For a sequence of functions $f_{k}(z,s)=\frac{1}{k} \sqrt[s]{Li_s(z^k)}$ with $s>2$ and $Li_{s}(s)$ is the Polylogarithm, I am trying to show

If $\Re f_{1}(e^{\frac{2\pi i}{3}},s) > \Re f_3(1,s)$ then for every $z\in \mathbb{D}-\{0\},$ $\max \Re f_{1}(z,s), \Re f_2(z,s) > \Re f_3(z,s).$

My question is are there anything known about this sequence of functions or their derivatives which could help me in my endeavor?

For a sequence of functions $f_{k}(z,s)=\frac{1}{k} \sqrt[s]{Li_s(z^k)}$ with $s>2$ and $Li_{s}(s)$ is the Polylogarithm, I am trying to show

If $\Re f_{1}(e^{\frac{2\pi i}{3}},s) > \Re f_3(1,s)$ then for every $z\in \mathbb{D}-\lbrace 0\rbrace,$ $\max \Re f_{1}(z,s), \Re f_2(z,s) > \Re f_3(z,s).$

My question is are there anything known about this sequence of functions or their derivatives which could help me in my endeavor?