On a conjecture of Fuchs

2001 ◽  
Vol 131 (5) ◽  
pp. 1209-1216 ◽  
Author(s):  
Joseph Miles ◽  
John Rossi
Keyword(s):  
Entire Function ◽  
Logarithmic Derivative ◽  
Image Position ◽  
Nevanlinna Deficiency

If f is an entire function of order ρ, 0 < ρ < 2−11, it is shown that the Nevanlinna deficiency d(0, f′/f) of the logarithmic derivative of f satisfies For small positive ρ, this result strengthens an earlier estimate of Eremenko et al. concerning a conjecture of Fuchs.

scholarly journals Oscillation results on y″ + Ay = 0 in the complex domain with transcendental entire coefficients which have extremal deficiencies

1995 ◽  
Vol 38 (1) ◽  
pp. 13-34 ◽  
Author(s):  
Y. M. Chiang
Keyword(s):  
Entire Function ◽  
Complex Domain ◽  
Transcendental Entire Function ◽  
Exponent Of Convergence ◽  
Linearly Independent ◽  
Image Position ◽  
Nevanlinna Deficiency

Let A(z) be a transcendental entire function and f1, f2 be linearly independent solutions ofWe prove that if A(z) has Nevanlinna deficiency δ(0, A) = 1, then the exponent of convergence of E: = flf2 is infinite. The theorems that we prove here are similar to those in Bank, Laine and Langley [3].

On the Properties of an Entire Function of Two Complex Variables

1968 ◽  
Vol 20 ◽  
pp. 51-57
Author(s):  
Arun Kumar Agarwal
Keyword(s):  
Entire Function ◽  
Double Series ◽  
Complex Variables ◽  
Maximum Term ◽  
Image Position

1. Letbe an entire function of two complex variables z1 and z2, holomorphic in the closed polydisk . LetFollowing S. K. Bose (1, pp. 214-215), μ(r1, r2; ƒ ) denotes the maximum term in the double series (1.1) for given values of r1 and r2 and v1{m2; r1, r2) or v1(r1, r2), r2 fixed, v2(m1, r1, r2) or v2(r1, r2), r1 fixed and v(r1r2) denote the ranks of the maximum term of the double series (1.1).

On the Maximum Modulus and the Mean Modulus of an Entire Function

1969 ◽  
Vol 12 (6) ◽  
pp. 869-872 ◽  
Author(s):  
A.R. Reddy
Keyword(s):  
Entire Function ◽  
Maximum Modulus ◽  
The Mean ◽  
Image Position

Let be an entire function, but not a polynomial. As usual let,1

scholarly journals A Class of functional equations which have entire solutions

1988 ◽  
Vol 38 (3) ◽  
pp. 351-356 ◽  
Author(s):  
Peter L. Walker
Keyword(s):  
Functional Equation ◽  
Entire Function ◽  
Wide Class ◽  
Functional Equations ◽  
Entire Functions ◽  
Inverse Function ◽  
Entire Solution ◽  
Entire Solutions ◽  
Image Position

We consider the Abelian functional equationwhere φ is a given entire function and g is to be found. The inverse function f = g−1 (if one exists) must satisfyWe show that for a wide class of entire functions, which includes φ(z) = ez − 1, the latter equation has a non-constant entire solution.

scholarly journals A Remark on Small Values of Entire Functions

1966 ◽  
Vol 15 (2) ◽  
pp. 121-123 ◽  
Author(s):  
S. L. Segal
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Image Position

Let f(z) be an entire function, M(r) the maximum of f(z) on ∣z∣=r, and λ>1. Let Eλ=Eλ(f{z:log∣f(z)≦(1-λ)log(M∣z∣)}, and denote the density of Eλbywhere m is planar measure.

Generating Functions for Hermite Functions

1959 ◽  
Vol 11 ◽  
pp. 141-147 ◽  
Author(s):  
Louis Weisner
Keyword(s):  
Differential Equation ◽  
Entire Function ◽  
Generating Functions ◽  
Hermite Functions ◽  
Kummer’S Function ◽  
Image Position

Hermite's function Hn(x) is denned for all complex values of x and n bywhere F (α; γ; x) is Kummer's function with the customary indices omitted. It satisfies the differential equation1.1of whichis a second solution. Every solution of (1.1) is an entire function.

Smooth majorants for functions of arbitrarily rapid growth

1991 ◽  
Vol 109 (3) ◽  
pp. 565-569 ◽  
Author(s):  
J. P. Earl ◽  
W. K. Hayman
Keyword(s):  
Entire Function ◽  
Rapid Growth ◽  
Image Position

Valiron[3] showed that if f(z) is an entire function for whichwhere A(r) = log(sup{|f(z)|:|z| = r}), then there is a function x(r) = rρ(r) satisfying both

On the zeros and Fourier transforms of entire functions in the Paley–Wiener space

1996 ◽  
Vol 119 (2) ◽  
pp. 357-362 ◽  
Author(s):  
Konstantin M. Dyakonov
Keyword(s):  
Entire Function ◽  
Complex Number ◽  
Compact Support ◽  
Fourier Transforms ◽  
Entire Functions ◽  
Wiener Space ◽  
Dirichlet Integral ◽  
Fixed Complex ◽  
Image Position

AbstractLet f be an entire function of the formwhere ø is a function in L2(ℝ) with compact support. If f|ℝ is real-valued then, for obvious reasons, (a) the supporting interval for ø is symmetric with respect to the origin, andAssuming that f has no zeros in {Im z > 0}, we prove that the converse is also true: (a) and (b) together imply that f|ℝ takes values in αℝ, where α is a fixed complex number.The proof relies on a certain formula involving the Dirichlet integral, which may be interesting on its own.

scholarly journals Dirichlet and Poincaré series

1985 ◽  
Vol 27 ◽  
pp. 39-56 ◽  
Author(s):  
A. Good
Keyword(s):  
Functional Equation ◽  
Number Theory ◽  
Algebraic Geometry ◽  
Entire Function ◽  
Cusp Form ◽  
Modular Forms ◽  
Modular Group ◽  
Positive Proportion ◽  
Fundamental Discriminant ◽  
Image Position

The study of modular forms has been deeply influenced by famous conjectures and hypotheses concerningwhere T(n) denotes Ramanujan's function. The fundamental discriminant Δ is a cusp form of weight 12 with respect to the modular group. Its associated Dirichlet seriesdefines an entire function of s and satisfies the functional equationThe most penetrating statements that have been made on T(n) and LΔ(s)are:Of these four problems only A1 has been established so far. This was done by Deligne [1] using methods from algebraic geometry and number theory. While B1 trivially holds with ε > 1/2, it was established in [2] for every ε>1/3. Serre [12] proved A2 for a positive proportion of the integers and Hafner [5] showed that LΔ has a positive proportion of its non-trivial zeros on the line σ=6. The proofs of the last three results are largely analytic in nature.

scholarly journals Some precisions on the Fourier-Borel transform and infinite order differential equations

1973 ◽  
Vol 14 (2) ◽  
pp. 161-167 ◽  
Author(s):  
Lawrence Gruman
Keyword(s):  
Entire Function ◽  
Differential Equations ◽  
Infinite Order ◽  
Borel Transform ◽  
Several Variables ◽  
Function Of Several Variables ◽  
Image Position

Let f(z) be an entire function (of several variables). We define the functionwhich is increasing. The orderof f(z) is the constant (perhaps infinite)

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