Research Interests                    Norbert Steinmetz

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     Complex Differential Equations  [2, 6, 7, 14, 16, 19, 24, 27, 30, 31, 32, 34, 35, 39, 40, 42, 44, 45, 46, 50, 54, 55, 56, 58, 59]
In [2, 4] Malmquist's Theorem is extended to (and Yosida's Theorem is completed for) binomial differential equations  w'n=R(z,w). In [33, 35]  several conjectures due to  Wiman,  Perron and  Wittich concerning the distribution of zeros of solutions of linear differential equations and fundamental sets are proven.  [40, 42,45, 46, 54, 56, 58] deal with Painleve's and related equations, and contain new existence proofs for the Painleve transcendents I, II, IV,  sharp order estimates (stated by Boutroux), and a description of the solutions of small order of growth (sub-normal solutions) to  II and IV.

     Nevanlinna Theory  [1, 3, 5, 8, 9, 10, 11, 12, 13, 17, 18, 23, 26, 28, 29, 52, 53, 57, 61]
In [9] the Tumura-Clunie Theorem is extended from entire to meromorphic functions;  [23, 29] contain a proof  and extensions of Nevanlinna's Second Main Theorem for small functions. [28] contains the classification of those function triples (f,g,h) sharing four values; this also disproves a conjecture (actually a ''theorem'') of H. Cartan.
 

     Holomorphic Dynamics  [31, 36, 37, 38, 41, 43, 51, 60]
 [31] contains a proof of Sullivan's (actually the Cremer-Fatou) classification of periodic stable domains. In [39] the  quasiconformal conjugacy classes of rational functions whose Julia sets are Jordan curves or arcs are determined.

    Geometric Function Theory   [20, 22, 25]
 
 
 

    Miscellanea

          
       Ableitungen meromorpher Funktionen, Diploma Thesis,  Karlsruhe (1975).  pdf
                   Eigenschaften  eindeutiger Lösungen gewöhnlicher Differentialgleichungen im Komplexen, PhD Thesis, Karlsruhe (1978).  pdf

   Rational Iteration. Complex Analytic Dynamical Systems. de Gruyter Studies in Mathematics 16  (1993). 

   Painleve differential equationation in the complex plane by V.I. Gromak, I. Laine and S. Shimomura,  
    de Gruyter studies 
in Mathematics 28  (2002).   Bull. Amer. Math. Soc  41,  523-528 (2004) pdf                                                                                                        
 

Normal families and complex dynamics. Abstracts from the mini-workshop held February 18-24, 2007
(Ed. Ph. Rippon, N. Steinmetz, L. Zalcman),  Oberwolfach Rep. 4, No. 1, 487-548 (2007).
 


    Papers and Preprints
 

[1]   Zur Wertverteilung von Exponentialpolynomen.Manuscripta Math.  6, 155-167 (1978).  ZBL 0395.30023   pdf

[2]  Bemerkung zu einem Satz von Yosida. Complex Analysis, Joensuu 1978. ZBL 0 419.34004  Lecture   Notes in Mathematics, Vol. 747, 369-377. Springer 1979. pdf 

[3]  (with E. Mues) Meromorphe Funktionen, die mit ihrer Ableitung Werte teilen. Manuscripta Math.  29, 195-206 (1979). ZBL 0416.30028 pdf

[4]    Zur Theorie der binomischen Differentialgleichungen. Math. Annalen  244, 263-274   (1979).  ZBL 0422.34005 pdf

[5]  Über die faktorisierbaren Lösungen gewöhnlicher Differentialgleichungen.  Math. Zeitschr. 170, 169-180 (1980).  ZBL 0425.34006 pdf

[6]   Ein Malmquistscher Satz für algebraische Differentialgleichungen erster Ordnung.  Journal reine angew. Math.  316, 44-53 (1980). ZBL 0426.34011  pdf

[7] Über das Anwachsen der Lösungen homogener algebraischer Differentialgleichungen zweiter Ordnung. Manuscripta Math.  32,  303-308 (1980). ZBL 0444.34035 pdf

[8]   Zur Wertverteilung der Quotienten von Exponentialpolynomen. Archiv Math.  35, 461-470 (1980). ZBL 0461.30023 pdf

[9]  (with E. MuesThe theorem of Tumura-Clunie for meromorphic functions. Journal LMS 23, 113-122 (1981). ZBL 0466.30025 pdf

[10]   Über die Nullstellen von Differentialpolynomen. Math. Zeitschr. 176, 255-264 (1981).  pdf
[11]   On factorization of the solutions of the Schwarzian differential equation {w,z} = q(z). Funkc. Ekvac. 24, 307-315 (1981). ZBL 0485.30028  pdf

[12]  On the primeness of the Painlevé transcendents. Factorization  theory of meromorphic functions. Lect. Notes Pure Appl. Math., p. 119-128, New  York  1982. ZBL 0489.30021
[13]   (with C. C. YangOn common right factors of F and F(N). Factorization theory of meromorphic functions. Lect. Notes Pure Appl. Math., p. 129-138, New  York 1982. ZBL 0491.30016

[14]   Zur Wertverteilung der Lösungen der vierten Painlevéschen Differentialgleichung. Math. Zeitschr.   181,  553-561 (1982). ZBL 0528.30019 pdf

 [15]  On the functional equation f(x) = f(px) + f(qx+p). C.R. Math. Rep. Acad. Sci. Canada  4, 367-371 (1982) . 
  ZBL 0502.39006  pdf

  [16]  Über die eindeutigen Lösungen einer homogenen algebraischen Differentialgleichung zweiter Ordnung,   Ann. Acad. Sci. Fenn. 7. 177-188 (1982) . ZBL 0565.34005 pdf

[17] On the order and lower order of entire functions with radially distributed zeros.Proc AMS 87, 449-452 (1983). ZBL 0515.30017 pdf

[18] (with  E. Mues) Meromorphe Funktionen, die mit ihrer Ableitung zwei Werte teilen. Results Math. 6, 48-55 (1983). ZBL 0571.30030  pdf

[19]   Über eine Klasse von Painlevéschen Differentialgleichungen Archiv Math.  41, 261-266 (1983).    pdf  ZBL 0601.34005


[20]  
Locally univalent functions in the unit disk.   Ann. Acad. Sci. Fenn.  8, 325-332 (1983).
  ZBL 0575.30015 pdf

[21] (with  P. Volkmann Funktionalgleichungen für konstante Funktionen. Aequat. Math 27, 152-167 (1986). ZBL 0544.39005 pdf

[22] Homeomorphic extension of univalent functions. Complex Variables 6, 1-9 (1986).  pdf  ZBL 0609.30001
 

[23]  Eine Verallgemeinerung des zweiten Nevanlinnaschen Hauptsatzes. Journal reine angew. Math  368, 134-141 (1986).   pdf    ZBL 0598.30045

[24]  Ein Malmquistscher Satz für algebraische Differentialgleichungen zweiter Ordnung.  Results Math.  10 , 152-167 (1986). pdf   ZBL 0652.34007

[25] De Branges´ proof of the Bieberbach conjecture. General Inequalities 5, pp. 3-16. Birkhäuser 1987. ZBL 0624.30025   pdf

[26]   On the zeros of  (f(p)+ ap-1f(p-1)+ ...+ a0 f)f.  Analysis 7, 375-389 (1987). ZBL 0638.30033 pdf

[27]   Meromorphe Lösungen der Differentialgleichung Q(z,w)d2w/dz2 = P(z,w)(dw/dz)2. Complex Variables  10,  31-41 (1988). ZBL 0646.30030  pdf
 

[28]  A uniqueness theorem for three meromorphic functions. Ann. Acad. Sci. Fenn.  13, 93-110 (1988). ZBL 0665.30033  pdf

[29] On the zeros of a certain Wronskian. Bulletin LMS 20, 525-531 (1988).  ZBL 0661.30025  pdf

[30]  Exceptional values of the solutions of linear differential equations. Math. Zeitschr. 201, 317-326 (1989). ZBL 0692.34006  pdf

[31]  Algebraische Differentialgleichungen erster und zweiter Ordnung. (In: Complex Methods on   Partial Differential Equations. Akademie-Verlag Berlin 1989,  pp. 187-192).  ZBL 0692.30023  pdf

[32]  Meromorphic solutions of second order algebraic differential equations. Complex Variables (Special Issue dedicated to A. Edrei and W. Fuchs)  13,  75-83 (1989). ZBL 0692.30022  pdf
[33]  On Sullivan's classification of periodic stable domains. Complex Variables 14,  211-214 (1990).  ZBL 0708.30027  pdf

[34]  Linear differential equations with exceptional fundamental sets. Analysis 11, 119-128 (1991). ZBL 0744.34014  pdf

[35]  Linear differential equations with exceptional fundamental sets, II.  Proc. AMS  117, 355-358 (1993). ZBL 0770.30027   pdf

[36] The formula of Riemann-Hurwitz and iteration of rational functions. Complex Variables  22, 203-206 (1993). ZBL 0805.30030  pdf

[37]  (with W. Schmidt) The polynomials associated with a Julia set. Bulletin LMS  27, 239-241 (1995). ZBL 0826.30020  pdf

[38]  (with A. Baesch) Exceptional solutions of n-th order periodic linear differential equations.  Complex Variables  34, 7-17 (1997). ZBL 0913.34007  pdf
 

[39] Jordan and Julia. Math. Annalen  307, 531-541 (1997).  pdf  ZBL 0879.30015
 
Comments on Jordan and Julia  pdf

[40]  On Painlevé's equations I, II and IV, Journal d'Analyse Math. 82, 363-377 (2000). ZBL 0989.34073 pdf

[41]   Sub-hyperbolic rational maps and algebraic differential equations, Complex Variables 43, 451-456 (2001). ZBL 1022.30022   pdf

[42]   Value distribution of the Painlevé transcendents, Israel Journal of Math. 128, 29-52 (2002).  ZBL 1016.34091 pdf

[43]  Zalcman functions and rational dynamics, New Zealand J. Math 32, 1-14 (2003) pdf
 

[44]  On Airy solutions of Painlevé's second equation, Comput. Methods & Function Theory  3, 117-126 (2003)      pdf  

  [45] Boutroux's method vs. Re-scaling. Lower estimates for the orders of growth of the second and fourth  
Painleve transcendents,   Portugaliae Math. 61, 369-374 (2004).  pdf 

[46]  Global properties of the Painlevé transcendents. New results and open questions.
 Ann. Acad. Sci. Fenn. 30 71-98 (2005). pdf
 
   [47]  On  the dynamics of the McMullen family  R(z)=z^m+lambda/z^l,   Conformal Geometry and Dynamics 10, 159-183  (2006).     pdf

  [48] Sierpinski curve Julia sets of rational functions  Comput. Methods & Function Theory  6, 317-327 (2006)   pdf                                    

[49]  Sierpinski and non-Sierpinski curve Julia sets in families of rational maps, Journal London Math. Soc78290-304         (2008)Dynamical Planes    pdf
                                                                                                                                     
    
[50]  Normal families and linear differential equations,  Journal d'Analyse Math. 117, 129-132  (2012). pdf
  
 
[51]   (with Hye Gyong Jang )   On the dynamics of the rational family f_t = -t(z²-2)²/(4(z²-1)),  
   Comput. Methods & Function Theory 12, 1-17   (2012)  pdf
                                      

[52]  Reminiscence of an open problem: Remarks on Nevanlinna's Four-Value-Theorem,  South East Asian Bull. Math. 36, 399-417 (2012)   SEABoM
       and  Corrigendum
  
[53] 
The Yosida class is universal,  Journal d'Analyse Math. 117, 347-364 (2012)
pdf                                                         
                     

[54]  Sub-normal solutions to Painleve's second differential ,  Bull. LMS 45, 225-235 (2013)  pdf


[55]
Complex Riccati differential equations revisited, Ann. Acad. Sci. Fenn. 39, 503-511  (2014)  pdf   Addendum 

              
  [56]  An old new class of meromorphic functions  (2014) to appear in Journal d'Analyse pdf


[57]   Remark on meromorphic functions that share five pairs,  Analysis 36 (2016), 195-198.    pdf

 [58]   A unified approach to the Painleve transcendents (2015),  Ann. Acad. Sci.  Fenn42 (2017), 17-49pdf

                                    

[59]  First order algebraic differential equations of genus zero,  to appear in Bull. LMS (2017)   pdf



[60]
   (with Hye Gyong Jang)  On the dynamics of rational maps with two free critical points, submitted (2016)  pdf

                [61] (with Gary Gundersen and Kazuya Tohge) Meromorphic functions that share four or five values of pairs, submitted 2016 pdf

Note that the pdf-files may differ from the published versions! 
 


Norbert Steinmetz