  The Hardy-Littlewood majorant problem has a positive answer only for expo-
nents p which are even integers, while there are counterexamples for all p =2
2N. Montgomery conjectured that there exist counterexamples even among
idempotent polynomials. This was proved recently by Mockenhaupt and Schlag with
some four-term idempotents. However, Mockenhaupt conjectured that even the
classical 1 + e^{2\piix} \pm e^{2\pii(k+2)x} three- term character sums, should
work for all 2k < p < 2k+2 and for all k \in N. In two previous papers we
proved this conjecture for k = 0; 1; 2; 3; 4, i.e. in the range 0 < p < 10, p
\notin 2N. Here we demonstrate that even the k = 5 case holds true. Refinements
in the technical features of our approach include use of total variation and
integral mean estimates in error bounds for a certain fourth order quadrature.
Our estimates make good use of the special forms of functions we encounter:
linear combinations of powers and powers of logarithms of absolute value
squares of trigonometric polynomials of given degree. Thus the quadrature error
estimates are less general, but we can find better constants which are of
practical use for us.
