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.

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

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.

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