Flett’s mean value theorem for holomorphic functions. (English) Zbl 1021.30003

Davitt, R. M.; Powers, R. C.; Riedel, T.; Sahoo, P. K.

Math. Mag. 72, No. 4, 304-307 (1999).

In 1958, T.M. Flett proved the following mean value theorem: if \(f:[a,b]\rightarrow \mathbb{R}\) is differentiable and \(f'(a)=f'(b)\) then there exists \(\eta\in(a,b)\) such that \(f(\eta)-f(a)=(\eta-a)f'(\eta).\) Its complex version is not valid as the function \(f(z)=e^z-z\) shows.
In this paper the authors give a generalization of Flett’s mean value theorem for holomorphic functions.

Reviewer: Nikola Tuneski (Skopje)

MSC:

30A99	General properties of functions of one complex variable
26A24	Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems

Keywords:

Flett’s mean value theorem; generalization; holomorphic function