Let D be a very general curve of degree d = 2ℓ − ε in P 2 , with ε ∈ {0, 1}. Let Γ ⊂ P 2 be an integral curve of geometric genus g and degree m different from D, and let ν : C → Γ be the normalization. Let δ be the degree of the reduction modulo 2 of the divisor ν * (D) of C (see § 2.1). In this paper we prove the inequality 4g + δ >= m(d − 8 + 2ε) + 5. We compare this with similar inequalities due to Geng Xu ([10, 11]) and Xi Chen ([1, 2]).