INTERACTION BETWEEN COEFFICIENT CONDITIONS AND SOLUTION CONDITIONS OF DIFFERENTIAL EQUATIONS IN THE UNIT DISK

where f #(z)= | f ′(z)|/(1 + | f (z)|2) is the spherical derivative of f . In Pommerenke [12] and more recent studies of Heittokangas [8], Benbourenane [3], and Chen and Shon [5], investigations have been made for linear differential equations in the unit disk D regarding the interplay between the behavior of the equation’s coefficients and that of its solutions. In this paper we consider in D the equation


Introduction and statement of results
The concept of a normal function in the unit disk D was introduced by Noshiro [11].He defined a function f to be normal in D if it is a meromorphic function in D for which the set of functions f • S is a normal family in D where S ranges over the conformal mappings of D onto itself.He also characterized normal functions as those meromorphic functions in D for which where is the spherical derivative of f .In Pommerenke [12] and more recent studies of Heittokangas [8], Benbourenane [3], and Chen and Shon [5], investigations have been made for linear differential equations in the unit disk D regarding the interplay between the behavior of the equation's coefficients and that of its solutions.
In this paper we consider in D the equation where k is a positive integer and A is a meromorphic function in D. We note that examples show that if A is a normal function in D, a solution f to (1.2) need not be a normal function.Conversely, if f is a normal function in D which is a solution to (1.2), then the coefficient function A need not be a normal function.However, while resulting functions in these two settings need not be normal, we can establish a measure of their closeness to being normal by considering the wider class of the so-called α-normal functions.
If α is a positive real number, then a meromorphic function (1.3) We denote the set of such α-normal functions by ᏺ α and observe that ᏺ 1 is the set of normal functions.A result of Heittokangas [8,Theorem 5.2] can be used to see that if A is an analytic coefficient in (1.2) and f ≡ 0 is a normal solution of (1.2), then A is in a set of analytic functions shown by Zhu [17,Proposition 7] to be a subset of ᏺ α for some α ≥ 1.
Our first theorem gives estimates restricting the growth of the coefficient A in (1.2) when f is an α-normal solution of (1.2) in D.
Theorem 1.1.Let f be an α-normal solution of (1.2), where the coefficient A is a meromorphic function in D. Then there exist constants C( f ) and P k ( f ) such that (i) (ii) for all z ∈ D.
The estimates in Theorem 1.1 enable us to determine a specific β for which A is βnormal when the behavior of f is restricted further.Theorem 1.2 provides such results.Theorem 1.2.For α ≥ 1, suppose f is an α-normal function which is a solution of (1.2) where A is a meromorphic function in D.
(i) If there exist constants and L such that 0 (ii) If for each compact set K ⊂ D, there exists a constant C(K) such that for all conformal mappings T of D onto itself and all z ∈ K, then A is an (1 + α(k + 1))normal function in D.
(iii) If there exists a number R > 0 and a constant M(R) such that (2) Parts (ii) and (iii) follow from some characterizations of α-normal functions by Wulan [15].
The estimates involved in the proof of Theorem 1.1 lead to the following more generally applicable result.
Theorem 1.4.Let f be an α-normal meromorphic function which is a solution in D of (1.2) where A is a meromorphic function in D. Then (i) for 0 < r < 1 there exist constants C 1 and C 2 depending on f such that where T(r, f ) is the Nevanlinna characteristic function of f at r, (ii) for 0 < r < 1 there exist constants K 1 and K 2 depending on f such that where T(r, f ) is the Nevanlinna characteristic function of f at r.
Lehto and Virtanen [10] showed that if f is a normal meromorphic function in D, then there is a constant K so (1.10) In [14] Shea and Sons studied the class F of functions defined as meromorphic in D for which lim sup If f is a normal function in D which satisfies (1.2) and is bounded, then the integrals in Theorem 1.4 when α = 1 are bounded.Also, if f is a normal function in D which satisfies (1.2) and is of bounded characteristic, then the integrals in Theorem 1.4 when α = 1 are bounded.These latter considerations may be compared with [8, Theorem 4.5] which states the following.
Theorem 1.5.Let A be the analytic coefficient of (1.2) Then any solution f of (1.2) is a function of bounded characteristic in D.
Using Nevanlinna's theory one can also see that for k = 2 and A = (−2(5 − 6z + 3z 2 ))/ (1 − z) 6 , the function f defined by f (z) = exp(1/(1 − z) 2 ) satisfies (1.2) and is not in class F, so f is certainly not a normal function in D. This example provides the expectation that in contrast with the integrand of the double integral in Theorem 1.5, the integrand in Theorem 1.4 involves a logarithm.Additional considerations along the lines of this example may be found in Benbourenane [3], Benbourenane and Sons [4], and Heittokangas [8].In [8] Theorems 3.1.4and 4.3 give restrictions on the growth of a solution f of (1.2) when A is an analytic function for which |A(z)| ≤ α/(1 − |z|) β for z in D where α > 0 and β ≥ 0.
The remaining sections of this paper proceed as follows.Section 2 provides some examples which further illuminate the theorems.Section 3 contains the proof of Theorem 1.1 which relies on a generalization of a result of Lappan [9].The proof of Theorem 1.4 is in Section 4. Finally, Section 5 gives a proof for Theorem 1.2, additional results, and some concluding discussion.
Earlier versions of Theorems 1.1 and 1.2 appeared in Fowler [6].

Examples
Some (2.1) (See Schiff [13] for discussions of these classes of functions.) We relate some examples to (1.2).
Example 2.1.The analytic function f defined in D by K. E. Fowler and L. R. Sons 5 satisfies the equation (2. 3) The function A(z) = −(2 + z)/(1 − z) 2 satisfies |A(z)| ≥ 1/4 in D and is thus a normal function, while f is not a normal function (cf.Hayman and Storvick [7]).So, even for k = 1 a normal coefficient in (1.2) need not lead to a normal function.
Example 2.2.Bagemihl and Seidel noted in [2] that the function f defined in D by where  No information regarding Example 2.2 above is a consequence of parts (i) and (ii) of Theorem 1.2, but part (iii) applies when α = 1 to give A(z) is 2-normal.
Details related to all of the above examples appear in Fowler [6].

Proof of Theorem 1.1
Our proof of Theorem 1.1 will use the following theorem which is a generalization of Lappan [9,Theorem 4].
If f is an α-normal meromorphic function in D, then for each positive integer n, there exists a positive constant P n ( f ) such that for each z ∈ D.
Proof of the lemma.We proceed by induction on n.
For n = 1, the result is trivially true by the definition of an α-normal function.
So we suppose the lemma is true for k < n.Then by Xu [16,Lemma 2] there exists a constant E n ( f ,α,1) such that , g is also an α-normal function in D. We differentiate the equation f (z)g(z) ≡ 1 n times to get and thus where P 0 ( f ) = 1 and we further use Xu [16, Lemma 2].It is easy to see that Hence, setting completes the proof of the lemma.
Proof of Theorem 1.1.Equation (1.2) gives f (k+1) + A f + A f = 0 for all z ∈ D, and thus K. E. Fowler and L. R. Sons 7 We then see for all z ∈ D, (3.8) Hence, using the lemma above, we get for all z ∈ D, and further, To see part (ii) we note that for all z ∈ D (1.2) gives so the lemma shows for all z ∈ D.

Proof of Theorem 1.4
Since f is an α-normal meromorphic function satisfying (1.2), we have f (k+1) + A f + A f = 0 for all z ∈ D and thus 8 Coefficient conditions and solution conditions for all z ∈ D. The lemma in Section 3 implies for z ∈ D, (4.2) Using properties of log + , we then see for z = re iθ in D where K 1 and K 2 are positive constants.
If f has no zeros or poles on |z| = r, we observe that The first fundamental theorem of Nevanlinna's theory gives where K is a constant.Combining this fact with (4.3) and (4.4) gives part (i) of the theorem upon integration.For part (ii) we observe that for z ∈ D, Then using the lemma in Section 3 we have for z ∈ D, for some positive constants C 1 , C 2 , and C 3 .Hence K. E. Fowler and L. R. Sons 9 We note as in part (i) that if f has no zeros or poles on |z| = r, then (4.10) Using the first fundamental theorem of Nevanlinna's theory and combining (4.9) and (4.10) upon integration, we get part (ii) of Theorem 1.4.

Proof of Theorem 1.2 and discussion
The proof of part (ii) of Theorem 1.2 is based on the following theorem of H. Wulan.(5.4)

1 −
well-known examples of normal functions in D are (i) bounded analytic functions in D; (ii) analytic univalent functions in D; (iii) analytic functions in D which omit two values; (iv) meromorphic functions in D which omit three values; and (v) Bloch functions which are analytic functions f in D for which sup z∈D |z| 2 f (z) < ∞.

. 5 )
It is easy to see that 1 ≤ | f (z)| ≤ 3 for z ∈ D, and thus f is a normal function in D. Thus by part (i) of Theorem 1.2, A(z) = − f (k) (z)/ f (z) is (k + 1)-normal in D. It also follows from part (iii) of Theorem 1.2 that A is (k + 1)-normal in D, whereas part (ii) of Theorem 1.2 gives A to be (k + 2)-normal in D.