An equitable coloring of a graph is a proper vertex coloring such that the sizes of every two color classes differ by at most 1. Chen, Lih, and Wu conjectured that every connected graph G with maximum degree ∆ ≥ 2 has an equitable coloring with ∆ colors, except when G is a complete graph or an odd cycle or ∆ is odd and G = K ∆,∆ . Nakprasit proved the conjecture holds for planar graphs with maximum degree at least 9. Zhu and Bu proved that the conjecture holds for every C 3 -free planar graph with maximum degree at least 8 and for every planar graph without C 4 and C 5 with maximum degree at least 7.In this paper, we prove that the conjecture holds for planar graphs in various settings, especially for every C 3 -free planar graph with maximum degree at least 6 and for every planar graph without C 4 with maximum degree at least 7, which * Corresponding Author improve or generalize results on equitable coloring by Zhu and Bu. Moreover, we prove that the conjecture holds for every planar graph of girth at least 6 with maximum degree at least 5.

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In this paper, we use (p,q)-integral to establish some Feje´r type inequalities. In particular, we generalize and correct existing results of quantum Feje´r 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.

The polar derivative of a polynomial p(z) of degree n with respect to a complex number α is a polynomial np(z)+α-zp′(z), denoted by Dαp(z). Let 1≤R≤k. For a polynomial p(z) of degree n having all its zeros in z≤k, we investigate a lower bound of modulus of Dαp(z) on z=R. Furthermore, we present an upper bound of modulus of Dαp(z) on z=R for a polynomial p(z) of degree n having no zero in z<k. In particular, our results in case R=1 generalize some well-known inequalities.