Let $f(x)=(1-x)^e (1+x)^{(n-e)}= \sum_{i=0}^n a_ix^i$, where $n$ is a positive integer and $e$ is a non-negative integer less than $n$. I want to find an upper bound on $\sum_{i=0}^n |a_i|$ other than the trivial upper bound $2^n$. Also for $e=0,1,\frac{n}{2}$, it is easy. Is there any integration type of approach for this problem?

Let $f(x)=(1-x)^b (1+x)^{(n-b)}= \sum_{i=0}^n a_ix^i$, where $n$ is a positive integer and $b$ is a non-negative integer less than $n$. I want to find an upper bound on $\sum_{i=0}^n |a_i|$ other than the trivial upper bound $2^n$. Also for $b=0,1,\frac{n}{2}$, it is easy. Is there any integration type of approach for this problem?

Let f(x)=(1-x)^e (1+x)^{(n-e)}= \sum_{i=0}^n a_ix^i, where n is a positive integer and e is a non-negative integer less than n. I want to find an upper bound on \sum_{i=0}^n |a_i| other than the trivial upper bound 2^n. Also for e=0,1,\frac{n}{2}, it is easy. Is there any integration type of approach for this problem?

Let $f(x)=(1-x)^e (1+x)^{(n-e)}= \sum_{i=0}^n a_ix^i$, where $n$ is a positive integer and $e$ is a non-negative integer less than $n$. I want to find an upper bound on $\sum_{i=0}^n |a_i|$ other than the trivial upper bound $2^n$. Also for $e=0,1,\frac{n}{2}$, it is easy. Is there any integration type of approach for this problem?