Are there any known bounds on this function? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T15:01:19Z http://mathoverflow.net/feeds/question/90005 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90005/are-there-any-known-bounds-on-this-function Are there any known bounds on this function? Daniel Parry 2012-03-02T01:05:56Z 2012-03-02T02:01:18Z <p>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 <a href="http://en.wikipedia.org/wiki/Polylogarithm" rel="nofollow">Polylogarithm</a>, I am trying to show </p> <p>If $\Re f_{1}(e^{\frac{2\pi i}{3}},s) > \Re f_3(1,s)$ then for every $z\in \mathbb{D}-\lbrace 0\rbrace,$ $\max \Re f_{1}(z,s), \Re f_2(z,s) > \Re f_3(z,s).$</p> <p>My question is are there anything known about this sequence of functions or their derivatives which could help me in my endeavor?</p>