Let $E$ be an elliptic curve over $\mathbb{Q}$. It is known from Gross and Zagier that if $\textrm{rank}\_{\textrm{an}}(E) \leq 1$, then


$$\textrm{rank}(E) \geq \textrm{rank}\_{\textrm{an}}(E).$$ 


Instead, suppose I know that $\textrm{rank}(E) \leq 1$, are there any (unconditional) inequalities that relate rank and analytic rank of $E$?



Inequality relating rank and analytic rank
This is not really the right question. I think it's easier to just list the two things that are known unconditionally and then you can figure out if this answers your question.


1) If analytic rank is zero then algebraic rank is zero.


2) If analytic rank is 1 then algebraic rank is 1.


Proof: see quid's comment.


One now deduces easily that if the algebraic rank is at most 1 then the analytic rank is at least the algebraic rank.


