New integral inequalities and their applications to convex functions with a continuous Caputo fractional derivative

We say that a function f : [a,b]→ R is (φ, δ)-Lipschitzian, where δ > 0 and φ : [0,∞)→ [0,∞), if |f(x) − f(y)| 6 φ(|x− y|) + δ, (x,y) ∈ [a,b]× [a,b]. In this work, some Hadamard’s type inequalities are established for the class of (φ, δ)-Lipschitzian mappings. Moreover, some applications to convex functions with a continuous Caputo fractional derivative are also discussed.