1. For the case of a single variable an obvious condition on polynomials is that a ratio
$$
\frac{P_{n-1}(x)P_{n+1}(x)}{(P_n(x))^2}
$$ 
is monotone. Then  sharp estimates hold true with limits via $P_n(0),P_n(\infty)$ and so polynomial's coefficients.

2. Consider also a reference http://arxiv.org/abs/1207.0936
with examples for sections of exponential series.