Let $cd(G)$ be the set of degrees of the irreducible complex characters of the finite group $G$.


It is conjectured that if $G$ is solvable then $dl(G)\leq |cd(G)|$ and it is a result by Gluck that $dl(G) \leq 2|cd(G)|$.


I have managed to show that $dl(G)\leq 2|cd(G)|-3$ and I was wondering if this is a well-known bound and whether even better bounds are known.
Edit Forgot to mention that this is for $|cd(G)|\geq 3$.



Is this a well-known bound for the derived length of a finite group?
In order to try getting this question off the unanswered list, here is an answer:


The result turned out not to be previously known, and has now been published in Communications in Algebra 40 (2012), no. 5, 1856–1859, "Bounding the derived length of a solvable group: an improvement on a result by Gluck" (<http://ams.org/mathscinet-getitem?mr=2924487>).


For large values of $|cd(G)|$ however, a better bound is known, namely $dl(G)\leq |cd(G)| + 24\rm{log}\_2(|cd(G)|) + 364$ which gives a better bound when $|cd(G)|\geq 588$ (this bound is due to Thomas Keller).


There are also better bounds known if one puts various assumptions on the group $G$. For example, if $|G|$ is odd it is a result of Berger that $dl(G)\leq |cd(G)|$, and as mentioned in the question, this bound is in fact conjectured to hold for all finite solvable groups (I just picked what is probably the simplest of the conditions known to be sufficient for the inequality to hold, as there are many others, even strictly weaker ones that $|G|$ being odd).


