Inequalities for exponential polynomials with applications to moment sequences

Let $\Phi_{\Lambda_{n}}$ be the unique solution of the differential operator $L=\prod_{j=0}^{n}\left( \frac{d}{dx}-\lambda_{j}\right)$ such that $\Phi_{\Lambda_{n}}^{\left( j\right) }\left( 0\right) =0$ for $j=0,...,n-1,$ and $\Phi_{\Lambda_{n}}^{\left( n\right) }\left( 0\right) =1.$ Assume that $\Phi_{\Lambda_{n}}$ is real-valued and $\Phi_{\Lambda_{n} }^{\left( n+1\right) }\left( x\right) \geq0$ for all $x\in\left[ 0,B\right] .$ Then, if a polynomial $R\left( x\right) = {\displaystyle\sum_{k=0}^{n}} a_{k}x^{k}$ is non-negative on the interval $\left[ 0,B\right] ,$ the inequality $${\displaystyle\sum_{k=0}^{n}} a_{k}k!\Phi_{\Lambda_{n}}^{\left( n-k\right) }\left( x\right) \geq R\left( x\right)$$ holds for $x\in\left[ 0,B\right]$. From this we derive several interesting inequalities for exponential polynomials. An important consequence is that for a non-negative measure $\mu$ over the interval $\left[ a,b\right]$ with $b-a<B$ the sequence defined by $$s_{k}:=\int_{a}^{b}k!\Phi_{\Lambda_{n}}^{\left( n-k\right) }\left( x-a\right) d\mu\left( x\right)$$ for $k=0,...,n$ is a moment sequence, i.e. there exists a non-negative measure $\nu$ with support in $\left[ a,b\right]$ such that $s_{k}=\int_{a} ^{b}\left( t-a\right) ^{k}d\nu\left( t\right)$ for $k=0,....,n.$