On Generalized Flett's Mean Value Theorem

We present a new proof of generalized Flett's mean value theorem due to Pawlikowska (from 1999) using only the original Flett's mean value theorem. Also, a Trahan-type condition is established in general case. 1. Introduction Mean value theorems play an essential role in analysis. The simplest form of the mean value theorem due to Rolle is well known. Theorem 1.1 (Rolle’s mean value theorem). If is continuous on and differentiable on and , then there exist a number such that . A geometric interpretation of Theorem 1.1 states that if the curve has a tangent at each point in and , then there exists a point such that the tangent at is parallel to the -axis. One may ask a natural question: What if we remove the boundary condition ? The answer is well known as the Lagrange mean value theorem. For the sake of brevity put for functions defined on (for which the expression has a sense). If , we simply write . Theorem 1.2 (Lagrange’s mean value theorem). If is continuous on and differentiable on , then there exist a number such that . Clearly, Theorem 1.2 reduces to Theorem 1.1 if . Geometrically, Theorem 1.2 states that given a line ？？joining two points on the graph of a differentiable function , namely, and , then there exists a point such that the tangent at is parallel to the given line . In connection with Theorem 1.1, the following question may arise: Are there changes if in Theorem 1.1 the hypothesis refers to higher-order derivatives? Flett, see [1], first proved in 1958 the following answer to this question for which gives a variant of Lagrange’s mean value theorem with the Rolle-type condition. Theorem 1.3 (Flett’s mean value theorem). If is a differentiable function on and , then there exists a number such that Flett’s original proof, see [1], uses Theorem 1.1. A slightly different proof which uses Fermat’s theorem instead of Rolle’s can be found in [2]. There is a nice geometric interpretation of Theorem 1.3: if the curve has a tangent at each point in and if the tangents at and are parallel, then there exists a point such that the tangent at passes through the point ; see Figure 1. Figure 1: Geometric interpretation of Flett’s mean value theorem. Similarly as in the case of Rolle’s theorem, we may ask about possibility to remove the boundary assumption in Theorem 1.3. As far as we know, the first result of that kind is given in the book [3]. Theorem 1.4 (Riedel-Sahoo). If is a differentiable function on , then there exist a number such that We point out that there are also other sufficient conditions guaranteeing the existence of a point satisfying