It is assumed that the reader is familiar with the standard notations and basic results of Nevanlinna's value distribution theory in the complex plane C [11, 20]. The order and hyper order of an entire function f are defined as follows:

respectively, where M(r,f) denotes the maximum modulus of f on the circle |z|=r.

If f and g are two meromorphic functions in the complex plane, f and g share a constant a CM if f-a and g-a have the same zeros with the same multiplicities. Rubel and Yang [16] proved for a nonconstant entire function that if f and its derivative f' share two finite distinct values CM, then f≡ f'. Later on, Brück [1] constructed entire functions with integer or infinite hyper order to show that f and f' share 1 CM do not satisfy f≡ f'. Therefore, Brück proposed the following conjecture [1].

Brück Conjecture. Let f be a nonconstant entire function such that its hyper order is finite and not a positive integer. If f and f' share one finite value a CM, then f'-a=c(f-a), where c is a nonzero constant.

In general, the conjecture is false for a meromorphic function f (a counterpart is presented in [10]). Brück [1] showed that the conjecture is valid for the case a=0. Afterward, Gundersen and Yang [10] proved that the conjecture is true for the case in which f is of finite order. Furthermore, Chen and Shon [5] showed that the conjecture is also true when f is of hyper order strictly less than 1/2. Thus, to solve this conjecture, the remaining case in which the hyper order of f is between [1/2,+∞)\ℕ should be considered; however, many attempts have failed.

There are many results that are closely related to the Brück conjecture; they can be mainly classified into two types. One type comprises generalizing the shared value a to a nonconstant function, such as polynomial, entire small function of f, or entire functions with lower orders than that of f, (e. g. see [2, 3, 4, 12, 13, 17]); the other type comprises improving the first derivative of f to an arbitrary k-th derivative (e. g. see [2, 6, 7, 13, 18]).

To study this conjecture, F(z)=f(z)a−1 is chosen. Thus F(z) is an entire function, and ρ(F)=ρ(f),ρ2(F)=ρ2(f). Because f and f' share one finite value a CM, according to the Hadamard theorem,

where h(z) is an entire function. Hence, F satisfies the following linear differential equation:

Yang [19] converted the conjecture into a question: Is it true that if h(z) is a nonconstant entire function, the hyper order of F satisfying Equation (1.2) is a positive integer or infinity?

When h(z) is a polynomial or transcendental entire function with order less than 1/2, the case is true, see[2, 5]. However, the answer for the remaining case with $\rho(h)\geq 1/2$ is unknown.

The following theorem provides a partial answer to the previously presented question for the case ρ(h)=1/2.

Theorem 1.1 There exists an entire function h(z) with order 1/2. Thus -e^h(z) is of hyper order 1/2. Let it be the coefficient of a differential equation

Consequently, the hyper order of the solutions of Equation (1.3) is infinite.

The Koenigs function is considered first. For β∈(0,1/e), the function Eβ(z):=eβz has a repelling fixed point ξ on R with multiplier λ=βξ>1. Around this repelling fixed point, there exists a unique local holomorphic solution &#_120509; of the Schröder's functional equation according to Koenigs:

It is normalized by Φ(ξ)=0 and Φ′(ξ)=1 (e.g., [14]). Moreover, &#_120509; increases on the real axis and approaches infinite, i.e. limx→∞Φ(x)=∞; however, it grows more slowly than any iterate of the logarithm. In other words, the following expression can be obtained for all m∈ℕ:

where Łog^m denotes the m-th iterate of the logarithm [15]. For the purpose of this study, ε:(ξ,∞)→(0,+∞) is defined:

This function approaches zero more slowly than any of the function 1/logmx, where m∈ℕ. In the next step, the following expression is constructed:

for some r≥r0. Thus, ρ(r) is a proximate order (see [8], Lemma 3.5]). The definition of proximate order is as follow:

Definition 2.1.(Proximate order)

A function ρ(r) defined on [r0,∞), where r0>0, is called a proximate order if it satisfies the following conditions:

• (1) ρ(r)≥0;

• (2) limr→∞ρ(r)=ρ;

• (3) ρ(r) is continuously differentiable on [r0,∞);

• (4) limr→∞rρ′(r)logr=0.

In [8], the author constructed an entire function by using the Weierstrass canonical product:

where {an}n∈ℕ is a positive sequence tending to infinity such that 1≤a0≤a1≤⋯ and n(r)=rρ(r)+O(ε(r)3rρ(r)). Here n(r) is the number of the element of the sequence {a_k}, which are contained in the disc {z:|z|<r}; ρ(r) is defined in (2.2). According to the result of Borel (see [9, Theorem 3.4]), the order of the Weierstrass canonical product h(z) is equal to the order of n(r); thus the order of h(z) is 1/2.

Lemma 2.2.([8, Lemma 3.7, 3.8]) Based on the previously defined h(z),ε(r) and ρ(r), the following expression can be constructed:

for ε(r)≤θ≤2π−ε(r) as r→∞.

The following expressions are true for some r0>0:

and

The following fact is a consequence of Lemma 2.2.

Lemma 2.3.([8, Lemma 3.7, 3.8]) The function h(z) is bounded on γ+ and γ− and in G(γ).

Lemma 2.4.([6]) Let f(z) be an entire function of infinite order; the hyper order is ρ₂(f); note the central index of f by ν(r). Consequently, the following expression holds true:

Lemma 2.5.([6]) Let f(z) be an entire function of infinite order ρ(f)=∞ and ρ2(f)=α<+∞; moreover, the set E⊂(1,∞) has a finite logarithmic measure. Hence, there exists a sequence {zk=rkeiθk} such that

If α>0, for any given ε(0<ε<α), there is a sufficiently large r_k such that

If α=0, for any large N>0, there is a sufficiently large r_k such that

Evidently, the solutions of Equation (1.3) are of infinite order. The hyper order is assumed to be ρ2(F)=α<+∞, and the assertion is obtained through the reduction to a contradiction. As in Lemma 2.5, a sequence {zk=rkeiθk} can be chosen such that

According to Equation (1.3), the following expression holds true:

By applying Wiman-Valiron Theorem to Equation (3.1), the following expressions can be obtained:

and

If θ0=0, there exists a subsequence {zkj=rkjeiθkj} of {zk=rkeiθk} such that zkj∈G(γ). Because F is of infinite order ν(rjk)>|zjk|N for any large N>0. In addition, according to Lemma 2.3, |h(zjk)| is bounded. Thus, (3.3) presents a contradiction.

If θ0∈(0,2π), there exists a subsequence {zkl=rkleiθkl} of {zk=rkeiθk} such that zkl∈ℂ∖G(γ). Based on Lemma 2.4, the following expressions can be obtained:

and

Taking the principal branch of the logarithm of Equation (3.2) on both sides leads to the following term:

Thus, because ν(r)>|z|N for any large N>0,

Lemma 2.4 and ρ2(F)=α lead to the following equation:

for sufficiently large r. This results in the following expression:

for sufficiently large r. The combination of (3.4), (3.5) and (3.9) leads to a contradiction for l→+∞. Thus, the proof is completed.