International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 574634, 7 pages

doi:10.1155/2012/574634

Research Article

On Generalized Flett's Mean Value Theorem

Jana Molnárová

Institute of Mathematics, Faculty of Science, Pavol Jozef Šafárik University in Košice, Jesenná 5, 040 01 Košice, Slovakia

Received 27 March 2012; Accepted 20 September 2012

Academic Editor: Adolfo Ballester-Bolinches

Copyright © 2012 Jana Molnárová. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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 (1.2). First such a condition was published in Trahan’s work [4]. An interesting idea is presented in paper [5] where the discrete and integral arithmetic mean is used. We suppose that this idea may be further generalized for the case of means studied, for example, in [6–8].

In recent years there has been renewed interest in Flett’s mean value theorem; see a survey in [9]. Among the many other extensions and generalizations of Theorem 1.3, see, for example, [10–13], we focus on that of Iwona Pawlikowska [14] solving the question of Zsolt Pales raised at the 35th International Symposium on Functional Equations held in Graz in 1997.

Theorem 1.5 (Pawlikowska). Let be -times differentiable on and . Then there exists such that

Observe that Pawlikowska’s theorem has a close relationship with the th Taylor polynomial of . Indeed, for

Pawlikowska’s theorem has the following very easy form .

Pawlikowska’s proof follows up the original idea of Flett; see [1], considering the auxiliary function

where for and using Theorem 1.1. In what follows we provide a different proof of Theorem 1.5 which uses only iterations of an appropriate auxiliary function and Theorem 1.3. In Section 3 we give a general version of Trahan condition, compare [4] under which Pawlikowska’s theorem holds.

2. New Proof of Pawlikowska’s Theorem

The key tool in our proof consists in using the auxiliary function

Running through all indices , we show that its derivative fulfills assumptions of Flett’s mean value theorem and it implies the validity of Flett’s mean value theorem for the th derivative of , where .

Indeed, for we have and . Clearly, , so applying Flett’s mean value theorem for on there exists , such that ; that is,

Then for , we get

and by (2.2). So, by Flett’s mean value theorem for on , there exists such that , which is equivalent to

Continuing this way after steps, , there exists such that

Considering the function , we get

Clearly, by (2.5). Then by Flett’s mean value theorem for on , there exists such that

Since

the equality (2.7) yields

which corresponds to (1.4).

It is also possible to state the result which no longer requires any endpoint conditions. If we consider the auxiliary function

then the analogous way as in the proof of Theorem 1.5 yields the following result also given in [14] including Riedel-Sahoo’s Theorem 1.4 as a special case ().

Theorem 2.1. If is -times differentiable on , then there exists such that

Note that the case is used to extend Flett’s mean value theorem for holomorphic functions; see [10]. An easy generalization of Pawlikowska’s theorem involving two functions is the following one.

Theorem 2.2. Let , be -times differentiable on and . Then there exists such that

This may be verified applying Pawlikowska’s theorem to the auxiliary function

A different proof will be presented in the following section.

3. A Trahan-Type Condition

In [4] Trahan gave a sufficient condition for the existence of a point satisfying (1.2) under the assumptions of differentiability of on and inequality

Modifying Trahan’s original proof using Pawlikowska’s auxiliary function , we are able to state the following condition for validity (1.4).

Theorem 3.1. Let be -times differentiable on and

where . Then there exists satisfying (1.4).

Proof. Since is continuous on and differentiable on with

for , [14; page 282], then

According to [4, Lemma  1], there exists such that which corresponds to (1.4).

Now we provide an alternative proof of Theorem 2.2 which does not use, original Pawlikowska’s theorem.

Proof of Theorem 2.2. For put and . Define the auxiliary function as follows:

Clearly, is continuous on , differentiable on , and for there holds

Then

and by [[4]; Lemma  1] there exists such that ; that is,

which completes the proof.

Acknowledgment

The work was partially supported by the Internal Grant VVGS-PF-2012-36.

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