asymptotic behavior of the real part of the Riemann zeta function for $0<\sigma<1$

Question

consider the zeta function $\zeta(\sigma+it)$ for $\sigma>1$ : $$\zeta(\sigma+it)=\sum_{n=1}^{\infty}\frac{1}{n^{\sigma+it}}$$ And: $$\zeta(\sigma-it)=\sum_{n=1}^{\infty}\frac{1}{n^{\sigma-it}}$$ From the identity $\zeta\left(\overline{s} \right )=\overline{\zeta(s)}\;\;$, we have: $$\Re(\zeta(\sigma+it))=\sum_{n=1}^{\infty}\frac{\cos(t\ln n)}{n^{\sigma}}\leq\zeta(\sigma)$$ Thus, for $\sigma>1$ the behavior of $\Re(\zeta(\sigma+it))$ is largely governed by its values along the real line. Now, using the functional equation of the zeta function, can we obtain similar results, on the asymptotic behavior/upper bound of $\Re(\zeta(\sigma+it))$ for $0<\sigma<1$!?