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 A number in square brackets (e.g. [1]) indicates that the paper has been either superseded or recently checked.

back to Jim Langley's home page

[1] 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.

[2] The distribution of zeros of certain differential polynomials,
*Journal of the London
Math. Soc.* 29 (1984), 485-498.
PDF file

[3] 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*Comment.* Theorem 1 of this paper has now been superseded by paper [59] below.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.

[4] 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.

[5] (With S B Bank)
On the value distribution theory of elliptic functions,
*Monatshefte der Math.* 98 (1984), 1-20.
PDF file

[6] 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This paper has now been superseded by paper [52] below.

[7] On Hayman's alternative, *Mathematika* 32 (1984), 139-146.

*Comment.*
This paper has now been superseded by paper [59] below.

[8] Exceptional
sets for linear differential polynomials, *Annales Academiae Scientiarum
Fennicae*,
Ser A I, Vol II (1986), 137-154.
PDF file

[9] (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.

[10] 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.

[12] 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 [28] and [106] below.

[13] 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 [28] 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.

[15] The Tsuji characteristic and zeros of linear differential polynomials,
*Analysis* 9 (1989), 269-282.
*Comment.*
This paper has now been superseded by paper [106] below.

[16] 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 [28] below.

[17] (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 [106] 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.

[21] 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 [106] 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.

[28] 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 [9],
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

See also:
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 [46]. 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 [46] 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.