## THE UNIVERSITY OF NOTTINGHAM

### JIM LANGLEY'S RESEARCH PAPERS

Prof. J.K. Langley, School of Mathematical Sciences, University of Nottingham, NG7 2RD.

Here is a list of my publications: in some cases there is a link to a readable PDF file. A number in square brackets (e.g. ) indicates that the paper has been either superseded or recently checked.

JIM LANGLEY'S PUBLICATIONS

 Analogues of Picard sets for entire functions and their derivatives, American Math. Society Contemporary Mathematics 25 (1983), 75-84.
Comment. This paper was published in the proceedings of an A.M.S. special session (NY 1983) and there were no page proofs for authors. The paper then emerged with numerous typos and omissions; in particular several modulus signs were missing. The PDF file in this link contains a scan of a corrected version.
 The distribution of zeros of certain differential polynomials, Journal of the London Math. Soc. 29 (1984), 485-498. PDF file
 On differential polynomials and results of Hayman and Doeringer, Mathematische Zeitschrift 187 (1984), 1-11. PDF file
Comment. Theorem 1 of this paper has now been superseded by paper  below. The statement of Theorem 2 contains a gap: it needs to be assumed that in (1.6) at least one of w', w'', ... has non-zero coefficient, in order to get a non-trivial linear differential equation from (4.4). Without this assumption one can take any w such that w-1 has no zeros and make the LHS of (1.6) just be w^2 - 2w + 1. Thanks are due to Michael Heckner for pointing this out.
 On normal families and a result of Drasin, Proc. Royal Society of Edinburgh 98A (1984), 385-393.
Comment. The rather heavy calculations in this paper are now redundant, since the result can be deduced quite straightforwardly from the Pang-Zalcman re-scaling lemma: choose alpha = 1/(n-1) in Lemma 1 of Larry Zalcman's survey PDF file.
 (With S B Bank) On the value distribution theory of elliptic functions, Monatshefte der Math. 98 (1984), 1-20. PDF file
 The distribution of finite values of meromorphic functions with few poles, Annali di Scuola Normale Superiore di Pisa, Ser IV, Vol XII, No 1 (1985), 91-104. PDF file
Comment. This paper has now been superseded by paper  below.
 On Hayman's alternative, Mathematika 32 (1984), 139-146.
Comment. This paper has now been superseded by paper  below.
 Exceptional sets for linear differential polynomials, Annales Academiae Scientiarum Fennicae, Ser A I, Vol II (1986), 137-154. PDF file
 (With S B Bank and Ilpo Laine) On the frequency of zeros of solutions of second order linear differential equations, Resultate der Mathematik 10 (1986), 9-24.
 On complex oscillation and a problem of Ozawa, Kodai Math Journal 9 (1986), 430-439.
11. (With S B Bank) On the oscillation of solutions of certain linear differential equations in the complex domain, Proc. Edinburgh Math. Soc. 30 (1987), 455-469.
 On the zeros of linear differential polynomials with small rational coefficients, Journal of the London Math. Society 36 (1987), 445-457.
Comment. This paper has now been superseded by papers  and  below.
 On the zeros of (f'' + a f)f and a result of Steinmetz, Proc. Roy. Soc. Edin. 108A (1988), 241-247.
Comment. This paper has now been superseded by paper  below.
14. (With S B Bank and I Laine) Oscillation results for solutions of linear differential equations in the complex domain, Resultate der Mathematik 16 (1989), 3-15.
 The Tsuji characteristic and zeros of linear differential polynomials, Analysis 9 (1989), 269-282.
Comment. This paper has now been superseded by paper  below.
 On the zeros of second order linear differential polynomials, Proc. Edin. Math. Soc. 33 (1990), 265-285.
Comment. This paper has now been superseded by paper  below.
 (With Gao Shian) On the zeros of certain linear differential polynomials, Journal of Math. Analysis and Applications 153 (1990), 159-178.
Comment. This paper has now been superseded by paper  below.
18. (With S B Bank) On the zeros of the solutions of the equation $w^{(k)} + (e^P + Q)w = 0$, Kodai Math Journal 13 (1990), 298-309.
19. (With S B Bank) Oscillation theory for higher order linear differential equations with entire coefficients, Complex Variables 16 (1991), 163-175.
20. Some oscillation theorems for higher order linear differential equations with entire coefficients of small growth, Resultate der Mathematik 20 (1991), 517-529.
 An application of the Tsuji characteristic, Journal of the Faculty of Science of the Univ. of Tokyo 38 (1991), 299-318.
Comment. This paper has now been superseded by paper  below.
22. (With S B Bank) Oscillation theorems for higher order linear differential equations with entire periodic coefficients, Commentarii Universitatis Sancti Pauli 41 (1992), 65-85.
23. On the deficiencies of composite entire functions, Proc. Edinburgh Math. Society 36 (1992), 151-164.
24. Proof of a conjecture of Hayman concerning $f$ and $f''$, Journal of the London Mathematical Society 48 (1993), 500-514.
25. (With A Eremenko and J Rossi) On the zeros of meromorphic functions of the form $\sum_{k=1}^{\infty} a_k/(z - z_k)$, Journal d'Analyse Math. 62 (1994), 271-286.
26. On linear differential equations and gap series, Complex Variables 24 (1994), 59-66.
27. On the fixpoints of composite entire functions of finite order, Proc. Roy. Soc. Edin. 124A (1994), 995-1001.
 On second order linear differential polynomials, Resultate der Mathematik 26 (1994), 51-82.
Remarks:
in the proof of Lemma 5 (ii) the constant K should be chosen to be a positive integer (so that Kn etc. are integers);
in the statement of Lemma 1 there is a typo- it should be f = Af_1 - Bf_2 (in keeping with the previous paragraph);
on p.69, the term \lambda_1 should be either -A/C or -C/A.
p.73, L16. Replace \phi by g^2.
None of these minor typos/errors affects the method.
29. (With W. Bergweiler and J. Clunie) Proof of a conjecture of Baker concerning the distribution of fixpoints, Bulletin of the London Mathematical Society 27 (1995), 148-154.
30. On the multiple points of certain meromorphic functions, Proceedings of the American Mathematical Society 123 (1995), 1787-1795.
31. On entire solutions of linear differential equations with one dominant coefficient, Analysis 15 (1995), 187-204 (see also p.433 for corrections).
Note that Theorem A(i) is an old result of Wittich, for which it should be assumed that A_0 is not identically zero.
32. On the zeros of solutions of certain linear differential equations, Resultate der Math. 29 (1996), 276-279.
33. A lower bound for the number of zeros of a meromorphic function and its second derivative, Proc. Edinburgh Math. Soc. 39 (1996), 171-185. PDF file
34. The zeros of the first two derivatives of a meromorphic function, Proc. Amer. Math. Soc. 124, no. 8 (1996), 2439-2441. PDF file
35. Two results related to a question of Hinkkanen, Kodai Math. Journal 19 (1996), 52-61. PDF file
36. The zeros of the second derivative of a meromorphic function, XVIth Rolf Nevanlinna Colloquium, Eds.: Laine/Martio, Walter de Gruyter, Berlin 1996.
37. On the zeros of the second derivative, Proc. Roy. Soc. Edinburgh 127A (1997), 359-368.
38. (With D.F. Shea) On multiple points of meromorphic functions, J. London Math. Soc. (2) 57 (1998), 371-384. PDF file
Note that Theorem B as stated for the case where the logarithmic derivative has lower order 1/2 does not follow from the result in reference , at least not directly. However, this assertion is not used in the paper and so does not affect the results proved therein. The assertion for lower order less than 1/2 is valid and follows from the argument in the subsequent paragraph.
The PDF file in this link contains a proof of a more precise version of Lemma 3, which suffices for this and a number of subsequent papers.
39. (With J.H. Zheng) On the fixpoints, multipliers and value distribution of certain classes of meromorphic functions, Ann. Acad. Sci. Fenn. 23 (1998), 133-150. PDF file
40. (With D. Drasin) On deficiencies and fixpoints of composite meromorphic functions, Complex Variables 34 (1997), 63-82.
Note: there is a slight error on p.79, in that we omitted to consider the possibility that n might be negative in (47). This only requires changing slightly the estimate for the derivative of the inverse function, and this is done in a PDF file
41. On the zeros of $f^{(k)}/f$, Complex Variables 37 (1998), 385-394. PDF file
42. Permutable entire functions and Baker domains, Math. Proc. Camb. Phil. Soc. 125 (1999), 199-202.
43. Quasiconformal modifications and Bank-Laine functions, Archiv der Mathematik 71 (1998), 233-239.
Correction: delete the term A(z)^{-1/2} in formula (18) (compare (17)).
44. (With G. Frank) On the zeros of pairs of linear differential polynomials, Ann. Acad. Sci. Fenn. 24 (1999), 409-436. PDF file
45. On differential polynomials, fixpoints and critical values of meromorphic functions, Resultate der Mathematik 35 (1999), 284-309.
Typo: in Lemma 1 s should satisfy exp( (1/4) log h) r < s < r exp ( (3/4) log h ). PDF file
46. (With G. Frank) Pairs of linear differential polynomials, Analysis 19 (1999), 173-194.
Note that there are a number of slight errors in . First, on p.176: given a fundamental set for (11), we have solutions of (9) if and only if G, H solve (12). This suffices for all uses of Lemma C.
In Theorem 3, f'/f needs to be transcendental, not just f.
Next, p.179: in Theorem 2 it is necessary to assume that $H$ is not identically zero. This causes no problem for the proof of Theorem 1, for if H vanishes identically f has finite order and finitely many poles. Thus F and G have finitely many zeros and so f has the form (5), by (19). Note that formula numbers above refer to the published version.
The following example is relevant here: let h be a transcendental solution of the homomogeneous linear ODE L(w) = 0 with rational coefficients, and let N be greater than the order of h. Let P be a polynomial of degree N, and let f = h + exp (P), F = L(f), G = (D - P')f. Then f'/f has order N and N(r, 1/F) + N(r, 1/G) = o( T(r, f'/f) ). This shows that the theorem is to some extent sharp. Also H vanishes identically in this case.
The pdf file of  below also contains these corrections and some proofs omitted from the published version. PDF file
47. Complex oscillation and removable sets, J. Math. Anal. Appl. 235 (1999), 227-236. PDF file
48. A certain functional-differential equation, J. Math. Anal. Appl. 244 (2000), 564-567. PDF file
49. Bank-Laine functions with sparse zeros, Proc. Amer. Math. Soc. 129 (2001), 1969-1978. PDF file
50. The second derivative of a meromorphic function, Proc. Edinburgh Math. Soc. 44 (2001), 455-478. PDF file
51. Linear differential equations with entire coefficients of small growth, Arch. Math. (Basel) 78 (2002), 291-296. PDF file
52. The distribution of finite values of meromorphic functions with deficient poles, Math. Proc. Camb. Phil. Soc. 132 (2002), 311-317. PDF file
53. Composite Bank-Laine functions and a question of Rubel, Trans. Amer. Math. Soc. 354 (2002), 1177-1191. PDF file
54. The second derivative of a meromorphic function of finite order, Bulletin London Math. Soc. 35 (2003), 97-108. PDF file
55. (with W. Bergweiler) Nonvanishing derivatives and normal families, J. d'Analyse 91 (2003), 353-367. PDF file
56. (with W. Bergweiler and A. Eremenko) Real entire functions of infinite order and a conjecture of Wiman, Geometric and Functional Analysis 13 (2003), 975-991. PDF file
57. (with John Rossi) Meromorphic functions of the form f(z) = \sum a_n/(z - a_n), Revista Math. Iberoamericana 20 (2004), 285-314. PDF file
Note that Lemma 4.7 only holds for large r as in Lemma 4.5, but this makes no difference to the proof of the theorem.
58. The zeros of ff'' - b, Resultate der Mathematik 44 (2003), 130-140. PDF file
59. (with W. Bergweiler) Multiplicities in Hayman's alternative, J. Australian Math. Soc. 78 (2005), 37-57. PDF file
60. Critical values of slowly growing meromorphic functions, Comput. Methods Funct. Theory. 2 (2002), 539-547. PDF file
61. (with John Rossi) Critical points of certain discrete potentials, Complex Variables 49 (special issue in memory of Matts Essen) (2004), 621-637. PDF file
62. Deficient values of derivatives of meromorphic functions in the class S, Comput. Methods Funct. Theory 4 (2004), 237-247. PDF file
Remark. Paper 62 uses a version of Fuchs' small arcs lemma from Hayman's Subharmonic Functions Vol. II, p.721 (the same lemma is also used in papers 65 and 74 below). Fuchs' small arcs lemma gives an upper bound for the integral of |zf'(z)/f(z)| with respect to arg z over small intervals, in terms of T(R, f), when f is meromorphic, but the exceptional sets and constants which arise are not particularly easy to read off from Fuchs' paper, whereas they are given explicitly by Hayman in his book. Hayman only states the estimate for the integral of the modulus of the partial derivative with respect to arg z of a delta-subharmonic function u, but examination of his proof shows that the same upper bound arises for the integral of |zf'(z)/f(z)| when f is meromorphic. This is because using the standard differentiated Poisson-Jensen formula for f'/f gives an integral term which can be estimated as on p.723 of Hayman's book, and a sum for which the estimates of p.724 are valid. The details may be found in the attached PDF file
63. (With W. Bergweiler and A. Eremenko) Zeros of differential polynomials in real meromorphic functions, Proc. Edinburgh Math. Soc. 48 (2005), 279-293. PDF file
64. Integer points of entire functions, Bulletin London Math. Soc. 38 (2006), 239-249. PDF file
65. Nonreal zeros of higher derivatives of real entire functions of infinite order, J. d'Analyse 97 (2005), 357-396. PDF file
Remarks: (a) see the remark following paper 62; (b) the observation that if f is in LP then all derivatives of f have only real zeros clearly only applies when f is transcendental; (c) Lemma 2.5 is for k at least 2.
66. Integer points of meromorphic functions, Comput. Methods Funct. Theory. 5 (2005), 253-262. PDF file
67. (With S.M. ElZaidi) Bank-Laine functions with periodic zero-sequences, Resultate der Mathematik 48 (2005), 34-43. PDF file
68. Pairs of nonhomogeneous linear differential polynomials, Royal Society of Edinburgh, Proceedings A (Mathematics) 136A, 785-794, 2006. PDF file
69. Solution of a problem of Edwards and Hellerstein, Comput. Methods Funct. Theory 6 (2006), No. 1, 243--252. PDF file
70. (With Walter Bergweiler), Zeros of differences of meromorphic functions, Math. Proc. Camb. Phil. Soc. 142 (2007), 133-147. PDF file
71. (With D. Drasin), Bank-Laine functions via quasiconformal surgery, Transcendental Dynamics and Complex Analysis, London Mathematical Society Lecture Notes 348 (2008), Cambridge University Press, 165-178. PDF file
72. Equilibrium points of logarithmic potentials on convex domains, Proc. Amer. Math. Soc. 135 (2007), 2821-2826. PDF file
73. (With W. Bergweiler and D. Drasin) Baker domains for Newton's method, Ann. Inst. Fourier 57 (2007), 803-814. PDF file
74. Meromorphic functions in the class S and the zeros of the second derivative, Comput. Methods Funct. Theory. 8 (2008), 73-84. PDF file
See the remark following paper 62.
75. Integer-valued analytic functions in a half-plane, Comput. Methods Funct. Theory 7 (2007), 433-442. PDF file
76. (With Eleanor Lingham) On the derivatives of composite functions, New Zealand J. Math. 36 (2007), 57-61. PDF file Online edition
77. (With Abdullah Alotaibi), The zeros of certain differential polynomials, Analysis 28 (2008), 269-282. PDF file
78. Zeros of meromorphic functions with poles close to the real axis, Result. Math. 51 (2007), 87-96. PDF file
79. Non-real zeros of linear differential polynomials, Journal d'Analyse Math. 107 (2009), 107-140. PDF file
For an example showing that Theorem 1.4 is sharp in the finite order case, see PDF file
80. (With A. Eremenko) Meromorphic functions of one complex variable. A survey, in Distribution of values of meromorphic functions by A.A. Gol'dberg and I.V. Ostrovskii, translated by Mikhail Ostrovskii, Translations of Mathematical Monographs 236, Amer. Math. Soc. Providence 2008. PDF file
81. (With J. Meyer), A generalisation of the Bank-Laine property, Comput. Methods Funct. Theory 9 (2009), 213-225. PDF file
Note that, contrary to what is stated following Theorem 1.2, the example f(z) = sin P(z) does not show that Theorem 1.2 fails for k=m=n-1=1, since f=1,-1 gives f'=0. In fact, if f is entire and f = a, b implies f'=c, where a, b are distinct and c is non-zero, then f has the "double Bank-Laine property" and so is determined by Theorem 1.1.
82. (With W. Bergweiler, A. Fletcher and J. Meyer), The escaping set of a quasiregular mapping, Proc. Amer. Math. Soc. 137 (2009), 641-651. PDF file
83. (With A. Fletcher) Integer points of analytic functions in a half-plane, Proc. Edinburgh Math. Soc. 52 (2009), 631-651. PDF file
84. Value distribution of differences of meromorphic functions, Rocky Mountain J. Math. 41 (2011), 275-291. PDF file
85. (With A. Fletcher and J. Meyer) Slowly growing meromorphic functions and the zeros of differences, Math. Proc. R. Ir. Acad. 109A(2) (2009), 147-154. PDF file
86. (With A. Fletcher and J. Meyer) Nonvanishing derivatives and the MacLane class A, Illinois J. Math. 53 (2009) 379-390. PDF file
87. Logarithmic singularities and the zeros of the second derivative, Comput. Methods Funct. Theory 9 (2009), No. 2, 565--578. PDF file
88. (With A. Fletcher) Meromorphic compositions and target functions, Ann. Acad. Sci. Fenn. 34 (2009), 615-636. PDF file
89. Non-real zeros of derivatives of real meromorphic functions, Proc. Amer. Math. Soc. 137 (2009), 3355-3367. PDF file
Remark: the link includes some additional detail in the proof of Lemma 4.3, which applies a normal families argument to a family of meromorphic functions which have no zeros.
90. Zeros of derivatives of meromorphic functions, Comput. Methods Funct. Theory 10 (2010), No. 2, 421--439. (submitted 22/7/09, accepted 14/09/09) PDF file
91. Real meromorphic functions and linear differential polynomials, Science China (Mathematics) 53 (2010), 739-748 (Yang Lo special issue). PDF file
92. (With A. Alotaibi) On solutions of linear differential equations with entire coefficients, J. Math. Analysis Appl. 367 (2010), 451-460. PDF file
93. Non-real zeros of derivatives of real meromorphic functions of infinite order, Math. Proc. Camb. Phil. Soc. 150 (2011), 343-351. PDF file
94. Non-real zeros of real differential polynomials, Proc. Roy. Soc. Edinburgh Sect. A. 141 (2011), 631-639. PDF file
95. An inequality of Frank, Steinmetz and Weissenborn, Kodai Math. Journal 34 (2011), 383-389. PDF file
96. Second order linear differential polynomials and real meromorphic functions, Resultate der Mathematik 63 (2013), 151-169. PDF file (The final publication is available at www.springerlink.com.)
97. Zeros of derivatives of real meromorphic functions, Comput. Methods Funct. Theory 12 (2012), 241-256. PDF file
98. (with Abdullah Alotaibi) The separation of zeros of solutions of higher order linear differential equations with entire coefficients, Resultate der Mathematik 63 (2013), 1365-1373.
99. The reciprocal of a real entire function and non-real zeros of higher derivatives, Ann. Acad. Sci. Fenn. 38 (2013), 855-871. PDF file
100. Zeros of the derivatives of the reciprocal of a real entire function, Proceedings of the Workshop on Complex Analysis and its Applications to Differential and Functional Equations, University of Eastern Finland (2014), ISBN 978-952-61-1353-1. PDF file
101. Derivatives of meromorphic functions of finite order, Comput. Methods Funct. Theory 14 (2014), 195-207. PDF file Arxiv version
102. The Schwarzian derivative and the Wiman-Valiron property, J. Analyse Math. 130 (2016), 71-89. PDF file (The final publication is available at www.springerlink.com.)
103. (With John Rossi) Wiman-Valiron theory for a class of functions meromorphic in the unit disc, Math. Proc. R. Ir. Acad. 114A (2014), 137-148. PDF file
104. Non-real zeros of derivatives of meromorphic functions, J. Analyse Math. 133 (2017), 183-228. PDF file (The final publication is available at www.springerlink.com.)
105. Trajectories escaping to infinity in finite time, Proc. Amer. Math. Soc. 145 (2017), 2107-2117. PDF file
106. Linear differential polynomials in zero-free meromorphic functions, Ann. Acad. Sci. Fenn. 43 (2018), 693-735. PDF file
107. Transcendental singularities for a meromorphic function with logarithmic derivative of finite lower order, Comput. Methods Funct. Theory 19 (2019), 117-133. PDF file
108. Bank-Laine functions, the Liouville transformation and the Eremenko-Lyubich class, J. Analyse Math. 141 (2020), 225–246. PDF file (The final publication is available at www.springerlink.com.)
109. Zeros of derivatives of strictly non-real meromorphic functions, Illinois J. Math. 64 (2020), 261-290. PDF file
110. Bank-Laine functions with real zeros, Comput. Methods Funct. Theory 20 (2020), 653-665. PDF file (The final publication is available at www.springerlink.com.)
111. Complex flows, escape to infinity and a question of Rubel, Ann. Fenn. Math. 47 (2022), 885-894. PDF file
More recent papers, including some which have not (yet) been submitted to journals, can be found on arxiv.org.