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 $|f|\_r=\sup\_{|z|<r}\{|f(z)|\}$? 



growth of an entire solution of a differential equation
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.


