Composition of entire functions with finite iterated order

In this paper, we investigate the growth of two composite entire functions of finite iterated order and a series of comparative growths of logp+qT(r,f(g)) (p,q∈N) with logpT(r,f) and logqT(r,g). At the end of this paper, we apply some growth results into the factorization of the solutions of linear differential equation. We achieve some results which are the improvements and extensions of the previous results.     

Uniqueness of entire functions

Let f be a nonconstant entire function and let a be a meromorphic function satisfying T(r,a)=S(r,f) and a?a′. If f(z)=a(z)⇔f′(z)=a(z) and f(z)=a(z)⇒f″(z)=a(z), then f≡f′, and a?a′ is necessary. This extended a result due to Jank, Mues and Volkmann.     

Approximation numbers for composition operators on spaces of entire functions

We study the degree of compactness of composition operators $C_{\phi }$ acting on weighted Hilbert spaces of entire functions, which include (i) the space of entire Dirichlet series, (ii) the space of entire power series, and (iii) the Fock space (we must have $\phi \left(z\right)=az+b$, and it is known that $C_{\phi }$ is compact if and only if $\left|a\right|<1$). More precisely, the sequence $\left(a_n\right)$ of approximation numbers of $C_{\phi }$ is investigated: for (i), we give the exact formula for $\left(a_n\right)$, while for (ii) and (iii) we give upper and lower estimates for $a_n$, showing that $a_n$ behaves like ${\left|a\right|}^n$ up to a subexponential factor. In particular, $C_{\phi }$ belongs to all Schatten classes $S_p,p>0$ as soon as it is compact.     

Uniqueness and value-sharing of entire functions

In this paper, we study the uniqueness problems on entire functions sharing one value with the same multiplicities. We generalize and unify some previous results of Fang and Hua [M.L. Fang, X.H. Hua, Entire functions that share one value, J. Nanjing Univ. Math. Biquarterly 13 (1) (1996) 44-48], Yang and Hua [C.C. Yang, X.H. Hua, Uniqueness and value-sharing of meromorphic functions, Ann. Acad. Sci. Fenn. Math. 22 (2) (1997) 395-406] and Fang [M.L. Fang, Uniqueness and value-sharing of entire functions, Comput. Math. Appl. 44 (2002) 828-831].     

Uniqueness of entire function sharing a small function with its derivative

This paper is devoted to studying the relationship between an entire function and its derivative when they share one small function. We generalize some previous results of Gundersen and Yang [G. Gundersen, L.Z. Yang, Entire functions that share one value with one or two of their derivatives, J. Math. Anal. Appl. 223 (1998) 85–95], Chang and Zhu [J. Chang, Y. Zhu, Entire functions that share a small function with their derivatives, J. Math. Anal. Appl. 351 (2009) 491–496].     

Normal families and uniqueness theorems for entire functions

There exists a set S with 3 elements such that if f is a non-constant entire function satisfying E(S,f)=E(S,f′), then f≡f′. The number 3 is best possible. The proof uses the theory of normal families in an essential way.     

Complete systems of partial derivatives of entire functions and frequently hypercyclic operators

We find some sufficient conditions for a system of partial derivatives of an entire function to be complete in the space H(C^d)$H({\mathbb{C}}^d)$ of all entire functions of d   variables. As an application of this result we describe new classes of frequently hypercyclic operators on H(C^d)$H({\mathbb{C}}^d)$.     

Entire functions sharing an entire function of smaller order with their difference operators

We study a uniqueness question of entire functions order with their difference operators, and deal with a question in this paper extend the corresponding results obtained by Liu Examples are provided to show that the results in this paper, in sharing an entire function of smaller posed by Liu and Yang. The results -Yang and by Liu-Laine respectively. a sense, are the best possible.     

Entire functions that share a small function with its derivative

In this paper, by studying the properties of meromorphic functions which have few zeros and poles, we find all the entire functions f(z) which share a small and finite order meromorphic function a(z) with its derivative, and f(n)(z)−a(z)=0 whenever f(z)−a(z)=0 (n?2). This result is a generalization of several previous results.     

On the uniqueness problems of entire functions and their derivatives

In this paper, we obtain some uniqueness theorems for entire functions and their derivatives sharing the same fixed points with the same multiplicities.     

On theorems of Gelfond and Selberg concerning integral-valued entire functions

For each sN define the constant θ_s with the following properties: if an entire function g(z) of type t(g)<θ_s satisfies then g is a polynomial; conversely, for any δ>0 there exists an entire transcendental function g(z) satisfying the display conditin and t(g)<θ_s+δ. The result θ₁=log2 is known due to Hardy and Pólya. We provide the upper bound θ_sπs/3 and improve earlier lower bounds due to Gelfond (1929) and Selberg (1941).     

Composite entire functions with no unbounded Fatou components

Let L be the set of all entire functions f such that for given ?>0,

logL(r,f)>(1−?)logM(r,f)     

The uniqueness of an entire function sharing a small entire function with its derivatives

In this paper, we prove a theorem on the growth of a solution of a linear differential equation. From this we obtain some uniqueness theorems concerning that a nonconstant entire function and its derivatives sharing a small entire function. The results in this paper improve many known results. Some examples are provided to show that the results in this paper are the best possible.     

Backward bifurcation of an epidemic model with saturated treatment function

An epidemic model with saturated incidence rate and saturated treatment function is studied. Here the treatment function adopts a continuous and differentiable function which can describe the effect of delayed treatment when the number of infected individuals is getting larger and the medical condition is limited. The global dynamics of the model indicate that the basic reproduction number being the unity is a strict threshold for disease eradication when such effect is weak. However, it is shown that a backward bifurcation will take place when this delayed effect for treatment is strong. Therefore, driving the basic reproduction number below the unity is not enough to eradicate the disease. And a critical value at the turning point is deduced as a new threshold. Some sufficient conditions for the disease-free equilibrium and the endemic equilibrium being globally asymptotically stable are also obtained. Mathematical results in this paper suggest that giving the patients timely treatment, improving the cure efficiency and decreasing the infective coefficient are all valid methods for the control of disease.     

Simultaneous approximation and interpolation of increasing functions by increasing entire functions

We prove that, under suitable assumptions, an isomorphism g of dense subsets A,B of the real line can be taken to approximate a given increasing Cⁿ surjection f with the derivatives of g agreeing with those of f on a closed discrete set. For example, we have the following theorem. Let be a nondecreasing Cⁿ surjection. Let be a positive continuous function. Let be a closed discrete set on which f is strictly increasing. Let each of {A_i}, {B_i} be a sequence of pairwise disjoint countable dense subsets of such that for each and xE we have xA_i if and only if f(x)B_i. Then there is an entire function such that and the following properties hold.

(a) For all , Dg(x)>0.

(b) For k=0,…,n and all , |D^kf(x)−D^kg(x)|<ε(x).

(c) For k=0,…,n and all xE, D^kf(x)=D^kg(x).

(d) For each , g[A_i]=B_i.

This provides a version for increasing functions of a theorem of Hoischen. In earlier work, we proved that it is consistent that a similar theorem, omitting clause (c), holds when the sets A_i,B_i are of cardinality ₁ and have second category intersection with every interval. (See the introduction for the exact statement.) In this paper, we show how to incorporate clause (c) into the statement of the earlier theorem.

On Bernstein type theorems concerning the growth of derivatives of entire functions

A subspace of which is invariant under all left translation operators is called admissible if is a Banach space satisfying the following properties:

(i) If then there exists a subsequence such that almost everywhere.

(ii) The group is a bounded strongly continuous group. In this case, let

Typical admissible spaces are and all spaces for More generally, all of the Peetre interpolation spaces of two admissible spaces are also admissible.

A function is called subexponential if for every With these definitions our main result goes as follows: . If is an entire function of exponential type such that its restriction to the real axis, denoted by , is subexponential and belongs to some admissible space then the derivative is also in Moreover,
for each real

This result yields as consequences and in a systematic way many new and old Bernstein type inequalities.

     

An entire function and its derivatives sharing a polynomial

In this paper, we study the growth of solutions of a first-order linear differential equation and that of a second-order linear differential equation. From this we obtain some uniqueness theorems of a nonconstant entire function and its first derivative having the same fixed points with the same multiplicities. The results in this paper also improve some known results. Some examples show that the results in this paper are best possible.     

Unicity of entire functions and their linear differential polynomials

We study the relationship between an entire function f and its certain type of linear differential polynomial L when f and L share one finite nonzero value under some additional conditions. The results improve and generalize some previous results obtained by C.C. Yang and some other authors.     

On a non-standard class of entire functions

In this article we construct a class of entire functions of arbitrary finite order and with non-zero Taylor coefficients, which is not written in the form of Hadamard's canonical product. Some basic properties of this class are also given.     

On sums of roots of infinite order entire functions

A class of infinite order entire functions is considered. Estimates for sums of the roots are derived. These estimates supplement the Hadamard theorem. Moreover, we establish a new estimate for the counting function of the roots, which in appropriate situations can be more useful than the Jensen inequality.