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On Flett’s mean value theorem. (English) Zbl 1325.26016

Mean value theorems play an important role in differential and integral calculus as they are a powerful tool for solving problems in mathematical analysis. The authors of the present paper provide a thorough study of Flett’s mean value theorem of a real-valued function of one real variable. More specifically, they study sufficient conditions under which the theorem is valid. In addition, they give geometric interpretations of all these conditions and prove two new extensions of the theorem. A detailed comparison of the corresponding classes of functions is offered and a possible generalization of Flett’s theorem to higher-order derivatives is examined. Their approach enables them to give new proofs of known results, such as Pawlikowska’s generalization of Flett’s theorem for higher-order derivatives. Finally, some interesting open problems are explicitly formulated at the end of the paper.

MSC:

26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26D20 Other analytical inequalities
39B22 Functional equations for real functions
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References:

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