Some inequalities between M(a, b, c; L; n) and the partition function p(n)

Let p(n) and M(m, L ; n) be the number of partitions of n and the number of partitions of n with crank congruent to m modulo L, respectively, and let M(a, b, c ; L ; n) := M(a, L ; n) + M(b, L ; n) + M(c, L ; n) . In this paper, we focus on some relations between M(m, L ; n) and p(n) , which Dyson, Andrews, and Garvan etc. have studied previously. By applying a modification of the circle method to estimate the Fourier coefficients of generating functions, we establish the following inequalities between M(a, b, c ; L ; n) and p(n) : for n >= 467, M(0, 1, 1; 9; n) > p(n)/3 when n equivalent to 0,1, 5, 8(mod 9), M(0, 1, 1; 9; n) < p(n)/3 when n equivalent to 2, 3, 4, 6, 7(mod 9), M(2, 3, 4; 9; n) < p(n)/3 when n equivalent to 0, 1, 5, 8(mod 9), M(2, 3, 4; 9; n) > p(n)/3 when n equivalent to 2,3,4, 6, 7(mod 9). In the proof of these inequalities, an inequality for the logarithm of the generating function for p(n) is derived and applied. Our method reduces the last possible counterexamples to 467 <= n <= 22471, and it will produce more effective estimates when proving inequalities of such types.