Research Questions on Meromorphic Functions and Complex Differential Equations

Abstract

Thirty research questions on meromorphic functions and complex differential equations are listed and discussed. The main purpose of this paper is to make this collection of problems available to everyone.MSC 2010: 30D20, 30D35, 34M05, 34M10. Complex analysis questionsSection 2 has notation, Section 3 has questions on meromorphic and entire functions, Sections 4 and 5 have questions on second order homogeneous linear complex differential equations, and Section 6 has questions on first order nonlinear complex differential equations. NotationIn this paper a meromorphic function means a function that is meromorphic in the whole complex plane. We assume the reader is familiar with the Nevanlinna theory of meromorphic functions; see [34], [66].For a meromorphic function f , we use the following notation.Let ρ(f ) denote the order of f .Let µ(f ) denote the lower order of f .Let λ(f ) denote the exponent of convergence of the sequence of zeros of f , counted according to the multiplicities.Let λ(f ) denote the exponent of convergence of the sequence of zeros of f , where each zero is counted once. Questions on meromorphic functionsThis section has questions on meromorphic and entire functions. Question 3.1. Does there exist an entire function that possesses an infinite number of positive real zeros and no other zeros and an infinite number of purely imaginary one-points and no other one-points?If such an entire function exists, then the function would have growth of at most order two, mean type; see [12], [52].Question 3.2. Does there exist an entire function that possesses an infinite number of positive real zeros and no other zeros, and an infinite number of onepoints such that all the one-points lie on a finite number of half-lines that start from the origin, where an infinite number of the one-points are not positive real numbers?If such an entire function f exists, then the zeros and one-points of f would all lie on two or more half-lines that start from the origin, and f would have finite

Entire functions, PT-symmetry and Voros’s quantization scheme

“…From the explicit form of f in (5) we see that the function θ → |f (re iθ )| is even, 2π-periodic, and strictly increasing on (0, π). Therefore (11) can hold only with real λ.…”

“…Proposition 1. Let f be an entire function of order less than 1 of the form (5), and suppose that (7) is satisfied with some entire function C which has no positive zeros. Then (10) is satisfied with D as in (9) and all zeros of C are negative.…”

Restrictions on meromorphic solutions of Fermat type equations

“…When k = 3 in (1.1), the open questions are as follows: meromorphic (n = 7, 8), rational (n = 6, 7), entire (n = 6), polynomial (n = 4, 5). Regarding these statements, see, for example, [2][3][4]6] and the references therein.…”

Relationship between entire functions and their derivatives sharing small function except a set

“…For example, the differential equation f + A 1 (z)e az + A 2 e bz = H, where A 1 , A 2 , H are three entire functions with order less than 1, and a, b are two constants. It is related to a famous differential equation question posed by Gundersen in [11]. We refer to [6,10,25] for some results of the above differential equation.…”