Extremal Functions in Geometric Function Theory. Higher Transcendental Functions. Inequalities

CHAPTER

15

Extremal Functions in Geometric Function Theory. Higher Transcendental Functions. Inequalities

R. Ktihnau Fachbereich Mathematik, Martin-Luther- Universitiit Halle- Wittenberg, D-06099 Halle an der Saale, Germany E-mail: kuehnau @mathematik, uni-halle, de

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Some examples of special functions in Geometric Function Theory . . . . . . . . . . . . . . . . . . . . 3. A curiosity: Deriving real inequalities from results in Geometric Function Theory . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

HANDBOOK OF COMPLEX ANALYSIS: GEOMETRIC FUNCTION THEORY, VOLUME 2 Edited by R. Ktihnau 9 2005 Elsevier B.V. All rights reserved 661

663 664 665 667

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1. Introduction

If we have an extremal problem in a class of schlicht conformal mappings w = w(z) of the unit disk Izl < 1 or of Izl > 1 (classes S and Z), then of special interest are those cases in which the extremal mappings are slit mappings, where the slits are prescribed by a quadratic differential Q(w) dw 2 in form of

Q(w) dw 2 >~O.

(1.1)

Here Q(w) is a rational function, where the poles and the order of the poles are in a simple way connected with the problem. This quadratic differential in these extremal problems of Gr6tzsch-Teichmtiller type appear also with the general variational method of Schiffer. Because the transformed quadratic differential Q(z)dz 2 in the z-plane satisfies at the unit circle the condition (Z) d z 2 >/O,

it can be constructed with a rational function Q (z). So we arrive at an ordinary differential equation for the extremal functions w(z) with the solution

f lld

v/Q(w)dw

~

f Z ,/o(zdz

(1.2)

(cf., e.g., [ 12]). This yields an implicit representation of w(z). The appearance of accessory parameters is of course a great difficulty. And the possibility to get in the following a final result in form of a concrete (sharp) inequality for the corresponding functional depends essentially on the character of the integrals in (1.2). Of course the right-hand side integral in (1.2) is more complicated because the integrand has generally more singularities than the integrand on the left-hand side. We have 3 possibilities for these integrals: (i) the integral represents an elementary function; (ii) the integral represents a "higher transcendental function"; (iii) for the integral does not exist a representation with these functions. Fortunately we have for the most interesting extremal problems generally the case (i). But there are many cases in which higher transcendental functions occur. An additional interesting thing is in many extremal problems the phenomenon that we obtain in the final (sharp) inequality different analytic expressions for different parameters. This is very surprising for the first moment. But the reason is simply the appearance of zeroes of the quadratic differential Q(w)dw 2 on the boundary slit as the image. Historically the first nice example was probably the famous Golusin "Drehungssatz" [3, p. 110], [5, Theorem 3.4]. If we replace the given domain Izl < 1 or Izl > 1 in the z-plane by an annulus, then the construction of Q(z) in (1.2) generally needs elliptic functions, and the things become much more complicated.

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There are many other situations in Geometric Function Theory in which special functions occur. In the case of classes of quasiconformal mappings the situation is a little bit another, although we have again a description of extremal mappings with quadratic differentials. Then a usual derivative of the mappings does not exist. But if we consider, e.g., the class S(Q) of those conformal mappings w(z) of the class S which permit a continuous Q-quasiconformal extension for Izl < 1, then in the "conformal part" of the mappings also derivatives (and higher coefficients) are possible. For extremal functions in these mapping classes, which are also linked with quadratic differentials, we have again a general representation, but a much more complicated one than in (1.2); cf. [10, Part II, Chapter III]. Only in a few cases elementary functions are sufficient for the representation of the extremal functions. In Section 2 we refer in examples about the appearance of higher transcendental functions in Geometric Function Theory, not only in connection with the above mentioned theory of extremal functions. A collection of such higher transcendental functions is listed in [30], with many references. We mention also some problems of another type in which higher transcendental functions occur. The calculation of the conformal module of some special ring domains needs often elliptic integrals and elliptic functions [2]. And there are of course many applications of higher transcendental functions in connection with the classical Schwarz-Christoffel integral for the conformal mapping of polygonal domains or circular polygons (cf., e.g., [9,12,20]; of special interest is the doubly-connected case which usually needs elliptic functions), or in connection with the conformal mapping of triangulated Riemannian manifolds, defined as a set of triangles with a corresponding set of linear sewings (cf. [13], where, e.g., hypergeometric functions occur). We will leave this aside here. Our collection in Section 2 is, of course, not exhaustive. Generally speaking, it is remarkable that elliptic integrals are the most frequently appearing higher transcendental functions in Geometric Function Theory. On the other side, it is striking that several higher transcendental functions (e.g., Bessel functions) practically did not appear until now.

2. Some examples of special functions in Geometric Function Theory

Elliptic integrals and elliptic functions. While the range of the values f (z), for a fixed z, in the class S can be prescribed with elementary functions [3, Chapter IV, Section 1], this range of values in the class Z: needs the complete elliptic integrals/C, C of the first and of the second kind; cf. (3.1). Further examples with/C, C and other elliptic integrals: [8, p. 424], [12, footnote 14], [21-23], [10, pp. 141-144], several papers of Kuz'mina (e.g., [26]) and of Solynin, e.g., [31,32]; in [24,25] in combination with the Jacobi functions sn, cn, etc. Of a special significance in Geometric Function Theory is the so-called Grrtzsch ring. This is the doubly connected domain between the unit circle and the segment 0 . . . . . r (0 < r < 1). The corresponding conformal module is given by zv/C(~/1 - r 2 ) # ( r ) = ~/C(r) "

(2.1)

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And therefore this yields an extremely important application of elliptic integrals. Here we have an exhaustive collection of properties, inequalities, applications, etc. in the books [2,29] and in the article [30]. Closely related to the Gr6tzsch ring are, e.g., the Teichmiiller ring and the Mori ring [2,29]. Here we mention also the application in the Teichmiiller Verschiebungssatz [34] (there is a simple error in the calculation on p. 343, lower half: e.g., in the formula before the footnote the factor 2 should be deleted). The elliptic modular function, elliptic integrals of the first kind and the Weierstraj3 G-function occur already in [4] to prescribe the range of the values of the cross-ratio of the images of 4 fixed points in classes of conformal mappings; later many papers on special cases and limit cases without knowledge of [4]. In [ 11 ] the concrete formulas (additionally with hyperelliptic integrals) in the case of the mappings of the class E; in [12, p. 107] for the mappings of an annulus. Also related is the analogous problem for quasiconformal mappings [33] (cf. also [1,15]). An application was given in [17]: Calculation of those Jordan curves through 4 given fixed points, for which the reflection coefficient has its smallest value. Hypergeometric functions (and more general functions) appear in [14] in connection with extremal problems for functions of the class E with a quasiconformal extension. But here the dilatation bound is not a constant but depends on Izl for Izl < 1. Euler's F-function and the Psi-function 7r = F t / F appear, e.g.,in [10, p. 129], [27] and in several other papers of Kuz' mina and Solynin. Orthogonal polynomials, hypergeometric series, Euler's beta integral and the gamma function appear also in the first proof of de Branges for the Bieberbach conjecture [6, p. 604]. Lobachevski's function, Euler's dilogarithm and Clausen's integral appear in [21 ]. Solutions of a Riccati differential equation occur in [19].

3. A curiosity: Deriving real inequalities from results in Geometric Function Theory Here we will sketch how it is possible to obtain inequalities for higher transcendental functions from inequalities in Geometric Function Theory. The idea is quite simple. If we have an inequality for a functional in a class of (conformal or quasiconformal) mappings, then we obtain an inequality for the involved, e.g., special higher transcendental functions, if we insert a special, not too complicated mapping. We can add an inequality in the other direction if we have an arbitrary nonsharp estimate of our functional. We will illustrate this with an example. In [3, Chapter IV, Section 3] the following sharp inequality for the mappings F ( ( ) of the class Z was given: F(() C

1 C(1/1(I) C(1/1(I) + 1 - ~ - 2 ~ ~<2-2~. I(I 2 /C(1/ICI) /C(1/1(I)

(3.1)

Here/C, s denote the complete elliptic integrals of the first kind and of the second kind. (For the history of (3.1), which goes back to GrOtzsch, cf. [16]; other proofs of (3.1)

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in [7, p. 97], [28, p. 251].) The inequality (3.1) shows that the possible values F ( ( ) for fixed ( = (* fill a closed disk. All boundary points are attained for exactly one mapping 9 The extremal mapping F ((), which corresponds to the point of this disk with the smallest real value, transforms I(I = 1 onto a parabolic slit symmetric to the real axis and with the focal point F ( ( * ) . By using the mapping F(()-

( -

-,

(3.2)

(

as a (very rough) approximation of this slit mapping, we obtain for ( - (* = 1 > 1 from (3.1) (used only with the real part) the rough inequality s

k2

/C(k)

~< 1

2 '

0 < k < 1.

(3.3)

We can expect that we will get a much more better estimate if we use a "better" mapping F ((), that means a slit mapping, for which the slit is a better approximation of the parabolic slit. For this reason we use now a circular slit mapping: (-(o +(0 F ( ( ) -- ( ( ~0

1 (0 ~ 2?

(3.4)

with a "suitable" (0 > 1 For (* -- 1+(~ that means by the choice (0 - 2(* - 1 we get a circular slit with the same curvature as our parabolic slit at the vertex, in the limit case (* -~ o0. This leaves us, by using of this "better" mapping (3.4), for ( - (* - ~ after (3.1) with 9

F((*) (,

-b 3 -

~

2

'

1 16( . 4 - 12('2 + 1 g(k) ~ = ~>4~ (,2 (4( ,2 - 1)( ,2 /C(k) '

therefore C(k)

1 16 - 12k 2 + k 4

/C(k) ~< 4

4 - k2

for 0 < k < 1.

(3.5)

Here the difference between both sides, e.g., for k = 0.5 is smaller than 4 . 1 0 -4, for k = 0.1 smaller than 2 . 1 0 -8. In a retrospect it is not immediately to answer the question: What did we use altogether in proving (3.5)? To obtain an estimate in the other direction we use the simple inequality

I F ( f ) - r ~< Ir3

for F ( ( ) 6 2?

(3.6)

This follows with the maximum principle by using the function ( [ F ( ( ) - (] (analytically also at ( = ~x~) and IF(()l ~< 2 for I l l - 1.

Extremal functions in Geometric Function Theory

667

If we use in (3.6) the above-mentioned parabolic slit mapping F ( ( ) with

1

s

F ( ( * ) - (* = - 4 ( * + ~-2 + 4 ( * ~

1

*= -

we obtain the desired rough estimate in the other direction: g(k) /C(k) ~> 1 - k2

for 0 < k < 1.

(3.7)

We add the general remark, that it is possible to prove inequalities for expressions in higher transcendental functions (e.g., also (3.5)), to get simpler expressions in elemetary functions, with a simple algorithm [18] by using a computer. (This yields a strong mathematical proof, not only for a finite number of parameters.) This sometimes can be useful if the sharp estimate for a functional in a class of mappings is too complicated. Further we add as a curiosity the remark that it is possible to use inequalities in Geometric Function Theory, in which higher transcendental functions are involved, also in the other direction, namely for a variational characterization of these functions. We explain this again with (3.1). This inequality can be read also as follows: The values of the function 4g(k)/1C(k) can be characterized for all fixed k (0 < k < 1) as the infimum of all values 3 - k 2 + kg~e. F ( 1 / k )

(3.8)

in the class of all mappings F 6 27. Finally we add the remark that often distinct expressions in higher transcendental functions appear if we solve an extremal problem in Geometric Function Theory with different methods. This means that we have proved an identity in higher transcendental functions, if both inequalities are sharp; cf. as an example (3.1) with Corollary 6.11 in [7].

References [ 1] L.V. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand, Princeton, NJ (1966). [2] G.D. Anderson, M.K. Vamanamurthy and M.K. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Mappings, Wiley, New York (1997). [3] G.M. Golusin, Geometrische Funktionentheorie, VEB Deutscher Verlag Wiss., Berlin (1957); 2nd edn of the Russian original: Nauka, Moskva (1966); English transl.: Amer. Math. Soc., Providence, RI (1969). [4] H. Gr6tzsch, Die Werte des Doppelverhiiltnisses bei schlichter konformer Abbildung, Sitz. Preul3. Akad. Wiss. Phys.-Math. K1. (1933), 501-515. [5] W.K. Hayman, Univalent and multivalent functions, Handbook of Complex Analysis: Geometric Function Theory, Vol. 1, Elsevier, Amsterdam (2002), 1-36. [6] P. Henrici, Applied and Computational Complex Analysis, Vol. 3, Wiley, New York (1986). [7] J.A. Jenkins, Univalent Functions and Conformal Mapping, Springer-Verlag, Berlin-G6ttingen-Heidelberg (1958). [8] J.A. Jenkins, The method of the extremal metric, Handbook of Complex Analysis: Geometric Function Theory, Vol. 1, Elsevier, Amsterdam (2002), 393-456.

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[9] W. von Koppenfels and E Stallmann, Praxis der konformen Abbildung, Springer-Verlag, Berlin-G6ttingenHeidelberg (1959). [ 10] S.L. Kruschkal and R. Ktihnau, Quasikonforme Abbildungen - neue Methoden und Anwendungen, Teubner, Leipzig (1983); in Russian: Nauka Sibirsk. Otd., Novosibirsk (1984). [ 11] R. Kiihnau, Berechnung einer Extremalfunktion der konformen Abbildung, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Nat. Reihe 9 (1960), 285-287. [12] R. Ktihnau, Ober die analytische Darstellung von Abbildungsfunktionen, insbesondere von Extremalfunktionen der Theorie der konformen Abbildung, J. Reine Angew. Math. 228 (1967), 93-132. [ 13] R. Ktihnau, Triangulierte riemannsche Mannigfaltigkeiten mit ganz-linearen Bezugssubstitutionen und quasikonforme Abbildungen mit stiickweise konstanter komplexer Dilation, Math. Nachr. 46 (1970), 243-261. [ 14] R. Ktihnau, Extremalprobleme bei quasikonformen Abbildungen mit kreisringweise konstanter Dilatationsbeschriinkung, Math. Nachr. 66 (1975), 269-282. [15] R. Ktihnau, Uber die Werte des Doppelverhiiltnisses bei quasikonformer Abbildung, Math. Nachr. 95 (1980), 237-251. [16] R. Ktihnau, Zur ebenen PotentialstrOmung um einen por6sen Kreiszylinder, Z. Angew. Math. Phys. 40 (1989), 395-409. [17] R. Ktihnau, M6glichst konforme Jordankurven durch vier Punkte, Rev. Roumaine Math. Pures Appl. 36 (1991), 383-393. [18] R. Ktihnau, Eine Methode, die Positivitiit einer Funktion zu priifen, Z. Angew. Math. Mech. 74 (1994), 140-142. [ 19] R. Ktihnau, Ersetzungssiitze bei quasikonformen Abbildungen, Ann. Univ. Mariae Curie-Sldodowska Lublin Sect. A 52 (1998), 65-72. [20] R. Ktihnau, Bibliography of Geometric Function Theory (GFT), Handbook of Complex Analysis: Geometric Function Theory, Vol. 2, Elsevier, Amsterdam (2005), this Volume. [21 ] R. Kiihnau and E. Hoy, Abschiitzung des Wertebereichs einiger Funktionale bei quasikonformen Abbildungen, Bull. Soc. Sci. Lett. L6di 29 (1979), 4, 1-9. [22] R. Ktihnau and E. Hoy, Bemerkungen iiber quasikonform fortsetzbare schlichte konforme Abbildungen, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Nat. Reihe 31 (1982), 129-133. [23] R. Ktihnau and W. Niske, Abschiitzung des dritten Koeffizienten bei den quasikonform fortsetzbaren schlichten Funktionen der Klasse 8, Math. Nachr. 78 (1977), 185-192. [24] R. Ktihnau and B. Thtiring, Berechnung einer quasikonformen Extremalfunktion, Math. Nachr. 79 (1977), 99-113. [25] G.V. Kuz'mina, Moduli of Families of Curves and Quadratic Differentials, Proc. Steklov Inst. Math. 139 (1982) (in Russian); English transl.: Amer. Math. Soc., Providence, RI (1980). [26] G.V. Kuz'mina, The module problem for families of classes of curves in a circular ring, Zap. Nauchn. Sem. Leningr. Otdel. Mat. Inst. Steklov 144 (1985), 115-127 (in Russian). [27] G.V. Kuz'mina, Methods of Geometric Function Theory, II, St. Petersburg Math. J. 9 (1998), 889-930. [28] N.A. Lebedev, The Area Principle in the Theory of Univalent Functions, Nauka, Moskva (1972) (in Russian). [29] O. Lehto and K.I. Virtanen, Quasiconformal Mappings in the Plane, 2nd edn, Springer-Verlag, BerlinHeidelberg-New York (1973). [30] S.-L. Qiu and M. Vuorinen, Special functions in Geometric Function Theory, Handbook of Complex Analysis: Geometric Function Theory, Vol. 2, Elsevier, Amsterdam (2005), 621-659 (this Volume). [31] A.Yu. Solynin, Solution of the P61ya-Szeg6 isoperimetric problem, Zap. Nauchn. Sem. LOMI 168 (1988), 140-153; J. Soviet. Math. 53 (1991), 311-320. [32] A.Yu. Solynin, Modules and extremal metric problems, St. Petersburg Math. J. 11 (1) (2000), 1-65. [33] O. Teichmtiller, Extremale quasikonforme Abbildungen und quadratische Differentiale, Abh. Preul~. Akad. Wiss. Math.-Nat. K1. 22 (1939), 1-179. [34] O. Teichmtiller, Ein Verschiebungssatz der quasikonformen Abbildung, Deutsche Math. 7 (1944), 336-343.