On some conjectures of Aygin and Chan on inequalities of linear combinations of cranks of partitions

Let N(a, L; n) (resp., M(a, L; n)) denote the number of partitions of n with rank (resp., crank) congruent to a modulo L. Recently, Aygin and Chan established the generating functions of some linear combinations of N(a, L; n) and M(a, L; n) and proved some inequalities on N(a, L; n), M(a, L; n) and p(n)/L, where p(n) is the ordinary partition function and \(L\in \{6,9,12\}\) $L\in \{6,9,12\}$ . At the end of their paper, Aygin and Chan posed several conjectures of inequalities on linear combinations of M(a, 12; n) and p(n). In this paper, we prove that Aygin and Chan’s conjectures are true for sufficiently large n by using asymptotic formulas for the Fourier coefficients of eta-quotients due to Chern.