lajos/Bounds on the coefficients of a polynomial having all derivatives non-negative

Proposition. Let \(s\) denote a fixed positive integer and set \(P(z):=1+\sum_{n=1}^s a_n z^n\) with some real coefficients \(a_n\). Fix any \(r>0\) and suppose that \(P^{(k)}(-r)\ge 0\) for all \(k=0,1,\ldots,s\). Then for each index \(1\le n\le s\) we have

\[ 0\le a_n \le \frac{\binom{s}{n}}{r^n}. \]

Moreover, if there is at least one \(n\) \((1\le n\le s)\) with \(a_n = \frac{\binom{s}{n}}{r^n}\), then \(P(z)=\left(1+\frac{z}{r}\right)^s.\)

Proof. By assumption, with \(\gamma_k :=\frac{P^{(k)}(-r)}{k!}r^k\) we have \(\gamma_k\ge 0\) for \(k=0,1,\ldots,s\). Taylor expansion, the binomial theorem and interchanging the order of summations show that

\[P(z)=\sum_{k=0}^s \gamma_k \left(1+\frac{z}{r}\right)^k= \sum_{k=0}^s \sum_{n=0}^{k}\gamma_k \binom{k}{n}\frac{z^n}{r^n}= \sum_{n=0}^s \sum_{k=n}^{s}\gamma_k \binom{k}{n}\frac{z^n}{r^n}= \sum_{n=0}^s \left(\frac{1}{r^n}\sum_{k=n}^{s}\gamma_k \binom{k}{n}\right)z^n.\]

Now fix \(1\le n\le s\). The coefficient of \(z^n\) in the leftmost and rightmost expressions is equal, hence

\[ a_n=\frac{1}{r^n}\sum_{k=n}^{s}\gamma_k \binom{k}{n}, \] further, from the equality of the constant term, we get \[\sum_{k=0}^{s}\gamma_k=1 \quad \quad\quad (*)\]

Non-negativity of the \(\gamma_k\) coefficients implies \(a_n\ge 0\) and (*) shows \(0\le \gamma_k\le 1\) for all \(0\le k\le s\).

Finally we seek the maximum of \(\sum_{k=n}^{s}\gamma_k \binom{k}{n}\) knowing \(0\le \gamma_k\) and (*). Suppose that \(\gamma_m>0\) for some \(m\) with \(0\le m\le s-1\). Then we define \(\widetilde{\gamma_s}:=\gamma_s+\gamma_m\), \(\widetilde{\gamma_m}:=0\) and \(\widetilde{\gamma_k}:=\gamma_k\) for \(s\ne k\ne m\). Clearly, \(0\le \widetilde{\gamma_k}\) for each \(0\le k\le s\) and \(\sum_{k=0}^{s}\widetilde{\gamma_k}=1\), but \(-\)by using the convention that \(\binom{m}{n}=0\) if \(m < n\) \(-\)

\[ \sum_{k=n}^{s}\widetilde{\gamma_k} \binom{k}{n}-\sum_{k=n}^{s}\gamma_k \binom{k}{n}=\binom{s}{n}\gamma_m+\binom{m}{n}(-\gamma_m)= \gamma_m \left(\binom{s}{n}-\binom{m}{n}\right), \] and this last expression is strictly positive, because \(1\le n\le s\) is fixed, \(s > m\) and \(\gamma_m>0\). We can therefore conclude that (for fixed \(s\ge 1\) and \(1\le n\le s\)) the value of \(\sum_{k=n}^{s}\gamma_k \binom{k}{n}\) over all \(0\le \gamma_k\) (\(k=0,1,\ldots,s\)) and under condition (*) is maximal if and only if \(\gamma_0=\gamma_1=\ldots=\gamma_{s-1}=0\) and \(\gamma_s=1\).

The above property establishes the upper bound on \(a_n\) and the uniqueness part as well.