Transcendental Singularities for a Meromorphic Function with Logarithmic Derivative of Finite Lower Order

Langley, J. K.

doi:10.1007/s40315-018-0253-3

Cited by 10 publications

(8 citation statements)

References 17 publications

(47 reference statements)

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“…Since lim S→+∞ Ψ(S) = 0 by L'Hôpital's rule, (21) implies that Ψ(S) < ε/4 if S 0 is large enough. Thus (19) follows from ( 16), ( 23) and (24), which proves (i).…”

Section: A Refinement Of Hille's Methodssupporting

confidence: 54%

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Bank-Laine functions, the Liouville transformation and the Eremenko-Lyubich class

Langley

2018

Preprint

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The Bank-Laine conjecture concerning the oscillation of solutions of second order homogeneous linear differential equations has recently been disproved by Bergweiler and Eremenko. It is shown here, however, that the conjecture is true if the set of finite critical and asymptotic values of the coefficient function is bounded. It is also shown that if E is a Bank-Laine function of finite order with infinitely many zeros, all real and positive, then its zeros must have exponent of convergence at least 3/2, and an example is constructed via quasiconformal surgery to demonstrate that this result is sharp. MSC 2000: 30D35.

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“…Since lim S→+∞ Ψ(S) = 0 by L'Hôpital's rule, (21) implies that Ψ(S) < ε/4 if S 0 is large enough. Thus (19) follows from ( 16), ( 23) and (24), which proves (i).…”

Section: A Refinement Of Hille's Methodssupporting

confidence: 54%

“…. denote positive constants which may depend on v 0 and φ, but not on m. By (19), these zeros X m satisfy, with ζ m ∈ K 3 and ε small,…”

Section: A Refinement Of Hille's Methodsmentioning

confidence: 99%

Bank-Laine functions, the Liouville transformation and the Eremenko-Lyubich class

Langley

2018

Preprint

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“…Now let F 4 be the closure of the union of E 4 and its reflections across the real and imaginary axes. Then Y is meromorphic off F 4 and (17) implies that the complex dilatation…”

Section: Lemma 41 the Möbius Transformationmentioning

confidence: 99%

Bank-Laine Functions with Real Zeros

Langley

2020

Comput. Methods Funct. Theory

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Suppose that E is a real entire function of finite order with zeros which are all real but neither bounded above nor bounded below, such that $$E'(z) = \pm 1$$ E ′ ( z ) = ± 1 whenever $$E(z) = 0$$ E ( z ) = 0 . Then either E has an explicit representation in terms of trigonometric functions or the zeros of E have exponent of convergence at least 3. An example constructed via quasiconformal surgery demonstrates the sharpness of this result.

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“…Because n is large and L m,n , L ′ m,n and L ′′ m,n have finitely many non-real zeros, Lemma 2.5 gives (28). Now let U n be the component of {z ∈ C : |zL ′ m,n (z)/L m,n (z)| < e −|am|rn/8 } which contains v n .…”

Section: Lemma 23 ([29]mentioning

confidence: 99%

“…Set h = H j and S n = e −βrn/8 . Then applying (28) with m = m k gives (8), while (27) with m = m j delivers (9). Lemma 2.6 now implies that for large n the set…”

Section: Lemma 23 ([29]mentioning

confidence: 99%