Solutions of Complex Differential Equation Having Pre-given Zeros in the Unit Disc

Abstract

Behavior of solutions of f ′′ + Af = 0 is discussed under the assumption that A is analytic in D and sup z∈D (1 − |z| 2 ) 2 |A(z)| < ∞, where D is the unit disc of the complex plane. As a main result it is shown that such differential equation may admit a non-trivial solution whose zero-sequence does not satisfy the Blaschke condition. This gives an answer to an open question in the literature.It is also proved that Λ ⊂ D is the zero-sequence of a non-trivial solution of f ′′ + Af = 0 where |A(z)| 2 (1 − |z| 2 ) 3 dm(z) is a Carleson measure if and only if Λ is uniformly separated. As an application an old result, according to which there exists a non-normal function which is uniformly locally univalent, is improved.2 > 1 then non-trivial solutions may have infinitely many zeros [13]. The condition A ∈ H ∞ 2 is equivalent to the fact that zero-sequences of non-trivial solutions of (1) are separated with respect

Converse growth estimates for ODEs with slowly growing solutions

“…Since |A| 2 is subharmonic for A ∈ Hol(D), we deduce A ∈ H ∞ 2 whenever |A(z)| 2 (1 − |z| 2 ) 3 dm(z) is a Carleson measure. This Carleson measure condition appears several times in the literature: in connection to solutions of (1) with uniformly separated zeros [11,15] and in relation to solutions in Hardy spaces [14,17].…”

“…2.5. Note that the coefficient condition A ∈ H ∞ 2 allows non-normal solutions by [10,Theorem 3] and [11,Theorem 1]; and even the normality of all solutions is not sufficient for A ∈ H ∞ 2 by Theorem 1(i) above. If f 1 , f 2 ∈ H ∞ are linearly independent solutions of (1) for A ∈ Hol(D), then A ∈ H ∞ 3 by a result of Steinmetz [44, p. 130].…”

“…By [18, Corollary 1.9], Theorem 7 allows us the prescribe any separated Blaschke sequence to be a zero-sequence of a non-trivial solution of (1). Theorem 7 should be compared to [11,Theorem 1], according to which any separated sequence of sufficiently small upper uniform density can appear as a subset of the zero-sequence of a non-trivial solution of (1) under the coefficient condition A ∈ H ∞ 2 . The coefficient condition in Theorem 7 is of different nature as it controls the growth in an average sense.…”

“…There are a lot of known results according to which zeros and critical points (i.e., zeros of the derivative) can be prescribed for solutions of (1) for A ∈ Hol(D). See [11,13,21,22] among many others. For example, the proof of Theorem 3 depends on such an argument.…”

“…Proof Let z ∈ Δ p (z n , δ) for some n, and let S be the supremum in (11). By the maximum modulus principle, there exists ζ δ) .…”