Inequalities of Jensen’s Type for \(\ K-\) Bounded Modulus Convex Complex Functions

Let \(\ D \subset \mathbb{C}\) be a convex domain of complex numbers and \(\ K > 0.\) We say that the function \(\ f : D \subset \mathbb{C} \to \mathbb{C}\) is called \(\ K-\) bounded modulus convex, for the given \(\ K > 0,\) if it satisfies the condition \(\|(1-\lambda) f(x) + \lambda f(y) - f((1-\lambda)x + \lambda y)| \leq \frac{1}{2}K\lambda(1-\lambda) |x-y|^2\) for any \(\ x, y \in D \text{ and } \lambda \in [0,1]\). In this paper we establish some new Jensen’s type inequalities for the complex integral on \(\gamma,\) a smooth path from \(\mathbb{C}\) and \(\ K-\) Bounded modulus convex functions. Some examples for the complex exponential and complex logarithm are also given.