A Generalisation of the Bank-Laine Property

Langley, J. K.; Meyer, Janis

doi:10.1007/bf03321723

Cited by 4 publications

(3 citation statements)

References 8 publications

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“…). The Bank-Laine functions arise in connection with solutions of second order homogeneous linear differential equations [2], and have been studied in many papers [2,8,9,11,12,13].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Shared values, Picard values and normality

Chang¹,

Wang

2011

Tohoku Math. J. (2)

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It is known that a family of meromorphic functions is normal if each function in the family shares a 3-element set with its derivative. In this paper we consider value distribution and normality problems with regard to 2-element shared sets. First we construct an example, by use of the Weierstrass doubly periodic functions, to show that a 3-element shared set can not be reduced to a 2-element shared set in general. We obtain a new criterion of normal families and new Picard-type theorems. The proofs make use of some results in complex dynamics. More examples are constructed to show that our assumptions are necessary. Recently, Liu-Pang [14] improved Schwick's result as follows. THEOREM B. Let F be a family of functions meromorphic in D and a, b, c distinct finite numbers. If f −1 ({a, b, c}; D) = (f ) −1 ({a, b, c}; D) for every f ∈ F, then F is normal in D.

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“…). The Bank-Laine functions arise in connection with solutions of second order homogeneous linear differential equations [2], and have been studied in many papers [2,8,9,11,12,13].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Shared values, Picard values and normality

Chang¹,

Wang

2011

Tohoku Math. J. (2)

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“…Here E = f 1 f 2 is a product of two linearly independent solutions f 1 , f 2 and the Wronskian determinant c = W (f 1 , f 2 ) = 0 is a constant. The Bank-Laine identity has led to the definitions of Bank-Laine functions and Bank-Laine sequences [2,26,68,69], and is also used in finding prescribed zeros for solutions [90]. Bank and Laine [6] have proved that if ρ(A) is not an integer, then λ(E) ≥ ρ(A), and if ρ(A) < 1/2, then λ(E) = ∞.…”

Section: Complex Linear Differential Equationsmentioning

confidence: 99%

Value distribution of exponential polynomials and their role in the theories of complex differential equations and oscillation theory

Heittokangas¹,

Ishizaki²,

Tohge³

et al. 2022

Preprint

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An exponential polynomial is a finite linear sum of terms P (z)e Q(z) , where P (z) and Q(z) are polynomials. The early results on the value distribution of exponential polynomials can be traced back to Georg Pólya's paper published in 1920, while the latest results have come out in 2021. Despite of over a century of research work, many intriguing problems on value distribution of exponential polynomials still remain unsolved. The role of exponential polynomials and their quotients in the theories of linear/non-linear differential equations, oscillation theory and differential-difference equations will also be discussed. Thirteen open problems are given to motivate the readers for further research in these topics.

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“…Here 𝐸 = 𝑓 1 𝑓 2 is a product of two linearly independent solutions 𝑓 1 , 𝑓 2 and the Wronskian determinant 𝑐 = 𝑊(𝑓 1 , 𝑓 2 ) ≠ 0 is a constant. The Bank-Laine identity has led to the definitions of Bank-Laine functions and Bank-Laine sequences [2,27,70,71], and is also used in finding prescribed zeros for solutions [92]. Bank and Laine [6] have proved that if 𝜌(𝐴) is not an integer, then 𝜆(𝐸) ⩾ 𝜌(𝐴), and if 𝜌(𝐴) < 1∕2, then 𝜆(𝐸) = ∞.…”

Section: Appendix D: Complex Linear Differential Equationsmentioning

confidence: 99%

Value distribution of exponential polynomials and their role in the theories of complex differential equations and oscillation theory

Heittokangas

Ishizaki

Tohge

et al. 2022

Bulletin of London Math Soc

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An exponential polynomial is a finite linear sum of terms P(z)eQfalse(zfalse)$P(z)e^{Q(z)}$, where Pfalse(zfalse)$P(z)$ and Qfalse(zfalse)$Q(z)$ are polynomials. The early results on the value distribution of exponential polynomials can be traced back to Georg Pólya's paper published in 1920, while the latest results have come out in 2022. Despite of over a century of research work, many intriguing problems on value distribution of exponential polynomials still remain unsolved. The role of exponential polynomials and their quotients in the theories of linear/non‐linear differential equations, oscillation theory and differential‐difference equations will also be discussed. Thirteen open problems are given to motivate the readers for further research in these topics.

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A Generalisation of the Bank-Laine Property

Cited by 4 publications

References 8 publications

Shared values, Picard values and normality

Shared values, Picard values and normality

Value distribution of exponential polynomials and their role in the theories of complex differential equations and oscillation theory

Value distribution of exponential polynomials and their role in the theories of complex differential equations and oscillation theory

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