Analysis of the Mean Value Theorem and Rolle’s Theorem in Holomorphic Function

The Mean Value Theorem and Rolle’s Theorem are one of the fundamental concepts in real analytical mathematics related to the derivative value of functions. This research examines both theorems on holomorphic functions in the complex plane. Holomorphic functions are also known as analytic functions, which are complex functions of one variable that have derivatives at every point in their domain. The results of the analysis show that there are fundamental differences in the application of the two theorems to holomorphic functions which can be seen in terms of the separation real and imaginary components, analytic property of the function, and the domain requirements that must be met. This research also discusses some variations of the Mean Value Theorem in holomorphic functions, such as Flett's Theorem and Myers Theorem and it can be shown that in complex functions the Mean Value Theorem and Rolle’s Theorem are mutually equivalent.