On the oscillation theory of 𝑓”+𝐴𝑓=0 where 𝐴 is entire

Oscillation results on y″ + Ay = 0 in the complex domain with transcendental entire coefficients which have extremal deficiencies

“…Here X(f) is the exponent of convergence of the zero sequence of / They proved the above conjecture in [1] if the order of A, denoted by p(A), is strictly less than \. We note that the order of any solution / to (1.1) must be infinite.…”

“…Let E: = e z " where n^2 is a positive integer. Then Bank and Laine [1] (see also [2]) show that £ is a product of two linearly independent solutions of 1 _ 2z n 2 2n-2 n-2 4 where A(z)= -\e~2 zn and P(z)= -i(n 2 z 2n~2 + 2n(2n-l)z n~2 ).…”

“…We assume on the contrary that E: = f t f 2 with A(£)<oo where f l and f 2 are linearly independent solutions of (3.1). It is well-known [1] that …”

“…Since 1982 there have been many efforts in order to settle the following conjecture of Bank and Laine [1]:…”

Proof of a conjecture of Bank and Laine regarding the product of two linearly independent solutions of 𝑦”+𝐴𝑦=0

“…Let the Wronskian of w, and vv2 be normalized so that it is identically one. Then the product / = wxw2 satisfies the differential equation (2). This implies that the order of / is > pA. Let p be the order of / and we assume that p < 1.…”