LV ANNALESPOLONICIMATHEMATICI

P O L S K A A K A D E M I A N A U K, I N S T Y T U T M A T E M A T Y C Z N Y

ANNALES

POLONICI MATHEMATICI

R E D A K T O R O W I E

S T A N I S L A W L O J A S I E W I C Z, J ´ O Z E F S I C I A K

KOM ITE T RE DAKCYJNY

A. BIELECKI, B. BOJARSKI, Z. CHARZY ´ NSKI, K. GE ¸ BA, P. JAK ´ OBCZAK (sekretarz), J. KRZY ˙Z, A. LASOTA,

K. MAURIN, W. MLAK, C. OLECH, W. PLE´ SNIAK, A. PLI´ S, B. SZAFIRSKI

LV

P R O C E E D I N G S

O F T H E T E N T H C O N F E R E N C E O N A N A L Y T I C F U N C T I O N S

SZCZYRK, APRIL 22–27, 1990

W A R S Z A W A 1991

Les ANNALES POLONICI MATHEMATICI constituent une continuation des ANNALES DE LA SOCI´ ET´ E POLONAISE DE MATH´ EMATIQUE (vol. I–XXV) fond´ees en 1921 par Stanis law Za r emba.

Les ANNALES POLONICI MATHEMATICI publient, en langues des congr`es internationaux, des travaux consacr´es ` a l’Analyse Math´ematique et ` a la G´eometrie.

Chaque volume paraˆ ı t en 3 fascicules.

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`

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Copyright by Instytut Matematyczny PAN, Warszawa 1991

ISBN 83-85116-21-4 ISSN 0066-2216

ANNALES

POLONICI MATHEMATICI 55 (1991)

FOREWORD

The Tenth Conference on Analytic Functions was held in Szczyrk from April 22 to April 27, 1990.

Organizing Committee: C. Andreian Cazacu (Bucharest), Z. Charzy´ nski ( L´ od´ z), P. Dolbeault (Paris), F. W. Gehring (Ann Arbor), A. A. Gonchar (Moscow), L. Iliev (Sofia), Z. J. Jakubowski ( L´ od´ z), M. Jarnicki (Krak´ ow;

Vice-Chairman), C. O. Kiselman (Uppsala), J. Krzy˙z (Lublin), O. Lehto (Helsinki), J. Leiterer (Berlin), P. Lelong (Paris), J. Lawrynowicz ( L´ od´ z), S. Lojasiewicz (Krak´ ow), P. Pflug (Vechta), W. Ple´ sniak (Krak´ ow), K. Rusek (Krak´ ow; Secretary), J. Siciak (Krak´ ow; Chairman), W. Tutschke (Halle), V. S. Vladimirow (Moscow), T. Winiarski (Krak´ ow).

The main topics were: geometric function theory of one complex vari- able, quasiconformal mappings, complex analysis in several variables and potential theory in C

.

140 mathematicians from 20 countries participated in the Conference.

They delivered 28 one-hour lectures and 87 short communications.

The organizer of the Conference was the Institute of Mathematics of the Jagiellonian University.

The Conference was sponsored by the Institute of Mathematics of the Polish Academy of Sciences as well as by the Universities of L´ od´ z and Lublin and by the Technical University of Rzesz´ ow.

The Organizing Committee expresses its gratitude to all these institu- tions and to all people who helped the organizers in their efforts.

∗

The Proceedings consist of the list of participants, the list of all lectures and communications and of 36 papers submitted to this volume. Most of the papers contain original results announced during the Conference; only a few of them are of survey character.

Kamil Rusek

Secretary

CONTENTS

List of participants . . . . iv–v Program:

I. Lectures (45 min.) . . . . vi II. Communications (20 min.) . . . . vii–ix Papers submitted to the Proceedings:

C. A n d r e i a n C a z a c u, On the disc theorem . . . . 1–10 A. B o i v i n and R. D w i l e w i c z, Holomorphic approximation of CR functions

on tubular submanifolds of C

. . . . 11–18 R. B r u m m e l h u i s and J. W i e g e r i n c k, Representing measures for the disc

algebra and for the ball algebra . . . . 19–35 P. C a r a m a n, New cases of equality between p-module and p-capacity . . . . 37–56 S. C y n k and P. T w o r z e w s k i, Diagonal series of rational functions . . . . 57–63 S. D i m i e v, Holomorphic non-holonomic differential systems on complex man-

ifolds . . . . 65–73 G. D i m k o v, On products of starlike functions. I . . . . 75–79 G. D i m k o v, J. S t a n k i e w i c z and Z. S t a n k i e w i c z, On a class of starlike

functions defined in a halfplane. . . . 81–86 M. M. D j r b a s h i a n and A. H. K a r a p e t y a n, Integral representations for some

weighted classes of functions holomorphic in matrix domains . . . . 87–94 L. M. D r u ˙z k o w s k i, A geometric approach to the Jacobian Conjecture

in C

. . . . 95–101 M. F r o n t c z a k and A. M i o d e k, Weil’s formulae and multiplicity . . . . 103–108 J. F u k a and Z. J. J a k u b o w s k i, On certain subclasses of bounded univalent

functions . . . . 109–115 A. Z. G r i n s h p a n, Univalent functions with logarithmic restrictions . . . . 117–138 C. D. H i l l and S. R. S i m a n c a, The super complex Frobenius theorem . . . . 139–155 W. H y b, On the spectral properties of translation operators in one-dimensional

tubes . . . . 157–161 A. J a n i k, On approximation of analytic functions and generalized orders . . . . 163–167 M. J a r n i c k i and P. P f l u g, Invariant pseudodistances and pseudometrics

—completeness and product property . . . . 169–189 M. J¸ edrzejowski, The homogeneous transfinite diameter of a compact subset of

C

. . . . 191–205 S. K o l o d z i e j, Jung’s type theorem for polynomial transformations of C

. . . . 207–212 T. K r a s i ´ n s k i, On branches at infinity of a pencil of polynomials in two com-

plex variables . . . . 213–220

A. K r o k and T. M a z u r, The kaehlerian structures and reproducing kernels . 221–224

Contents iii

W. K r ´ o l i k o w s k i, Anisotropic complex structure on the pseudo-Euclidean

Hurwitz pairs . . . . 225–240 K. K u r d y k a, A counterexample to subanalyticity of an arc-analytic function 241–243 J. L a w r y n o w i c z, K. K ¸ e d z i a and O. S u z u k i, Supercomplex structures, sur-

face soliton equations, and quasiconformal mappings . . . . 245–268 J. M. M y s z e w s k i, On roots of the automorphism group of a circular domain

in C

. . . . 269–276 D. P a r t y k a, A distortion theorem for quasiconformal automorphisms of the

unit disk . . . . 277–281 M. P a s s a r e, A new division formula for complete intersections . . . . 283–286 Z. P a s t e r n a k - W i n i a r s k i, On the dependence of the Bergman function on

deformations of the Hartogs domain. . . . 287–300 K. S p a l l e k, Continuous transformation groups on spaces . . . . 301–320 S. S p o d z i e j a, Some criteria for the injectivity of holomorphic mappings . . . . . 321–323 H. M. S r i v a s t a v a, Applications of certain linear operators in the theory of

analytic functions . . . . 325–330 A. S t r z e b o ´ n s k i, The growth of regular functions on algebraic sets . . . . 331–341 T. S z e m b e r g, Automorphisms of Riemann surfaces with two fixed

points . . . . 343–347 K. T o c h o w i c z, The classes of univalent functions connected with homogra-

phies . . . . 349–355

A. W r z e s i e ´ n, On some majorization of derivatives in the class S

(γ) . . . . 357–359

J. Z a j ¸ a c, Distortion function and quasisymmetric mappings . . . . 361–369

LIST OF PARTICIPANTS

BULGARIA Dimiev, Stancho Dimkov, Georgi M.

Hristov, Valentin Z.

Kiryakova, Virginia S.

Kovacheva, Ralitza Rusev, Peter CANADA Boivin, Andr´ e Gilligan, Bruce Srivastava, Hari M.

CHILE

C´ ordova, Antonio

CHINESE PEOPLE’S REPUBLIC Xing, Yang

CZECHOSLOVAKIA Fuka, Jaroslav

DDR

Achilles, R¨ udiger Peters, Klaus Schmalz, Gerd Tutschke, Wolfgang Wegert, Elias FINLAND Naekki, Raimo FRANCE Bonneau, Pierre Dloussky, Georges Dolbeault, Pierre GREAT BRITAIN Chisholm, John S. R.

ITALY

Gentili, Graziano Marchiafava, Stefano

Romani, Giuliano Succi, Francesco JAPAN Koshi, Shozo Morimoto, Mitsuo MEXICO

Ramirez de Arellano, Enrique NETHERLANDS

Korevaar, Jacob Wiegerinck, Jan POLAND Baran, Miros law B locki, Zbigniew Chmielowski, Jan Cygan, Ewa Cynk, S lawomir Czy ˙z, Janusz Da ¸browska, Anna

Downarowicz, Ma lgorzata Dronka, Janusz

Dru ˙zkowski, Ludwik Dwilewicz, Roman Frontczak, Maria Godula, Janusz Hyb, Wojciech Jak´ obczak, Piotr Jakubowski, Zbigniew J.

Janik, Adam Jarnicki, Marek Jelonek, Zbigniew Je ¸drzejowski, Mieczys law Jondro, Halina

Kaczmarek, Ludwika

Ko lodziej, S lawomir

Kopiecki, Ryszard

List of participants v

Krasi´ nski, Tadeusz Krok, Anna

Kr´ olikowski, Wies law Krzy ˙z, Jan

Kurdyka, Krzysztof Kwieci´ nski, Micha l Ligocka, Ewa Lisiecki, Wojciech Lawrynowicz, Julian Lojasiewicz, Stanis law Macura, Janina Majchrzak, Wies law Maszczyk, Tomasz Mazur, Tomasz Miodek, Andrzej Myszewski, Jan Partyka, Dariusz

Pasternak- Winiarski, Zbigniew Pethe, Karol

Pierzchalski, Antoni P loski, Arkadiusz Rembieli´ nski, Jakub Rusek, Kamil

Rza ¸dkowski, Grzegorz Siciak, J´ ozef

Sitarski, Ryszard Skalska, Krystyna Skibi´ nski, Przemys law Skwarczy´ nski, Maciej Sowa, Artur

Spodzieja, Stanis law Stankiewicz, Jan Strzebo´ nski, Adam Strzelecki, Pawe l Szapiel, Wojciech Szemberg, Tomasz

Sladkowska- Zahorska, Janina ´ Tochowicz, Kajetan

Tutaj, Halszka Tworzewski, Piotr Waniurski, J´ ozef Winiarski, Tadeusz W lodarczyk, Kazimierz Wojtaszczyk, Przemys law W´ ojcik, Adam

Wrzesie´ n, Andrzej Zaja ¸c, J´ ozef

Zwonek, W lodzimierz Zyskowska, Krystyna ROMANIA

Andreian Cazacu, Cabiria Caraman, Petru

Gussi, Gheorge Mihalache, N.

Pascu, Eugen Vijiitu, V.

SWEDEN Backlund, Ulf F¨ alstr¨ om, Anders Passare, Mikael SWITZERLAND Kaup, Burchard USA

Duren, Peter Hill, C. Denson McNeal, Jaffery USSR

A˘ızenberg, Lev I.

Chirka, Evgeni˘ı M.

Grinshpan, Arkadi˘ı Z.

Ivashkovich, Serge˘ı M.

Krasichkov- Ternovski˘ı, Igor F.

Kruzhilin, Nikola˘ı G.

Lin, Vladimir Ya.

Milin, Izaak M.

Napalkov, Valentin V.

Pokhilevich, V.

Ronkin, Lev I.

Sergeev, Armen G.

Starkov, Viktor V.

Tamrazov, Promarz M.

Tsikh, Avgust K.

Yuzhakov, Aleksandr P.

Za˘ıdenberg, Mikhail G.

WEST GERMANY Bingener, J¨ urgen Kaup, Ludger Kaup, Wilhelm Pflug, Peter

Ruscheweyh, Stephan

Spallek, Karlheinz

PROGRAM

I. LECTURES (45 min.)

1. R. A c h i l l e s, Excess intersections in complex analytic geometry.

2. L. I. A˘ız e n b e r g, On holomorphic continuation from a part of the boundary.

3. J. B i n g e n e r, Infinite-dimensional superanalysis.

4. P. C a r a m a n, New cases of equality between p-module and p-capacity in R

. 5. E. M. C h i r k a, CR-foliations.

6. G. D l o u s s k y, On filling holes in dimension two.

7. P. D o l b e a u l t, On CR-analytic chains with given boundaries.

8. P. D u r e n, Univalent functions with range of prescribed capacity.

9. Z. J. J a k u b o w s k i, Extremal problems in some classes of analytic functions.

10. B. K a u p, Hyperbolic C

-actions on affine algebraic surfaces.

11. J. K o r e v a a r, Fields of electrons on the sphere and quadrature problems.

12. J. K r z y ˙z, Generalized Neumann–Poincar´ e operator and chord-arc curves.

13. J. L a w r y n o w i c z (with K. K ¸ e d z i a and O. S u z u k i), Supercomplex structures, surface soliton equations, and quasiconformal mappings.

14. M. M o r i m o t o, Analytic functionals on the complex light cone and their Fourier–

Borel transformations.

15. I. M. M i l i n, Leningrad school of geometric theory of functions of one complex vari- able.

16. P. P f l u g (with M. J a r n i c k i), Properties of invariant distances—a survey.

17. A. P l o s k i, On the irreducibility of polynomials in several variables.

18. L. I. R o n k i n, Jenssen’s theorems for holomorphic almost-periodic functions in sev- eral variables.

19. S. R u s c h e w e y h, Multipliers for convex harmonic mappings.

20. A. G. S e r g e e v, On pseudoconvexity of matrix domains.

21. K. S p a l l e k, Transformation groups on spaces.

22. J. S t a n k i e w i c z (with Z. S t a n k i e w i c z), On classes of functions regular in a half- plane.

23. P. M. T a m r a z o v, Removable singularities of plurisubharmonic functions in topo- logical vector spaces.

24. A. K. T s i k h, The method of principal residue in Leray’s theory.

25. W. T u t s c h k e, The role of analyticity for solving Cauchy–Kovalevskaya problems.

26. T. W i n i a r s k i, Global and local criteria for algebraicity.

27. A. P. Y u z h a k o v, Calculation of the full sum of local residua with respect to a polynomial mapping.

28. M. G. Z a˘ıd e n b e r g, Ramanujan surfaces and exotic algebraic structures on C

.

Program vii

II. COMMUNICATIONS (20 min.)

1. C. A n d r e i a n C a z a c u, On the disc theorem.

2. —, Some generalizations of the Zorich Theorem by M. Cristea.

3. M. B a r a n, Bernstein type theorems for compact sets in C

.

4. A. B o i v i n, Holomorphic approximation in the complex plane—generalization of the Vitushkin Theorem.

5. P. B o n n e a u, On ∂-equations on weakly pseudoconvex domains.

6. J. S. R. C h i s h o l m, Clifford–Pad´ e approximation.

7. J. C h m i e l o w s k i, Quelques remarques sur les ensembles d´ eterminants.

8. S. C y n k, Diagonal series of rational functions.

9. J. C z y ˙z, Summation formula for the coefficients in the Hausdorff series.

10. S. D i m i e v, Local non-holonomic complex analytic geometry.

11. G. D i m k o v, On products of starlike functions.

12. M. D o w n a r o w i c z, On exceptional cones for partial type.

13. J. D r o n k a, On the location of critical points of a polynomial.

14. L. D r u ˙z k o w s k i, The Jacobian Conjecture—a few steps towards solution.

15. M. F r o n t c z a k, A new simple proof of the Weil Integral Formula.

16. J. F u k a (with Z. J a k u b o w s k i), On a class of univalent functions.

17. G. G e n t i l i, Regular quaternionic maps and ∂-type operators.

18. B. G i l l i g a n, Complex analysis on homogeneous manifolds.

19. A. Z. G r i n s h p a n, Univalent functions with logarithmic restrictions.

20. C. D. H i l l, Counterexamples to the Newlander–Nirenberg Theorem up to the bound- ary and positive results for complex supermanifolds.

21. V. Z. H r i s t o v, Remarks on Carath´ eodory and Kobayashi distances.

22. S. M. I v a s h k o v i c h, Rational curves, shells and extension of holomorphic maps.

23. P. J a k ´ o b c z a k, Division by a holomorphic matrix in strictly pseudoconvex domains.

24. A. J a n i k, On approximation of analytic functions and generalized orders.

25. Z. J e l o n e k, Sets determining polynomial automorphisms of C

.

26. M. J ¸ e d r z e j o w s k i, The homogeneous transfinite diameter for compact subsets of C

.

27. L. K a c z m a r e k, Bounds of univalent functions in terms of iterated extrema.

28. V. S. K i r y a k o v a, Generalized fractional differintegrals and their use in univalent function theory.

29. S. K o s h i, Extremal problems in H

-space.

30. R. K o v a c h e v a, Regions of uniform convergence of rational Chebyshev approxi- mants with an unbounded number of poles.

31. I. F. K r a s i c h k o v - T e r n o v s k i˘ı, General description of closed ideals and submod- ules of the analytic functions of one variable.

32. T. K r a s i ´ n s k i, Bifurcation points of polynomials and the Jacobian Conjecture in C

.

33. —, Properness and the Jacobian Conjecture in C

.

34. N. G. K r u z h i l i n, On hulls of two-dimensional spheres in C

.

35. K. K u r d y k a, A counterexample to subanalyticity of an arc-analytic function.

36. M. K w i e c i ´ n s k i, Polynomial mappings and Gr¨ obner bases—a criterion for isomor- phism of varieties.

37. E. L i g o c k a, A characterization of ellipsoids and balls in C

.

38. V. Ya. L i n, On the Liouville property connected with amenable group actions.

39. —, On superposition problem for algebraic functions of several complex variables.

viii Program

40. J. L a w r y n o w i c z (with J. R e m b i e l i ´ n s k i), On Fueter-like equations for J

-triples and regularity.

41. J. M a c u r a, On the second variation for univalent meromorphic functions.

42. T. M a s z c z y k, Noncommutative analyticity and instantons.

43. T. M a z u r, On complex manifolds of Bergman type.

44. J. M c N e a l, Necessary conditions for H¨ older estimates on solutions to ∂-equations.

45. N. M i h a l a c h e, Special neighbourhoods of analytic sets in Stein spaces.

46. A. M i o d e k (with J. C h ¸ a d z y ´ n s k i and T. K r a s i ´ n s k i), On radical complete inter- sections.

47. J. M y s z e w s k i, On the structure of the Lie algebra of the automorphism group of a circular domain in C

.

48. R. N a e k k i, One-sided quasidiscs.

49. V. V. N a p a l k o v, Some problems on convolution equations.

50. D. P a r t y k a, A distortion theorem for quasiconformal automorphisms of the unit disk.

51. E. P a s c u, On superanalytic algebras.

52. M. P a s s a r e, Polynomial interpolation and complex convexity.

53. Z. P a s t e r n a k - W i n i a r s k i, On dependence of the Bergman function on deformation of the Hartogs domain.

54. K. P e t e r s, C

-estimates for the ∂-equation of non-transversal intersections of strictly pseudoconvex domains.

55. K. P e t h e, On a functional defined in the class of p-valued functions.

56. A. P i e r z c h a l s k i, Quasiconformal deformations of manifolds with boundary.

57. V. P o k h i l e v i c h, On subclasses of regular univalent functions with a fixed coeffi- cient.

58. E. R a m i r e z d e A r e l l a n o (with W. K r ´ o l i k o w s k i), An integral representation for Fueter–Hurwitz regular functions.

59. P. R u s e v, M¨ untz type theorems for systems of degenerate hypergeometric functions.

60. G. S c h m a l z, Identification of pseudoconvex domains by the Bergman operator.

61. J. S i c i a k (with N g u y e n T h a n h V a n), Doubly orthogonal systems of holomorphic functions and extremal plurisubharmonic functions.

62. R. S i t a r s k i, On some L-analytic maps of two variables.

63. K. S k a l s k a, On typically-real functions meromorphic in the unit disc.

64. P. S k i b i ´ n s k i, Description of the set of critical points of a polynomial mapping in two variables.

65. M. S k w a r c z y ´ n s k i, Bergman function, alternating projections, L

-angles.

66. A. S o w a, Schwarz’s derivative in high dimensions.

67. S. S p o d z i e j a, A criterion of injectivity of holomorphic mappings.

68. H. M. S r i v a s t a v a, Applications of certain linear operators in the theory of analytic functions.

69. J. S t a n k i e w i c z (with Z. J a k u b o w s k i), On classes of functions with non-classical normalization.

70. V. V. S t a r k o v, Linear-invariant families with Stieltjes representation involving com- plex measures.

71. A. S t r z e b o ´ n s k i, Growth of regular functions on algebraic sets.

72. W. S z a p i e l (with W. H e n g a r t n e r), On univalent functions omitting a given value.

73. T. S z e m b e r g, Automorphisms of Riemann surfaces with two fixed points.

74. K. T o c h o w i c z, The classes of univalent functions connected with homographies.

75. P. T w o r z e w s k i, Improper isolated intersections in complex analytic geometry.

76. V. V i j i i t u, A remark on Runge domains.

Program ix

77. J. W a n i u r s k i, The Bloch constant for the M¨ obius transforms of convex mappings.

78. E. W e g e r t, On interpolation and approximation with families of holomorphic func- tions.

79. J. W i e g e r i n c k, Representing measures for the disc algebra and for the ball algebra.

80. K. W l o d a r c z y k, Angular derivative for holomorphic maps in complex Banach spaces.

81. P. W o j t a s z c z y k, Bases in the disc algebra.

82. A. W ´ o j c i k, Some remarks on Bernstein’s Theorems.

83. A. W r z e s i e ´ n, On some majorization of derivatives in the class S

(γ).

84. X i n g Y a n g, A generalization of the M¨ untz–Szasz Theorem.

85. M. G. Z a˘ıd e n b e r g, Normal mappings and growth estimates of Schottky–Landau type.

86. J. Zaj¸ ac, Distortion function and quasisymmetric mappings.

87. K. Z y s k o w s k a, On an estimate of some functional in the class of odd bounded

univalent functions.