Inequalities for exponential polynomials with applications to moment sequences

Let ${\mathrm{\Phi }}_{{\mathrm{\Lambda }}_n}$ be the unique solution of the differential operator $L={\prod }_{j=0}^n(\frac{d}{dx}-{\lambda }_j)$ such that ${\mathrm{\Phi }}_{{\mathrm{\Lambda }}_n}^{\left(j\right)}\left(0\right)=0$ for $j=0,\mathrm{.}\mathrm{.}\mathrm{.},n-1,$ and ${\mathrm{\Phi }}_{{\mathrm{\Lambda }}_n}^{\left(n\right)}\left(0\right)=1.$ Assume that ${\mathrm{\Phi }}_{{\mathrm{\Lambda }}_n}$ is real-valued and ${\mathrm{\Phi }}_{{\mathrm{\Lambda }}_n}^{(n+1)}\left(x\right)\ge 0$ for all $x\in [0,B]\mathrm{.}$ Then, if a polynomial $R\left(x\right)={\displaystyle \sum _{k=0}^n}{a}_k{x}^k$ is non-negative on the interval $[0,B],$ the inequality $${\displaystyle \sum _{k=0}^n}{a}_{k}k!{\mathrm{\Phi }}_{{\mathrm{\Lambda }}_n}^{(n-k)}\left(x\right)\ge R\left(x\right)$$ holds for $x\in [0,B]$. From this we derive several interesting inequalities for exponential polynomials. An important consequence is that for a non-negative measure $\mu$ over the interval $[a,b]$ with $b-a<B$ the sequence defined by $$s_k:={\int }_a^{b}k!{\mathrm{\Phi }}_{{\mathrm{\Lambda }}_n}^{(n-k)}(x-a)d\mu \left(x\right)$$ for $k=0,\mathrm{.}\mathrm{.}\mathrm{.},n$ is a moment sequence, i.e. there exists a non-negative measure $\nu$ with support in $[a,b]$ such that $s_k={\int }_a^b{(t-a)}^{k}d\nu \left(t\right)$ for $k=0,\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.},n\mathrm{.}$