On the relationship between the lower order of coefficients and the growth of solutions of differential equations

Some criteria for entire coefficients A(z)A(z) and B(z)B(z) are given in terms of the lower order forcing the solutions of f″+A(z)f′+B(z)f=0f″+A(z)f′+B(z)f=0 to grow fast. In the literature similar criteria have been published in terms of the usual order. The case when the coefficient A(z)A(z) has an asymptotic growth T(r,A)∼αlog⁡M(r,A)T(r,A)∼αlog⁡M(r,A), α∈(0,1)α∈(0,1), outside of an exceptional set is also discussed. Previously, Laine–Wu (2000) and Kim–Kwon (2001) have made use of this asymptotic growth in the case α=1α=1.