criterion. Our main tool is the local bifurcation research of M_{f} for the level set parametrization f_{r}(ζ) = f(rζ), r > 0.

Introduction

G.Polya and G.Szegö ([17]: Bd.2,Abschn.4, Kap.2) were apparently the first, who turned to study the extrema of the inner mapping radius

R_{f}(ζ) = |f′(ζ)|(1 − |ζ|^{2})  (1)

of the domain f(E), where E = {ζ ∈ ℂ : |ζ| < 1}, and f is the Riemann mapping function (see also [9], [18], Ch.7 and, of course, [8]). This problem attracted the attention periodically because of the appearance of (1) in certain topics of the complex function theory (the Hilbert problem, see [5]) and mathematical physics (e.g., the plasma problem [4]; see bibliography in [7] for the other problems). Now it may be considered both as a part of research of the level sets and gradient zeros in PDE (see [11] and bibliography there, [3], [7]), and as a part of the Bloch function theory (see, e.g., [19]).

The most bright achievement in the above Polya-Szegö problem is the result contained in [6], which may be packed in the following

Theorem GP. Let f be a meromorphic function in E satisfying the Nehari univalence criterion [16]

(1 − |ζ|^{2})^{2}|{f, ζ}| ≤ 2, ζ ∈ E,  (2)

where {f, ζ} = (f′′(ζ)/f′(ζ))′ − (f′′(ζ)/f′(ζ))^{2}/2 is the Schwarzian derivative of f. Then the critical points set of (1), M_{f} ∪ P_{f}, where M_{f} = {a ∈ E : (∂R_{f}/∂ζ)(a) = 0}, P_{f} = {poles of f in E}, has k_{f} := #(M_{f} ∪ P_{f}) ≤ 1 elements, unless f(E) = strip. The constant ”2” at the right-hand side of (2) is sharp.

This theorem is essentially the part of [6]’s Theorem 2, and their proof demonstrates that (in two principal alternatives, f′′(0) = 0 and f(0) = ∞) the function (1) decreases along the radii of E. (The same is demonstrated in [20] for the holomorphic case, where the same idea is exploited with some ”injections” from [19]. Another method based on [15] was developed in [1].) It is already possible to guess the Theorem GP in Haegi’s M_{f}-classification [8], the invariant form of which is stated as follows (we try to reconstruct Haegi’s style here).

Theorem H. Necessary and sufficient condition for a ∈ M_{f} to have the following character is

|{f, a}| < 2/(1 − |a|^{2})^{2} for the local maximum,
|{f, a}| > 2/(1 − |a|^{2})^{2} for the saddle point,
|{f, a}| = 2/(1 − |a|^{2})^{2} for the parabolic point,

{f, a} = 0 for the umbilical point.

(Haegi have observed also the non-existence of the flattening M_{f}–points in view of the superharmonicity of log R_{f}.)

Theorem H classifies the critical points a ∈ M_{f} according to the sign of Gaussian curvature of R_{f},

K_{f}(a) = R_{f}^{2}(a)[4/(1 − |a|^{2})^{4} − |{f, a}|^{2}].  (3)

Another M_{f}–classification, where the geometry of (1) is controlled by the index

γ_{f}(a) = −(2πi)^{−1} ∫_{|ζ−a|=ρ} d log(∂R_{f}/∂ζ), a ∈ M_{f},

is more sensitive (and, in the natural sense, the best possible):

”K − γ”–classification([13]).The correspondence between sgnK_{f} and γ_{f} on M_{f} is γ({sgnK_{f} = ±1}) = sgnK_{f} and γ_{f}({sgnK_{f} = 0}) = {−1, 0, +1}, and these are the only possibilities. Moreover, this correspondence is Mobius invariant, i.e.

sgnK_{f∘T} (a) = sgnK_{f}(T (a)), γ_{f∘T} (a) = γ_{f}(T (a)), a ∈ M_{f},  (4)

for any Mobius transformation T : E ⟶ E.

In particular,

k_{f} = m^{+} + m^{0} + m^{−},  (5)

where m^{±} = #{a ∈ M_{f} : γ(a) = ±1}, m^{0} = #{a ∈ M_{f} : γ(a) = 0}. (If no confusion can arise, we will drop the subscripts of M_{f}, R_{f}, K_{f} and γ_{f}, or will change them in an obvious manner.)

In the case when f satisfies the strong Nehari’s inequality

(1 − |ζ|^{2})^{2}|{f, ζ}| < 2, ζ ∈ E,  (6)

and belongs to the little Bloch class B_{0} with lim_{|ζ|→1} R_{f}(ζ) = 0, the proof of Theorem GP is almost trivial and is obtained by using (3), ”K − γ” and the following proposition, which connects m^{±} with the Euler characteristic of E:

If f ∈ B_{0} and k_{f} < ∞, then ∑_{a∈M_{f}} γ(a) = m^{+} − m^{−} = 1  (7)

This adaptation of the well-known results (see, e.g.,[2] or [14]) to ”our” case has been done in [13]. So

The proof of Theorem GP: the trivial case. Due to (6) and ”K − γ” we have γ_{|M_{f}} = +1, i.e. m^{−} = m^{0} = 0. Then (5) and (7) lead to the required statement: k_{f} = m^{+} = m^{+} − m^{−} = 1. The condition k_{f} < ∞ is established by the use of the following

Stability Lemma([12]). Let a ∈ M_{f}. Then the relation K_{f}(a) ≠ 0 is stable under the perturbation f_{r}(ζ) = f(rζ) of f for r near 1: K_{f_{r}}(a_{r}) ≠ 0, where a_{r} is the unique (by the Implicit Function Theorem) element of M_{f_{r}} near a. Moreover, when r decreases, a_{r} drifts to ζ = 0, being the (local) maximum, if K_{f}(a) > 0, and a_{r} moves to ∂E, being the saddle, if K_{f}(a) < 0.

Our goal is to get the proof of Theorem GP, ”overbuilding” the particular case just considered. As the above Lemma indicates, the crucial role will be played by the local bifurcation research of the elements of M_{f_{r}} for f_{r}(ζ) = f(rζ), |ζ| < 1/r, r varies over (0, +∞). Such a research is contained in the Main Lemma in the next section and can have an independent interest. In passing we encounter a new phenomenon: the complete analysis of bifurcations, as the final result, is connected with the extraction of the zeros of g′(ζ), where g(ζ) = ζf′′(ζ)/f′(ζ). The proof of Theorem GP is distributed over the section 2 (holomorphic case in detail) and section 3 (meromorphic case and sharpness in sketch).

1. M_{f_{r}}–dynamics

M_{f} is imbed as a slice (for r = 1) in the foliation M = ⋃_{0<r<∞} M_{f_{r}} × {r}, where M_{f_{r}} ⊂ E_{r} is the critical points set of

R_{f_{r}}(ζ) = |f′_{r}(ζ)|(1 − |ζ|^{2}),  (8)

the inner mapping radius (1) for the function f_{r}(ζ) = f(rζ) with natural domain of definition E_{r} = {ζ ∈ ℂ : |ζ| < 1/r}, 0 < r < ∞ (the prime, ’, denotes the differentiation with respect to ζ). In other words, M is exactly the set of solutions of the equation

f′′_{r}(ζ)/f′_{r}(ζ) = 2ζ̄/(1 − |ζ|^{2}),  (9)

(ζ, r) ∈ E = ⋃_{0<r<∞} E_{r} × {r}, derived from ∂R_{f_{r}}/∂ζ = 0 in an elementary manner. (It was (9), to which we referred in [12] as to the Gahov equation; see also the equation (11) in [19].)

Our goal here is to describe the local structure of M, for which we are needed in the following notations. First of all, we will schrink the subscript ”f_{r}” till ”r”, so R_{f_{r}} = R_{r}, M_{f_{r}} = M_{r}, etc. Next, we introduce M_{r}(D) := M_{r} ∩ D = {a ∈ D : (a, r) ∈ M} and k_{r}(D) := #M_{r}(D) for D ⊆ E_{r}, thus M_{r} = M_{r}(E_{r}) with k_{r} = #M_{r}, and M_{1} = M_{f}. The functional γ_{r} : M_{r} → {−1, 0, +1} is defined to assign the index γ_{r}(a) to the point a ∈ M_{r}. Finally, the fundamental role in the following research belongs to the function g(ζ) = ζf′′(ζ)/f′(ζ); if we set g_{r}(ζ) = g(rζ), then g_{r}(ζ) = ζf′′_{r}(ζ)/f′_{r}(ζ). For the brevity we denote b_{0} = g_{r_{0}}(ζ_{0}), γ_{0} = γ_{r_{0}}(ζ_{0}), where (ζ_{0}, r_{0}) ∈ M.

Since our subject is concerned with the isolated points of the slice M_{r}, we are needed in the following statement about

Isolated γ_{0}-bifurcation (consequence of the homotopy properties from [14], p. 34). Let D ⊂ E be the closed Jordan domain and τ = [r_{0}, r_{1}]. Suppose that the graphs of the continuous functions ζ_{n} : τ → D with ζ_{n}(r_{0}) = ζ_{0}, n = \overline{0, l}, compose the set M ∩ (D × τ). Then ∑_{n=0}^{l} γ_{r}(ζ_{n}(r)) = γ_{0}, r ∈ τ.

(This result may be interpreted as the parametrical localization of (7).) Now we are ready to formulate our

Main Lemma. Suppose f is a meromorphic function in E, and let ζ_{0} be an isolated element of M_{r_{0}} with K_{r_{0}}(ζ_{0}) = 0. Let k denotes the multiplicity of ζ_{0} as the zero of the function g_{r_{0}}(ζ)−b_{0}. Then there is a polydisk D ×τ ×W ⊂ E×ℝ×ℝ centered at (ζ_{0}, r_{0}, 0), such that the set M∩(D×τ) consists of k(= 2, if ζ_{0} = 0) analytic curves (ζ_{n}(v), r_{n}(v)), v ∈ W, n = \overline{0, k − 1}, intersecting at the point (ζ_{0}, r_{0}) only. Moreover, for every r ∈ τ \ {r_{0}} the inequality γ_{r} ≠ 0 holds on M_{r}(D), and k_{r}(D) = k, or k_{r}(D) = k + (−1)^{m}sgn(r − r_{0}), where m = 1, or m = 2.

In particular case k = 1(ζ_{0} ≠ 0) we have two following possibilities: k_{r}(D) = 1, r ∈ τ, when |γ_{0}| = 1, and the ”birth” or ”annihilation” of one maximum and one saddle at r = r_{0}, when γ_{0} = 0. In the case ζ_{0} = 0(k = 2) we have 0 ∈ M_{r} for all r ∈ ℝ_{+} with γ_{r}(0) = sgn(r_{0} − r), r ∈ ℝ_{+} \ {r_{0}}, and k_{r}(D) = 2 + γ_{0} · sgn(r − r_{0}), r ∈ τ \ {r_{0}}.

For the sake of the brevity we reproduce neither general dependence of k_{r} on γ_{0} nor realizations of corresponding possibilities; the most of them will be excluded in our uniqueness result. We remark that the computation of k_{r} may be carried out without the use of Isolated γ_{0}-bifurcation during the study of deformation of M ∩ (D × τ), when arg b_{k} varies at the expansion g_{r_{0}}(ζ) = b_{0} + b_{k}(ζ − ζ_{0})^{k} + . . .. Stability Lemma plays more essential role in the approach just indicated than at the appropriate steps in the following

Proof of Main Lemma. Since the situation is local, it would be sufficient to confine oneself to the holomorphic f near ζ_{0}.

The desired structure of M near (ζ_{0}, r_{0}) is provided by the local uniformization z = α(ξ), w = b_{0} + ξ^{k} of the map w = g(z), where α(ξ) = r_{0}ζ_{0} + α_{1}ξ + …, α_{1} ≠ 0, is holomorphic in the disk V centered at ξ = 0, and by k parametrizations ξ = e^{iφ_{n}}v, φ_{n} = πn/k, n = \overline{0, k − 1}, v ∈ W = ReV, of the inclusion b_{0} + ξ^{k} ∈ ℝ_{+} imposed by the following form of (9),

g_{r}(ζ) = 2|ζ|^{2}/(1 − |ζ|^{2}).  (10)

The latter equation, therefore, falls into k systems

\{ \begin{matrix}rζ & = & α(e^{iφ_{n}}v) \\ 2ρ^{2}/(1 − ρ^{2}) & = & b_{0} + (−1)^{n}v^{k} (ζ = ρe^{iθ}), \end{matrix}  (11)

v ∈ W, n = \overline{0, k − 1}, whence (for the case ζ_{0} ≠ 0)

ζ = ζ_{n}(v) = α(e^{iφ_{n}}v)/r_{n}(v),  (12)
r = r_{n}(v) = |α(e^{iφ_{n}}v)|/ρ_{n}(v),  (13)

where ρ_{n}(v) is the (unique) solution of the second equation of (11).

The construction of the polydisk D×τ ×W accompanies the above scheme in the natural manner. D is chosen as a disk centered at ζ_{0} with D ∩ M_{r_{0}} = {ζ_{0}}, V is adjusted so that α(V)/τ = {α(v)/r : v ∈ V, r ∈ τ} ⊂ D, and the elimination of superfluous solutions from the set M ∩ (D × τ) is achieved by the shortening of τ. The underlying technique is supplied by the Weierstrass Preparation Theorem and the Implicit Function Theorem.

We note that the slight generalization of Stability Lemma yields us immediately

γ_{r_{n}(v)}(ζ_{n}(v)) ≠ 0, v ∈ W \ \{0\}, n = \overline{0, k − 1}.  (14)

Now, we must convert (13) and substitute it into (12). For the best survey we consider the following cases.

Case 1: ζ_{0} ≠ 0, k ≥ 2.

The inconvenience appeared in the substitution (13) into (12) is that r′_{n}(v) may change its sign at v = 0 for some n (otherwise, obviously, k_{r} = k, r ∈ τ \ {r_{0}}). However, such a n is at most unique, because the differentiation of log(12) with respect to v implies

α′(0)e^{iφ_{n}}/ζ_{0} = r′_{n}(0) + ir_{0}θ′_{n}(0),  (15)

so r′_{n}(0) = c · cos(β + φ_{n}), c ≠ 0, and r′_{n}(0) = 0 for no more than one of n’s. (Here we are served by the branch of log with Imlog α(0) = θ_{0}, where ζ_{0} = ρ_{0}e^{iθ_{0}}.)

If exactly one of r_{n}’s, say r_{0}(v), is inconvenient in the above sense, then we are need to convert both of its branches over two halves of W. As a result, besides k − 1 different solutions ζ_{n}(v_{n}(r)), n = \overline{1, k − 1}, for r ∈ τ \ \{r_{0}\}, we have two solutions ζ_{0}(v_{0}^{+}(r)) and ζ_{0}(v_{0}^{−}(r)) ”over” their own halves of τ, which are differ from each other (and from the previous k − 1), since r′_{n}(0) and θ′_{n}(0) can’t vanish simultaneously in view of (15). Thus, k_{r} = k + (−1)^{m}sgn(r − r_{0}), where m = 1, or m = 2.

Case 2: ζ_{0} ≠ 0, k = 1

By virtue of (14) and Isolated γ_{0}-bifurcation we have

\{ \begin{matrix}m_{r} + s_{r} & = & k_{r} \\ m_{r} − s_{r} & = & γ_{0}, \end{matrix}

r ∈ τ, where k_{r} = 1(+(−1)^{m}sgn(r − r_{0})) is obtained as in the Case 1. If k_{r} = 1, then γ_{0} ≠ 0 (otherwise 2m_{r} = 1, the contradiction), i.e. |γ_{0}| = 1. If k_{r} has the jump= 2 at r = r_{0}, then m_{r} = s_{r} = 0 for those r, where k_{r} = 0, and so γ_{0} = 0. Thus we have

\{ \begin{matrix}m_{r} + s_{r} & = & 2 \\ m_{r} − s_{r} & = & 0 \end{matrix}

on the other half of τ, hence m_{r} = s_{r} = 1 there.

Case 3: ζ_{0} = 0.

It is obvious that 0 ∈ M_{r} for all r ∈ ℝ_{+}. We have g(ζ) = {f, 0}ζ^{2} + …, where {f, 0} ≠ 0 due to K_{f_{r_{0}}}(0) = 0 and (3). (It turns out, in passing, that r_{0} = \sqrt{2/|\{f, 0\}|} and that γ_{r}(0) = sgn(r_{0} − r), r ∈ ℝ_{+} \ \{r_{0}\}.) Thus, k = 2. Hence, the equation (10) now falls into two systems

\{ \begin{matrix}rζ & = & α(v) \\ 2ρ^{2}/(1 − ρ^{2}) & = & v^{2} \end{matrix} and \{ \begin{matrix}rζ & = & α(iv) \\ 2ρ^{2}/(1 − ρ^{2}) & = & −v^{2}. \end{matrix}  (16)

The second system (16) makes sense only for ζ = ρ = v = 0. We consider it as an excess, corresponding to the ”excessive” multiplication (9) by ζ, which leads to (10). A priori solution ζ = 0 of (9) (for any r) is taken into account by the first system (16).

In order to solve this system we permit ρ to take the negative values. The status of ρ such as ”oriented polar radius” equalizes (by the change ρ to −ρ) both factors at the left-hand side of the second equation

(v − ρ\sqrt{2}/\sqrt{1 − ρ^{2}})(v + ρ\sqrt{2}/\sqrt{1 − ρ^{2}}) = 0

of the first system of (16), implying the same result:

r = r(ρ) = |α(ρ\sqrt{2}/\sqrt{1 − ρ^{2}})/ρ|,
ζ = ζ(ρ) = α(ρ\sqrt{2}/\sqrt{1 − ρ^{2}})/r(ρ).

The computation of k_{r} is similar to the Case 2. This completes the proof of Main Lemma.

It should be noted that Main Lemma is the generic prototype for its generalization proved in [12], where the treatment of Cases 2 and 3 is quite different from the above. The underlying result for the Case 2 to treat along the lines of [12] is the following statement, in which we shall need in the sequel.

Jacobians Lemma[12]. If ζ_{0} = ρ_{0}e^{iθ_{0}} ≠ 0, then the following conditions are equivalent:

i) all of the three real Jacobians of the equation (9) vanish at (ρ_{0}, θ_{0}, r_{0}) simultaneously;

ii) (1 − |ζ_{0}|^{2})^{2}{f_{r_{0}}, ζ_{0}} = −2ζ̄_{0}/ζ_{0};

iii) g′_{r_{0}}(ζ_{0}) = 0.

In conclusion of this Section we propose some remarks concerning the nonisolated critical points of the inner mapping radius.

As Theorem GP states, the only non-isolated case in the Nehari class corresponds to the strip with the canonical form of the mapping function,

f_{s}(ζ) = \frac{1}{2} log \frac{1 + ζ}{1 − ζ} .  (17)

The latter supplies an example of the ”degenerated” bifurcation r(ρ) ≡ r_{0} for the Case 3. In fact, the equation (9) for f_{s,r} is r^{2}ζ/(1−r^{2}ζ^{2}) = ζ̄/(1−|ζ|^{2}) and has the following solutions: the only (elliptic) root ζ = 0 in E with γ(0) = +1 for r < 1 (global maximum), the diameter of (parabolic) roots E ∩ ℝ for r = 1 (the ”groove”), and if r > 1, then (8) has the only (hyperbolic) root r = 0 in E_{r} with γ(0) = −1 (the saddle ζ = 0 ”hanges” between two infinite R_{r}-maxima ζ = ±1/r lying on ∂E_{r}).

By the use of (17) the example of ”degenerated” bifurcation may be constructed also for the Case 2. Indeed, let us consider the function f = f_{s} ∘ \frac{ζ−iα}{1+iαζ} , α > 0. As is easily seen, M_{f} = \{\frac{ζ+iα}{1−iαζ} ∘ M_{f_{s}}\} ∩ E and g(ζ) = 2α^{2}/(1 − α^{2}) + 2(1 + α^{2})(1 − α^{2})^{−3}(ζ − iα)^{2} + . . .. Thus, g′_{|M_{f}} = 0 (only) at ζ = iα, and k = 2. The bifurcation picture is similar to that of f_{s}; the details we refer to the reader.

It follows from [19] that no more than one of k analytic curves of M∩(D×τ) can generate the critical points continuum for r = r_{0}. It is interesting to ask about the inverse: does there exists a bifurcation point on such a continuum? As an answer we offer the following

Proposition. Let γ be the non-isolated component of M_{f} for g ≢ const, and a ∈ γ. If a = 0, then a is always the bifurcation point. If a ≠ 0, then the following conditions are equivalent:

(a) a is the bifurcation point for f_{r} at r = 1;

(b) |γ| is stationary at a, i.e. (d/dt)|γ(t)|_{|t=0} = 0, where γ = γ(t) is local representation of γ near γ(0) = a;

(c) g′(a) = 0.

The proof is simple and is based on Jacobians Lemma.

It should be pointed out that this Proposition may be considered as a dual in some sense to Stability Lemma.

As a final remark we note that we can construct the examples of ”nonisolated” bifurcations with prescribed k ≥ 3 in a standard manner, going back to [19].

2. Theorem GP: holomorphic case

First of all, we should remark that our conclusion in the form of inequality k_{f} ≤ 1 is contensive. In fact, the possibility k_{f} = 0 is realized, e.g., by the function f(ζ) = (1+ζ)/(1−ζ), satisfying (2). Now, suppose that M_{f} ≠ ∅ and, of course, f ∈ Nehari (the figurative synonim of (2)). We divide the proof into two cases.

1) M_{f} have no limit points.

a) As is easily checked, if f satisfies (2), then f_{r} satisfies (6) for any r ∈ (0, 1). By Trivial Case of Theorem GP it follows immediately that k_{r} := k_{f_{r}} = 1.

b) Here we show that γ_{f}(M_{f}) = {+1}.

Indeed, if K_{f}(a) ≠ 0, then γ_{f}(a) = +1 owing to ”K − γ”-classification. If K_{f}(a) = 0, then, according to (4), K_{h}(0) = 0 for h(ζ) = f ∘ \frac{ζ+a}{1+āζ}, and also 0 ∈ M_{h} and h ∈ Nehari. Thus, r = 1 is the bifurcation parameter value of the equation (9) for h_{r}. By Main Lemma the number b_{r} of the solutions branching from zero as r < 1 is equal to b_{r} = 1 + γ_{h}(0)sgn(r − 1) = 0, if γ_{h}(0) = +1, or 1, if γ_{h}(0) = 0, or 2, if γ_{h}(0) = −1. We claim that two latter alternatives fail. In fact, if it should not be the case, then for r < 1 near 1 we should have k_{h_{r}} ≥ 1 (contribution of zero point) +b_{r} = 2 − γ_{h}(0) = 2 or 3, the contradiction with a). Thus, γ_{h}(0) = +1, and then, by (4), γ_{f}(a) = +1.

c) Let a and (without loss of generality) 0 are elements of M_{f}, and a ≠ 0. In order to establish the desired conclusion k_{f} = 1 we must bring the following alternate versions to contradiction with a).

If K_{f}(a) ≠ 0, or K_{f}(a) = 0 and g′(a) ≠ 0, then we have a_{r} ∈ M_{f_{r}} near a for r < 1 near 1. In the first case it is provided by the Implicit Function Theorem, in the second — by the Case 2 of Main Lemma and b). As a result we obtain k_{f_{r}} ≥ 2, since 0 ∈ M_{f_{r}}.

If g′(a) = 0, then, by Main Lemma, we have k_{r}(D) ≥ k − 1 ≥ 1 for r < 1 near 1 and for some neighborhood D ⊂ E of the point a. As in the above, the contradiction is attained at the expence of 0 ∈ M_{f}: k_{r} ≥ 1 + k_{r}(D) ≥ 2.

The case 1) is complete.

2)M_{f} contains a limit point.

According to [19] in this case there exists an analytic arc ζ = ζ(t), t ∈ T ⊂ ℝ, passing through the origin (without loss of generality) and such that ζ(t) ∈ M_{f}:

(f′′/f′)(ζ(t)) = 2ζ̄(t)/(1 − |ζ(t)|^{2}), t ∈ T,

whence

{f, ζ(t)}ζ′(t) = 2ζ̄′(t)/(1 − |ζ(t)|^{2})^{2}, t ∈ T,  (18)

i.e. ρ_{f}(ζ(t)) := (1 − |ζ(t)|^{2})^{2}|{f, ζ(t)}| ≡ 2, t ∈ T. Thus, the function ρ_{f}(ζ), which is not greater than 2 in E, attains its maximum at ζ(t), t ∈ T. This means that for every t ∈ T the point ζ(t) is the root of

{f, ζ}′/{f, ζ} = 4ζ̄/(1 − |ζ|^{2})  (19)

(the analogue of (9) for ρ_{f}(ζ)).

Functions ζ(t) and ζ̄(t) are analytic in T, hence they are extended to certain strip 𝔗 ⊂ ℂ containing T. The extended functions u(τ) and v(τ), respectively, are holomorphic in 𝔗 with u(t) = ζ(t), v(t) = ζ̄(t), t ∈ T. Without loss of generality we assume

v(0) = 0  (20)

(so u(0) = 0). Furthermore, in view of the analyticity we have u′(0) = ζ′(0) ≠ 0. This allows us to correctly define the function v(u) := v(τ(u)) in some neighborhood of u = 0. Then we can also refer to this neighborhood the equations (18) and (19), which have been extended to 𝔗 and transformed by means of τ = τ(u) beforehand:

{f, u} = 2v′(u)/(1 − uv(u))^{2},  (18a)
{f, u}′/{f, u} = 4v(u)/(1 − uv(u)).  (19a)

In particular, it follows from (18a) that |v′(0)| = \frac{1}{2}(1 − uv(u))^{2}|\{f, u\}|_{|u=0} = 1. Hence we can achieve

v′(0) = 1.  (21)

Equating the logarithmic derivative of (18a) to (19a), we obtain v′′(u)/v′(u) = 2(v(u) − uv′(u))/(1 − uv(u)), whence

v′′(0) = 0  (22)

and {v, u} = 0. Solving this equation with the initial data (20)–(22), we have, as is well-known, v(u) = u, which we must substitute into (18a) to obtain f = f_{s}.

ON A PROBLEM OF POLYA AND SZEGÖ 45

The case 2), and, therefore, the proof of holomorphic part of Theorem GP, are complete.

3. Concluding remarks: meromorphic case and sharpness problem

The proof of the Theorem GP for the meromorphic functions is essentially the copy of that for holomorphic case. The only supplement is the relation γ_{f}(P_{f}) = {+1}, which is established in an elementary manner by the extraction of the principal part of corresponding vector field near the pole of f. Thus the latter falls under the ”jurisdiction” of the proposition (7). (The methodology under consideration doesn’t leave the contents of Chapter 1 of [14].)

Sharpness of the constant ”2” in (2) means that for every α > 2 there exists a holo- (or mero-)morphism f_{α} : E → ℂ with sup_{ζ∈E} ρ_{f_{α}}(ζ) = α and k_{f_{α}} > 1. The simplest holomorphic example is furnished by the family f_{r}(ζ) = f(rζ), 0 < r < 1, where f(ζ) = ζ/(1 − ζ^{2}) (the bifurcation value of the parameter here is r = 1/\sqrt{3}). The sharpness for the meromorphic functions may be provided by the ”meromorphization”, f_{γ} = (ω_{γ} + 1)/(ω_{γ} − 1), of the well-known Hille family [10], ω_{γ}(ζ) = (1 + ζ)^{iγ}(1 − ζ)^{−iγ}, γ > 0. In addition we meet an interesting phenomenon: the level set parametrization, f_{r}, of meromorphic function f in E with #P_{f} = 1 and sup_{ζ∈E} ρ_{f}(ζ) > 2 doesn’t admit the simultaneous leaving the Nehari class and the region {f : k_{f} ≤ 1}, when r increases. Unfortunately, the investigation of the nature of this phenomenon falls outside the limits of this paper.

References

[1] F.G. Avhadiev, Conformal mappings and boundary value problems, (in Russian), Kazan, 1996.

[2] I.Ya. Bakelman, A.L. Verner and B.E. Kantor, Introduction to Differential Geometry ”in large”, (in Russian), Moscow, Nauka, 1973.

[3] L.A. Caffarelli and A. Friedman, Convexity of solutions of semilinear elliptic equations, Duke Math.J., V. 52, No. 2 (1985), 431-456.

[4] A. Friedman, Variation principles and free-boundary problems, John Wiley and Sons, New York, 1982.

[5] F.D. Gahov, Boundary value problems, (in Russian), Moscow, Nauka, 1977.

[6] F.W. Gehring and Ch. Pommerenke, On the Nehari univalence criterion and quasicircles, Comment. Math. Helv., 59 (1984), 226-242.

[7] B. Gustafsson, On the convexity of a solution of Liouville’s equation, Duke Math. J., V.60, No. 2 (1990), 303-311.

[8] H.R. Haegi, Extremalprobleme und Ungleichungen konformer Gebietsgrossen, Compos. Math., V.8, Fasc.2 (1950), 81-111.

[9] H.R. Haegi, Sur le maximum du rayon interieur, C. R. Acad. Sci., 228 (1949), 891-892.

[10] E. Hille, Remarks on a paper by Zeev Nehari, Bull. Amer. Math. Soc., V. 55, No. 6 (1949), 552-553.

[11] B. Kawohl, Rearrangements and convexity of level sets in PDE, Lecture Notes in Math., 1150, Springer-Verlag, Berlin Heidelberg, 1985.

[12] A.V. Kazantsev, Bifurcations of Gahov equation with Lowner left side, (in Russian), Izvestija VUZ. Matematika, No. 6 (1993), 69-73.

[13] M.I. Kinder, On the number of solutions for Gahov equation in the case of multiply connected domain, (in Russian), Izvestija VUZ. Matematika, No. 8 (1984), 69-72.

46 A. V. KAZANTSEV

[14] M.A. Krasnoselsky, A.I. Perov, A.I. Povolotsky and P.P. Zabreiko, Vector fields on the plane, (in Russian), Moscow, Gosizdat, 1963.

[15] O. Martio and J. Sarvas, Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn., Ser. AI Math., V. 4 (1978/1979), 383-401.

[16] Z. Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc., V. 55, No. 6 (1949), 545-551.

[17] G. Polya and G. Szegö, Aufgaben und Lehrsatze aus der Analysis, Bd. 2, Berlin, Springer, 1925.

[18] G. Polya and G. Szegö, Isoperimetric inequalities in mathematical physics, Princeton Univ. Press, 1951.

[19] St. Rusheweih and K.-J. Wirths, On extreme Bloch functions with prescribed critical points, Math. Z., Bd. 180 (1982), 91-106.

[20] S. Yamashita, The Schwarzian derivative and local maxima of the Bloch derivative, Math. Japonica, V. 37, No. 6 (1992), 1117-1128.

Department of Computational Mathematics and Cybernetics, Kazan State University, Kremlevskaya 18, 420008, Kazan, Russia

E-mail address: Andrei.Kazantsev@ksu.ru

Received 17 June 2001