Let $h(t)=\sqrt{2t\log\log t^{-1}}$, and for $\theta \in (0,1) $ consider the affine function $$l_n(t):=\beta_n+\alpha_n\frac{t}{2}$$ where $\alpha_n:=h\left(\theta^n\right)/\theta^n$ and $\beta_n:=h(\theta^n)/2$. Than for any $t\in\left[\theta^n,\theta^{n-1}\right]$ we have $l_n(t) \le h(t)/\theta$.
I've tried monotonicity and derivative arguments, but than I saw that h is not monotone, and I'm out of ideas, so if anyone can hel it would be greatly appreciated.