Order of growth of solutions to algebraic differential equations in the unit disk

where I is a finite set of distinct tuples (α0,α1, . . . ,αk) for which eachαi is a nonnegative integer, and the aᾱ are meromorphic functions in D = {z | |z| < 1}. For some index sets I, we determine conditions on aᾱ, whereby a meromorphic solution f of (1.1) in D will have finite order of growth as measured by the Ahlfors-Shimizu characteristic function. In [1], Bank investigated (1.1) where I consists of 2-tuples and the aᾱ are arbitrary analytic functions of finite order in the unit disk. He observed that such equations could possess analytic solutions of infinite order in the unit disk, but obtained a uniform growth estimate for all such solutions. He further noted that for arbitrary meromorphic solutions in the disk, no such uniform growth estimate is possible. Recently, Heittokangas [3] showed for certain sets I that each meromorphic solution of (1.1) has finite order when the aᾱ are polynomial functions. Further, he and Wulan [5] studied the equation