Sufficient conditions for a linear operator on $\mathbb{R}[x]$ to be   monotone
  We demonstrate that being a hyperbolicity preserver does not imply monotonicity for infinite order differential operators on $\mathbb{R}[x]$, thereby settling a recent conjecture in the negative. We also give some sufficient conditions for such operators to be monotone. 
Leah Buck, Kelly Emmrich and Tam\'as Forg\'acs
math.CV