On the growth of a meromorphic function and its derivatives

We investigate the uniqueness of transcendental analytic fun- ctions that share three values DM in one angular domain instead of the whole complex plane. 1. Introduction and main results In this paper, a transcendental meromorphic (analytic) function is mero- morphic (analytic) in the whole complex plane C and not rational. We as- sume that the reader is familiar with the Nevanlinna's theory of meromorphic functions and the standard notations such as m(r, f ), T (r, f ). For references, see (2). We say that two meromorphic functions f and g share the value a (a ∈ C = C ∪ {∞}) in X ⊆ C provided that in X, we have f (z) = a if and only if g(z) = a. We will state whether a shared value is by DM (differential mul- tiplicities), or by IM (ignoring multiplicities). R. Nevanlinna (see (4)) proved that if two meromorphic functions f and g have five distinct IM shared values in X = C, then f (z) ≡ g(z). After his very work, the uniqueness of meromor- phic functions with shared values in the whole complex plane attracted many investigations (for references, see (7)). E. Mues consider DM shared values and proved the following theorem.

We study the uniqueness of meromorphic functions concerning nonlinear differential polynomials sharing a nonzero polynomial IM.Though the main concern of the paper is to improve a recent result of the present author [12], as a consequence of the main result we also generalize two recent results of X. M. Li and L. Gao [11].

Let be a point sequence in the unit disc and the Nevanlinna characteristic of the sequence , . In terms of the Nevanlinna characteristic one finds estimates for the slowest possible growth of the characteristic as in the class of holomorphic functions in vanishing on . Let be a meromorphic function in . In terms of the Nevanlinna characteristic function of one finds estimates for the slowest possible growth of the characteristics and in the class of pairs of holomorphic functions and such that .

Let D denote the unit disc in the complex plane. If p and q are two complex numbers, let x ( p , q ) denote the chordal distance between p and q on the Riemann sphere. In particular, we have the formula and The following problem was posed by Paul Gauthier: if f (z) and g (z) are meromorphic functions in D such that Clunie [4] has answered this problem in the negative by constructing different meromorphic functions f (z) and g (z) with the desired property. However, the functions constructed by Clunie both have an infinity of poles in D . It is the purpose of this note to give an example of two analytic functions―which thus have no poles―which also give a negative answer to the problem of Gauthier.

We say two meromorphic functions f(z) and g(z) share the finite complex value c if f(z) — c and g(z) — c have the same zeros. We will state whether a value is shared CM (counting multiplicities), IM (ignoring multiplicities), or by DM (by different multiplicities at one point or more). In this paper all functions will be assumed to be meromoφhic in the whole complex plane, unless stated otherwise. R. Nevanlinna [7, p. 109] proved that if/and g share five values IM, then either f—govf and g are both constants. He also found [7, Chapter V] the particular form of all pairs /, g that share four values CM and all pairs/, g that share three values CM. L. A. Rubel and C. C. Yang proved the following result:

The Kac-Baker model describes a 1-dim. classical lattice spin system with exponentially fast decaying two body interaction. The model was introduced by M. Kac and G. Baker to investigate the phenomenon of phase transition in systems with weak long-range interactions like van der Waals gas. Ruelle's dynamical zeta function for this model can be expressed in terms of Fredholm determinants of two transfer operators and hence is a meromorphic function. One of the two operators, found by M. Kac, is an integral operator with symmetric kernel acting in the Hilbert space of square integrable functions on the line. The other one is Ruelle's transfer operator acting in some Banach space of holomorphic observables of the system. In this paper we show how the Kac operator can be explicitly related basically through the Segal-Bargmann transform to the Ruelle operator restricted to a certain Segal-Bargmann space of entire functions in the complex plane. This allows us to show that Ruelle's zeta function for the Kac-Baker model has infinitely many non-trivial zeros on the real axis. In a special case we can show the existence of also infinitely many trivial zeros on the line Re s = ln 2 in the complex s-plane. Hence some kind of Riemann hypothesis seems to hold for this dynamical zeta function.

In this paper, we will give suitable conditions on differential polynomials [Formula: see text] such that they take every finite nonzero value infinitely often, where [Formula: see text] is a meromorphic function in complex plane. These results are related to Problems 1.19 and 1.20 in a book of Hayman and Lingham [Research Problems in Function Theory, preprint (2018), https://arxiv.org/pdf/1809.07200.pdf ]. As consequences, we give a new proof of the Hayman conjecture. Moreover, our results allow differential polynomials [Formula: see text] to have some terms of any degree of [Formula: see text] and also the hypothesis [Formula: see text] in [Theorem 2 of W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana 11(2) (1995) 355–373] is replaced by [Formula: see text] in our result.

A high order expansion of the renewal function is provided under the assumption that the inter-renewal time distribution is light tailed with finite moment generating function $g$ on a neighborhood of $0$. This expansion relies on complex analysis and is expressed in terms of the residues of the function $1/(1-g)$. Under the assumption that $g$ can be extended into a meromorphic function on the complex plane and some technical conditions, we obtain even an exact expansion of the renewal function. An application to risk theory is given where we consider high order expansion of the ruin probability for the standard compound Poisson risk model. This precises the well- known Cramer-Lundberg approximation of the ruin probability when the initial reserve is large.

Using the techniques of the hypercyclicity criterion, we prove that there is a meromorphic function f(z) on the complex plane whose translates f(z + n) for all n ≥ 1 are dense in the metric space of meromorphic functions on any region in the plane. In additions, we prove the analogue of the result for non-Euclidean translation on the unit disk.

We study the value distribution of L-functions and uniqueness questions of meromorphic functions sharing one or two sets with L-functions. The results in this paper extend the corresponding results from [J. Steuding, Value Distribution of L-Functions, Lect. Notes Math., Vol. 1877, Springer-Verlag, Berlin, Heidelberg, 2007, p. 152] and [B.Q. Li, A result on value distribution of L-functions, Proc. Am. Math. Soc., 138(6):2071–2077, 2010]. The studied questions in this paper are related to the questions of Gross from [F. Gross, Factorization of meromorphic functions and some open problems, in Complex Analysis, Kentucky 1976. Proceedings of the Conference Held at the University of Kentucky, May 18–22, 1976, Lect. Notes Math., Vol. 599, Springer-Verlag, Berlin, Heidelberg, 1977, pp. 51–69]. Here the meromorphic functions are of finitely many poles in the complex plane.

Let f(z) be a transcendental meromorphic function, whose zeros have multiplicity at least 3. Set α(z): = β(z)exp (γ(z), where β(z) is a nonconstant elliptic function and γ(z) is an entire function. If σ(f(z)) > σ(α(z)), then f′(z) = α(z) has infinitely many solutions in the complex plane.

Recently, the degenerate gamma functions were introduced as a degenerate version of the usual gamma function. In this paper, we investigate several properties of these functions. Namely, we obtain an analytic continuation as a meromorphic function on the whole complex plane, the difference formula, the values at positive integers, some expressions following from the Weierstrass and Euler formulas for the ordinary gamma function, and an integral representation as an integral along a Hankel contour.

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Temperature is a critical environmental factor determining the upper limits of evergreen broadleaved tree taxa. However, whether carbon source or carbon sink limitation shapes this limit is not yet fully understood. We studied a subtropical evergreen oak (Cyclobalanopsis gracilis) at the northern limit of its distributional range. Along an elevational/temperature gradient towards its upper limit, we surveyed the variations in non-structural carbohydrate (NSC) concentrations of C. gracilis adults for 3 years. Additionally, a carbon balance manipulation experiment of debudding and defoliation was done to C. gracilis seedlings close to the upper distributional limit, aiming at investigating the changes in NSC concentrations and growth rates in different treatment groups. Our results showed that increasing elevation or decreasing temperature did not affect the trends of NSC concentration in twigs, old branches, or trunks of adults, nor did carbon balance manipulations (debudding or defoliation) of seedlings have a significant effect on the growth, while defoliation decreased NSC concentration in twigs. These results suggest that carbon sink limitation is the key physiological mechanism underlying low temperature in the shaping of this dominant evergreen broadleaved tree species in eastern China. Therefore, the formation of upper limits in evergreen oaks is most likely the result of a direct low-temperature restriction on meristematic activity and tissue formation instead of the result of insufficient carbon supply. More studies with expanded sample sizes are needed on other evergreen broadleaved tree species growing at their upper limits to confirm the carbon sink limitation hypothesis and reveal the detailed mechanisms.

The classical statement of the lemma in question [7], [3] is about meromorphic functions f on ℂ and says thatfor all r > 0, with the possible exception of a set of finite Lebesgue measure. Here T(r, f) is the Nevanlinna characteristic of f. The lemma plays an important role in value distribution theory.