Give bounds for degree of “decreasing” polynomial

Let $$p$$ be a polynomial of minimal degree to which the following is true:

$$p(0) > \sum\limits_{i=1}^n \lvert p(i) \rvert + \sum\limits_{i=1}^m \lvert p(-i) \rvert$$

Give upper and lower bounds for $$deg(p)$$ (for sufficently large $$n$$).

I constructed bounds for $$p(0) > \sum\limits_{i=1}^n \lvert p(i) \rvert$$ using Chebyshev-polynomials, which is $$c_1\sqrt{n} < deg(p) < c_2\sqrt{n}$$, where $$c_1$$ and $$c_2$$ are constants. However, I'm having trouble with both sums.

• Is there anything missing from the question? It is possible to get polynomials of arbitrarily high degree with $p(0)=1$ and $p(\pm i)=0$ for $|i|\leq\max\{m,n\}$. So there cannot be an upper bound. – Martin Argerami Mar 20 '14 at 1:44
• @MartinArgerami Sorry, you're right. Fixed. – mathaway__ Mar 20 '14 at 4:30