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GROWTH OF SOLUTIONS OF SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS WITH EXTREMAL FUNCTIONS FOR DENJOY’S CONJECTURE AS COEFFICIENTS

The classical problem of finding conditions on the entire coefficients A(z) and B(z) guaranteeing that all nontrivial solutions of f ′′ + A(z) f ′ +B(z) f = 0 are of infinite order is discussed. Some such conditions which involve deficient value, Borel exceptional value and extremal functions for Denjoy’s conjecture are obtained.


Introduction and main results
The growth of solutions of complex linear differential equations is one of interesting topic in complex analysis after Wittich's work published [24]. It is studied by using Nevanlinna theory of meromorphic functions. Here we assume that the reader is familiar with the fundamental results and standard notation in Nevanlinna theory, see [12,16,28] for more details.
In general, we use the order of growth to measure the growth of entire functions. For a function f entire in the complex plane C, the order of growth and lower order of growth are given What conditions on A(z) and B (z) will guarantee that every nontrivial solution of (1.1) is of infinite order? Many such results are obtained in the literature. The following theorem on this question is a summary of results due to Gundersen [10], Hellerstein, Miles and Rossi [11], and Ozawa [21].
Theorem A. Suppose that A(z) and B (z) are entire functions satisfying one of the following conditions.
Then every nontrivial solution of (1.1) is of infinite order.
By Theorem A, the main problem left to consider is that whether every nontrivial solution In general, the conclusions are false for these situations. For example, f (z) = exp(P (z)) satisfies the equation where A(z) is an entire function and P (z) is a nonconstant polynomial. For the case of ρ(B ) < ρ(A), there are also some examples [10] showing that a nontrivial solution of (1.1) has finite order.
The problem of finding conditions on A(z) and B (z) for the case ρ(A) ≥ ρ(B ) under which all nontrivial solutions of (1.1) are of infinite order has raised considerable interest, and many parallel results on this question written after Theorem A, see, for example, [3,4,14,15,17,18,22]. It would be interesting to get some relations between the growth of solutions of (1.1) and some deep results in Nevanlinna theory of meromorphic functions. It is well known that deficient value and Borel direction play fundamental role in Nevalinna theory of meromorphic functions and many interesting work done on this aspect, see, for example, [29]. Some relationships between the growth of solutions of (1.1) and deficient value of coefficients are found, see, for example, [15,25]. Borel direction plays a basic role in the theory of angular distribution of meromorphic functions, lots of results can be found, see, for example, [8,20,26,30,31]. In [18], some relationships between the growth of solutions of (1.1) and angular distribution of coefficients are found. To this end, we begin by recalling a conjecture due to Denjoy [6] from 1907, verified by Ahlfors [1] in 1930.
Denjoy's Conjecture. Let f be an entire function of finite order ρ. If f has k distinct finite asymptotic values, then k ≤ 2ρ.
An entire function f is called an extremal function for Denjoy's conjecture if it is of finite order ρ and has k = 2ρ distinct finite asymptotic values. These functions are investigated by Ahlfors [1], Drasin [7], Kennedy [13] and Zhang [32], to mention a few.
An example of an extremal function for Denjoy's conjecture is where q is a positive integer. Then ρ( f ) = q, and f has 2q distinct finite asymptotic values with its corresponding 2q asymptotic curves being where l = 1, 2, . . . , 2q, see [33, p. 210] for more details.
The first result shows the relationship between the fast growth of solutions of (1.1) and extremal functions for Denjoy's conjecture as coefficient. In order to state our results, we also need some definitions. For θ ∈ R, let ∆(θ) = {r e i θ : r ≥ 0}.
For α < β and r 1 , Here we obtain an analogue of Theorem B in which the assumption µ(B ) < 1 2 is replaced with extremal functions for Denjoy's conjecture, and an additional condition of angular distribution is needed.

Auxiliary results
We begin with recalling a lemma on logarithmic derivatives due to Gundersen [9], which plays an important role in proving our results.
such that for all z satisfying arg z = ψ 0 and |z| ≥ R 0 , we have We proceed to recall a property of extremal functions for Denjoy's conjecture, which is needed in proving our results.
where E denotes a subset of S(θ − σ, θ + σ), and satisfies The next lemma is related with Borel exceptional value. The following lemma due to Markushevich [19], see also [2, Lemma 2.3].
Finally, we also need the following result from [23, Corollary 2.3.6]. The proof relies on the Phragmén-Lindelöf principle.
We say that a half straight line L θ : arg z = θ from the origin is called a radial line of order can be found in [5]. There exists at least one the radial angular domain of order ρ( f ) for any entire function f of order ρ( f ) ∈ (0, ∞) by Lemma 2.5.

Proof of Theorem 1.1
If ρ(A) = ∞, then it is clear that ρ( f ) = ∞ for every nontrivial solution f of (1.1). Hence we may assume ρ(A) < ∞. Suppose on the contrary to the assertion that there is a nontrivial solution f of (1.1) with ρ( f ) < ∞. We aim for a contradiction. By our assumption, suppose that a is a finite Borel exceptional value of A(z). Set Then g (z) has two Borel exceptional values 0 and ∞. Applying Lemma 2.3 and Hadamard's factorization theory, g (z) takes the form where h(z) is an entire function satisfying and Q(z) is a polynomial satisfying where λ(g ) denotes the exponent of convergence of zeros of g (z). Therefore, . For any given where j = 0, 1, . . . , 2d − 1. By Lemma 2.4 and ρ(h) for all sufficiently large r , while for any arg for all sufficiently large r , where C is a positive constant. We consider one of the n sectors S i , i = 1, 3, . . . , 2d − 1. Without loss of generality, say S 1 = S(α 1 , β 1 ). This implies (3.2) holds for all z = r e i θ ∈ S(α 1 , β 1 ) and enough large r . Next we consider the two cases appearing in Lemma 2.2.
This is a contradiction for all sufficiently large n. Therefore Theorem 1.1 holds for the case arg z = θ is boundary of S(α, β).

Proof of Theorem 1.3
As reasoning of the proof of Theorem 1.1, we may assume ρ(A) < ∞. Assume on the contrary to the assertion that there is a nontrivial solution f of (1.1) with ρ( f ) < ∞. We aim for a contradiction. Suppose that a is a finite deficient value of A(z) with deficiency δ = δ(a, A). It is an open problem whether the conclusion of Theorem 1.3 still holds if the assumption m(E 0 ) = 0 is removed.