Consider the function given by $$f(x)=1-a_1x-a_2x^2-a_3x^3-\cdots$$ where each $a_k\geq0$ and some $a_j>0$. If $f(x)$ is a polynomial then Descartes' Rule of signs tells us there is exactly one positive zero, i.e. root of $f(x)=0$.

Assume $f(x)$ is a (real) power series with radius of convergence $0<R<\infty$.

Question.For which class or classes of such $f$ can we ensure that there is only one positive real root? This is asking for imposing condition(s).