Suppose that we have two implicit functions
$$f(t)=\log\frac{t}{2}+O\left(\frac{1}{t}\right),\, g(t)=\log t+O\left(\frac{1}{t}\right).$$
Let $z(t)$ be a function such that $f(t)< z(t) < g(t)$, when $t$ is sufficiently large.
Can the last inequalities be replaced by any known asymptotic notations?
A bit more general version of my question could be "Let $g(t)-f(t)=constant+o(1)$ and $f(t)< z(t) < g(t)$, as $t\to\infty$. Can the last inequalities be expressed by any known asymptotic notations?"
The idea of my question is that the constants implied by $O$ might be different and it is confusing to write $\log\frac{t}{2}+O\left(\frac{1}{t}\right)<z(t)<\log t+O\left(\frac{1}{t}\right)$ or $\log\frac{t}{2}<z(t)+O\left(\frac{1}{t}\right)<\log t$, as $t \to \infty$.