Let D ⊂ C be a convex domain of complex numbers and K > 0. We say that the function f: D ⊂ C → C is called K-bounded modulus convex, for the given K > 0, if it satisfies the condition |(1 − λ) f (x) + λf (y) − f ((1 − λ) x + λy)| ≤ 1 Kλ (1 − λ) |x − y|2 2 for any x, y ∈ D and λ ∈ [0, 1] . In this paper we establish some new Hermite-Hadamard type inequalities for the complex integral on γ, a smooth path from C, and K-bounded modulus convex functions. Some examples for integrals on segments and circular paths are also given.