Stieltjes moment sequences and positive definite matrix sequences

For a certain constant $\delta>0$ (a little less than $1/4$), every function $f\colon {\mathbb N}_0\to \left]0,\infty\right[$ satisfying $f(n)^2\leq\delta f(n-1)f(n+1)$, $n\in {\mathbb N}$, is a Stieltjes indeterminate Stieltjes moment sequence. For every indeterminate moment sequence $f\colon {\mathbb N}_0\to {\mathbb R}$ there is a positive definite matrix sequence $(a_n)$ which is not of positive type and which satisfies $\operatorname{tr}(a_{n+2})=f(n)$, $n\in {\mathbb N}_0$. For a certain constant $\varepsilon>0$ (a little greater than $1/6$), for every function $\varphi\colon {\mathbb N}_0\to \left]0,\infty\right[$ satisfying $\varphi(n)^2\leq\varepsilon\varphi(n-1)\varphi(n+1)$, $n\in {\mathbb N}$, there is a convolution semigroup $(\mu _t)_{t\geq 0}$ of measures on ${\mathbb R}_+$, with moments of all orders, such that $\varphi(n)=\int x^n\,d\mu _1(x)$, $n\in {\mathbb N}_0$, and for every such convolution semigroup $(\mu _t)$ the measure $\mu _t$ is Stieltjes indeterminate for all $t>0$.