Real entire functions of infinite order and a conjecture of Wiman
نویسندگان
چکیده
We prove that if f is a real entire function of infinite order, then ff ′′ has infinitely many non-real zeros. In conjunction with the result of Sheil-Small for functions of finite order this implies that if f is a real entire function such that ff ′′ has only real zeros, then f is in the Laguerre-Pólya class, the closure of the set of real polynomials with real zeros. This result completes a long line of development originating from a conjecture of Wiman of 1911.
منابع مشابه
Real entire functions with real zeros and a conjecture of Wiman
This conclusion is not true for the first derivative as the example exp(sin z) shows. For real entire functions with finitely many zeros, all of them real, Theorem 1.1 was proved in [3]. Theorem 1.1 can be considered as an extension to functions of infinite order of the following result of Sheil-Small [20], conjectured by Wiman in 1914 [1, 2]. For every integer p ≥ 0, denote by V2p the set of e...
متن کاملSome difference results on Hayman conjecture and uniqueness
In this paper, we show that for any finite order entire function $f(z)$, the function of the form $f(z)^{n}[f(z+c)-f(z)]^{s}$ has no nonzero finite Picard exceptional value for all nonnegative integers $n, s$ satisfying $ngeq 3$, which can be viewed as a different result on Hayman conjecture. We also obtain some uniqueness theorems for difference polynomials of entire functions sharing one comm...
متن کاملEntire functions sharing a small entire function with their difference operators
In this paper, we mainly investigate the uniqueness of the entire function sharing a small entire function with its high difference operators. We obtain one results, which can give a negative answer to an uniqueness question relate to the Bruck conjecture dealt by Liu and Yang. Meanwhile, we also establish a difference analogue of the Bruck conjecture for entire functions of order less than 2, ...
متن کاملOn the number of real critical points of logarithmic derivatives and the Hawaii conjecture
For a given real entire function φ with finitely many nonreal zeros, we establish a connection between the number of real zeros of the functions Q = (φ/φ) and Q1 = (φ /φ). This connection leads to a proof of the Hawaii conjecture [T.Craven, G.Csordas, and W. Smith, The zeros of derivatives of entire functions and the Pólya-Wiman conjecture, Ann. of Math. (2) 125 (1987), 405–431] stating that th...
متن کاملDifferential Operators of Infinite Order and the Distribution of Zeros of Entire Functions
whenever the right-hand side of (1.2) represents an analytic function in a neighborhood of the origin. When φ(x) is an entire function, the operator φ(D) has been studied by several authors (see, for example, [5, §11], [19, Chapter IX], [22] and [32]). The conjecture of Pólya and Wiman, proved in [8], [9] and [17], states that if f(x) ∈ L-P∗, then Df(x) is in the Laguerre-Pólya class for all su...
متن کامل