On generalized Flett's mean value theorem

We present a new proof of generalized Flett's mean value theorem due to Pawlikowska (from Demonstratio Math. 1999) using only the original Flett's mean value theorem. Also, a Trahan-type condition is established in general case.


Introduction
Mean value theorems play an essential role in analysis. The simplest form of the mean value theorem due to Rolle is well-known. The answer is well-known as the Lagrange's mean value theorem. For the sake of brevity put b a K f (n) , g (n) = , n ∈ N ∪ {0}, for functions f, g defined on a, b (for which the expression has a sense). If Theorem 1.2 (Lagrange's mean value theorem) If f : a, b → R is continuous on a, b and differentiable on (a, b), then there exists a number η ∈ (a, b) Clearly, Theorem 1.2 reduces to Theorem 1.1 if f (a) = f (b). Geometrically, Theorem 1.2 states that given a line ℓ joining two points on the graph of a differentiable function f , namely (a, f (a)) and (b, f (b)), then there exists a point η ∈ (a, b) such that the tangent at (η, f (η)) is parallel to the given line ℓ.  [3], first proved in 1958 the following answer to this question for n = 1 which gives a variant of Lagrange's mean value theorem with Rolle-type condition.
Flett's original proof, see [3], uses Theorem 1.1. A slightly different proof which uses Fermat's theorem instead of Rolle's can be found in [10]. There is a nice geometric interpretation of Theorem 1.3: if the curve y = f (x) has a tangent at each point in a, b and if the tangents at (a, f (a)) and (b, f (b)) are parallel, then there exists a point η ∈ (a, b) such that the tangent at (η, f (η)) passes through the point (a, f (a)), see Figure 1.
Similarly as in the case of Rolle's theorem we may ask about possibility to remove the boundary assumption f ′ (a) = f ′ (b) in Theorem 1.3. As far as we know the first result of that kind is given in book [11].
We point out that there are also other sufficient conditions guaranteeing the existence of a point η ∈ (a, b) satisfying (1). First such a condition was published in Trahan's work [13]. An interesting idea is presented in paper [12] where the discrete and integral arithmetic mean is used. We suppose that this idea may be further generalized for the case of means studied e.g. in [5,6,7].
In recent years there has been renewed interest in Flett's mean value theorem. Among the many other extensions and generalizations of Theorem 1.3, see e.g. [1], [2], [4], [9], we focus on that of Iwona Pawlikovska [8] solving the question of Zsolt Pales raised at the 35-th International Symposium on Functional Equations held in Graz in 1997.
Observe that the Pawlikowska's theorem has a close relationship with the n-th Taylor polynomial of f . Indeed, for the Pawlikowska's theorem has the following very easy form f (a) = T n (f, η)(a).
Pawlikowska's proof follows up the original idea of Flett, see [3], considering the auxiliary function a K(f ) for x ∈ (a, b 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, cf. [13] under which Pawlikowska's theorem holds.

New proof of Pawlikowska's theorem
The key tool in our proof consists in using the auxiliary function Running through all indices k = 1, 2, . . . , n 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 l-th derivative of f , where l = n − 1, n − 2, . . . , 1. Indeed, for k = 1 we have ϕ 1 (x) = −f (n−1) (x) + xf (n) (a) and ϕ ′ 1 (x) = −f (n) (x) + f (n) (a). Clearly, ϕ ′ 1 (a) = 0 = ϕ ′ 1 (b), so applying the Flett's mean value theorem for ϕ 1 on a, b there exists (3). So, by Flett's mean value theorem for ϕ 2 on a, u 1 there exists u 2 ∈ (a, u 1 ) ⊂ (a, b) such that ϕ ′ 2 (u 2 )(u 2 − a) = ϕ 2 (u 2 ) − ϕ 2 (a), which is equivalent to Continuing this way after n − 1 steps, n ≥ 2, there exists u n−1 ∈ (a, b) such that Considering the function ϕ n we get Clearly, ϕ ′ n (a) = 0 = ϕ ′ n (u n−1 ) by (4). Then by Flett's mean value theorem for ϕ n on a, u n−1 there exists η ∈ (a, u n−1 ) ⊂ (a, b) such that Since the equality (5) yields which corresponds to (2). ✷ 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 [8] including Riedel-Sahoo's Theorem 1.4 as a special case (n = 1).
Note that the case n = 1 is used to extend Flett's mean value theorem for holomorphic functions, see [1]. An easy generalization of Pawlikowska's theorem involving two functions is the following one.
Theorem 2.2 Let f , g be n-times differentiable on a, b and g (n) (a) = g (n) (b). Then there exists η ∈ (a, b) such that This may be verified applying the Pawlikowska's theorem to the auxiliary function A different proof will be presented in the following section.

A Trahan-type condition
In [13] Trahan gave a sufficient condition for the existence of a point η ∈ (a, b) satisfying (1) under the assumptions of differentiability of f on a, b and inequality Modifying the Trahan's original proof using the Pawlikowska's auxiliary function G f we are able to state the following condition for validity (2).
Proof of Theorem 2.2. For x ∈ (a, b put ϕ(x) = x a K(f ) and ψ(x) = x a K(g). Define the auxiliary function F as follows Clearly, F is continuous on a, b , differentiable on (a, b and for x ∈ (a, b there holds g(a)) . Then and by Lemma 1 in [13] there exists η ∈ (a, b) such that F ′ (η) = 0, i.e., f (a) − T n (f, η)(a) = b a K f (n) , g (n) · (g(a) − T n (g, η)(a)). ✷