In this paper, we use $(p,q)$-integral to establish some Fejér type inequalities. In particular, we generalize and correct existing results of quantum Fejér type inequalities by using new techniques and showing some problematic parts of those results. Most of the inequalities presented in this paper are significant extensions of results which appear in existing literatures.

A ( q , r ) -tree-coloring of a graph G is a q-coloring of vertices of G such that the subgraph induced by each color class is a forest of maximum degree at most r. An equitable ( q , r ) -tree-coloring of a graph G is a ( q , r ) -tree-coloring such that the sizes of any two color classes differ by at most one. Let the strong equitable vertex r-arboricity of G, denoted by v a r ź ( G ) , be the minimum p such that G has an equitable ( q , r ) -tree-coloring for every q ź p .Z. Guo, H. Zhao, Y. Mao 4 found the exact values of v a 2 ź ( K n , n , n ) for each n except when n is divisible by 20. In this paper, we find the exact value for each v a 2 ź ( K m , n ) and v a 2 ź ( K l , m , n ) . Equitable vertex arboricity has many applications such as scheduling and resource allocation.Z. Guo, H. Zhao, Y. Mao 4 found the exact values of equitable vertex 2-arboricity of K n , n , n except when n is divisible by 20.We find a linear-time algorithm to find the exact values of equitable vertex 2-arboricity for all K m , n and K l , m , n .

In this paper, we apply $(p,q)$-calculus to establish some new Chebyshev-type integral inequalities for synchronous functions. In particular, we generalize results of quantum Chebyshev-type integral inequalities by using $(p,q)$-integral. By taking $p=1$ and $q\to 1$, our results reduce to classical results on Chebyshev-type inequalities for synchronous functions. Furthermore, we consider their relevance with other related known results.