Let $d$ be an integer and $\sum_{k=0}^dP_k(z)y^{(k)}(z)=0$ be a differential equation over $\mathbb C$, where the $P_k$ are polynomials of degree $\le d$. Consider (if it exists) an entire solution $f$ solution of this equation. Can one assert that $\limsup_{r\to+\infty}\frac{\ln(\ln|f|_r)}{\ln r}\le d$, with $|f|_r=\sup_{|z|<r}\{|f(z)|\}$?
1 Answer
The counterexample is $y'+z^dy=0$, where $P_d=\ldots=P_2=0$. The order of solutions is $d+1$. There is a simple method to determine the orders of entire solutions. It is called the Newton polygon. Plot the points with coordinates $(k,\deg P_k-k)$, $k=0,\ldots,d$ in the plane. Newton's polygon is the smallest concave function on $[0,d]$ whose graph is nowhere below these points. (If $P_k=0$ the corresponding point is omitted). Consider the negative slopes $p_j$ of this function. The possible orders of $f$ are $-p_j$. For example, if all $P_k\neq 0$ then the order is at most $2d/(d+1)$, but in general, it is at most $d+1$ under your conditions.
See Valiron, Fonctions analytiques, or Wasow, Asymptotic expansions for odrinary differential equations, or Hayman MR1438606.