The inequality at the top of page 99 in Lectures on The Riemann Zeta–Function gives $$|\zeta(\tfrac{1}{2}+it)|<9t^{1/2},\;\;t\geq 1.$$ To cover the whole the interval $t\geq 0$, just add $3/2$ to the right-hand-side of the inequality.

The inequality at the top of page 99 in Lectures on The Riemann Zeta–Function gives $$|\zeta(\tfrac{1}{2}+it)|<9t^{1/2},\;\;t\geq 1.$$ To cover the whole the interval $t\geq 0$, just add $3/2$ to the right-hand-side of the inequality.

The inequality $|\zeta(\tfrac{1}{2}+it)|<\tfrac{1}{4}+t^2$ holds for $t> 0.8$.

The inequality at the top of page 99 in Lectures on The Riemann Zeta–Function gives $$|\zeta(\sigma+it)|<9t^{1/2},\;\;t\geq 1,\;\;\sigma\geq \tfrac{1}{2}.$$ This still excludes the interval $0\leq t<1$, but at least it's not a large-$t$ result.

The inequality at the top of page 99 in Lectures on The Riemann Zeta–Function gives $$|\zeta(\tfrac{1}{2}+it)|<9t^{1/2},\;\;t\geq 1.$$ To cover the whole the interval $t\geq 0$, just add $3/2$ to the right-hand-side of the inequality.