Polynomization of the Bessenrodt-Ono Type Inequalities for A-Partition Functions

For an arbitrary set or multiset A of positive integers, we associate the A-partition function pA (n) (that is the number of partitions of n whose parts belong to A). We also consider the analogue of the k-colored partition function, namely ,pA,-k(n). Further, we define a family of polynomials f(A,n)(x) which satisfy the equality f(A,n)(k)=pA,-k(n)forall n is an element of Z >= 0 and k is an element of N. This paper concerns a polynomialization of the Bessenrodt-Ono inequality, namely f(A,a)(x)f(A,b)(x)>f(A,a+b)(x), where a,b are positive integers. We determine efficient criteria for the solutions of this inequality. Moreover, we also investigate a few basic properties related to both functions f(A,n)(x) and f '(A,n)(x)