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

We say that a function \(f:[a,b]\to \mathbb{R}\) is \((\varphi,\delta)\)Lipschitzian, where \(\delta\geq 0\) and \(\varphi:[0,\infty)\to [0,\infty)\), if\[|f(x)f(y)|\leq \varphi(|xy|)+\delta,\quad (x,y)\in [a,b]\times [a,b].\]In this work, some Hadamard s type inequalities are established for the class of \((\varphi,\delta)\)Lipschitzian mappings. Moreover, some applications to convex functions with a continuous Caputo fractional derivative are also discussed.