scholarly journals Bank-Laine Functions with Real Zeros

2020 ◽  
Vol 20 (3-4) ◽  
pp. 653-665
Author(s):  
J. K. Langley
Keyword(s):  
Entire Function ◽  
Finite Order ◽  
Explicit Representation ◽  
Trigonometric Functions ◽  
Exponent Of Convergence ◽  
Real Zeros ◽  
Real Entire Function ◽  
Bounded Below ◽  
Quasiconformal Surgery

AbstractSuppose 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.

scholarly journals The radial oscillation of solutions to ode's in the complex domain

1996 ◽  
Vol 39 (3) ◽  
pp. 473-483 ◽  
Author(s):  
John Rossi ◽  
Shupei Wang
Keyword(s):  
Entire Function ◽  
Finite Order ◽  
Upper Bound ◽  
Complex Domain ◽  
Radial Oscillation ◽  
Real Zeros ◽  
The Third ◽  
Finite Set ◽  
Oscillation Of Solutions

We prove three results concerning the oscillation near a ray of solutions to (*)w″ + Aw = 0, where A is an entire function. The first result assumes that A is a polynomial and gives an upper bound on the number of its real zeros if (*) admits a solution with only real zeros and infinitely many. The second result proves that for A of finite order a solution w to (*) has “few” zeros “near” a ray if and only if the same is true for w′. The third result involves the density of the zeros of a solution to (*) “away” from a finite set of rays.

scholarly journals Value distribution of meromorphic functions concerning rational functions and differences

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mingliang Fang ◽  
Degui Yang ◽  
Dan Liu
Keyword(s):  
Positive Integer ◽  
Meromorphic Function ◽  
Rational Function ◽  
Finite Order ◽  
Meromorphic Functions ◽  
Rational Functions ◽  
Value Distribution ◽  
Exponent Of Convergence ◽  
Nonzero Constant ◽  
Zero Points

AbstractLet c be a nonzero constant and n a positive integer, let f be a transcendental meromorphic function of finite order, and let R be a nonconstant rational function. Under some conditions, we study the relationships between the exponent of convergence of zero points of $f-R$ f − R , its shift $f(z+nc)$ f ( z + n c ) and the differences $\Delta _{c}^{n} f$ Δ c n f .

scholarly journals The Spaces of Entire Function of Finite Order

2016 ◽  
Vol 7 (3) ◽  
Author(s):  
Malyutin KG ◽  
Studenikina IG
Keyword(s):  
Entire Function ◽  
Finite Order

scholarly journals On a partial theta function and its spectrum

2016 ◽  
Vol 146 (3) ◽  
pp. 609-623 ◽  
Author(s):  
Vladimir Petrov Kostov
Keyword(s):  
Entire Function ◽  
Theta Function ◽  
Local Minimum ◽  
Real Zero ◽  
Positive Real ◽  
The Real ◽  
Real Zeros ◽  
Partial Theta Function

The bivariate series defines a partial theta function. For fixed q (∣q∣ < 1), θ(q, ·) is an entire function. For q ∈ (–1, 0) the function θ(q, ·) has infinitely many negative and infinitely many positive real zeros. There exists a sequence of values of q tending to –1+ such that has a double real zero (the rest of its real zeros being simple). For k odd (respectively, k even) has a local minimum (respectively, maximum) at , and is the rightmost of the real negative zeros of (respectively, for k sufficiently large is the second from the left of the real negative zeros of ). For k sufficiently large one has . One has and .

On the question of whether f″+ e−zf′ + B(z)f = 0 can admit a solution f ≢ 0 of finite order

1986 ◽  
Vol 102 (1-2) ◽  
pp. 9-17 ◽  
Author(s):  
Gary G. Gundersen
Keyword(s):  
Entire Function ◽  
Finite Order ◽  
Infinite Order ◽  
Independent Interest ◽  
Partial Result ◽  
Transcendental Entire Function ◽  
Degree Polynomial ◽  
Odd Degree ◽  
Polynomial Of Odd Degree

SynopsisWe show that if B(z) is either (i) a transcendental entire function with order (B)≠1, or (ii) a polynomial of odd degree, then every solution f≠0 to the equation f″ + e−zf′ + B(z)f = 0 has infinite order. We obtain a partial result in the case when B(z) is an even degree polynomial. Our method of proof and lemmas for case (i) of the above result have independent interest.

scholarly journals A further result on the complex oscillation theory of periodic second order linear differential equations

1990 ◽  
Vol 33 (1) ◽  
pp. 143-158 ◽  
Author(s):  
Shian Gao
Keyword(s):  
Entire Function ◽  
Positive Integer ◽  
Oscillation Theory ◽  
Transcendental Entire Function ◽  
Arbitrary Positive Integer ◽  
Order Of Growth ◽  
Exponent Of Convergence ◽  
Complex Oscillation ◽  
Zero Sequence ◽  
Linear Differential

We prove the following: Assume that , where p is an odd positive integer, g(ζ is a transcendental entire function with order of growth less than 1, and set A(z) = B(ezz). Then for every solution , the exponent of convergence of the zero-sequence is infinite, and, in fact, the stronger conclusion holds. We also give an example to show that if the order of growth of g(ζ) equals 1 (or, in fact, equals an arbitrary positive integer), this conclusion doesn't hold.

The generalized Laguerre inequalities and functions in the Laguerre-Pólya class

2013 ◽  
Vol 11 (9) ◽  
Author(s):  
George Csordas ◽  
Anna Vishnyakova
Keyword(s):  
Entire Function ◽  
Open Problem ◽  
A Priori ◽  
Sufficient Conditions ◽  
Class Definition ◽  
Lindelöf Theorem ◽  
Order And Type ◽  
Geometric Result ◽  
Real Entire Function ◽  
Inequality Theorem

AbstractThe principal goal of this paper is to show that the various sufficient conditions for a real entire function, φ(x), to belong to the Laguerre-Pólya class (Definition 1.1), expressed in terms of Laguerre-type inequalities, do not require the a priori assumptions about the order and type of φ(x). The proof of the main theorem (Theorem 2.3) involving the generalized real Laguerre inequalities, is based on a beautiful geometric result, the Borel-Carathédodory Inequality (Theorem 2.1), and on a deep theorem of Lindelöf (Theorem 2.2). In case of the complex Laguerre inequalities (Theorem 3.2), the proof is sketched for it requires a slightly more delicate analysis. Section 3 concludes with some other cognate results, an open problem and a conjecture which is based on Cardon’s recent, ingenious extension of the Laguerre-type inequalities.

scholarly journals On entire solutions of a certain type of nonlinear differential equation

2001 ◽  
Vol 64 (3) ◽  
pp. 377-380 ◽  
Author(s):  
Chung-Chun Yang
Keyword(s):  
Differential Equation ◽  
Entire Function ◽  
Finite Order ◽  
Nonlinear Differential Equation ◽  
Entire Solution ◽  
Differential Polynomial ◽  
Entire Solutions ◽  
Value Distribution Theory ◽  
Linear Differential Polynomial ◽  
Linear Differential

In this note, we shall study, via Nevanlinna's value distribution theory, the uniqueness of transcendental entire solutions of the following type of nonlinear differential equation: (*) L (f (z)) – p (z) fn(z) = h (z), where L (f) denotes a linear differential polynomial in f with polynomials as its co-efficients, p (z) a polynomial (≢ 0), h an entire function, and n an integer ≥ 3. We show that if the equation (*) has a finite order transcendental entire solution, then it must be unique, unless L (f) ≡ 0.

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