International Journal of Mathematics and Mathematical Sciences Volume 2012, Article ID 574634,7pages

doi:10.1155/2012/574634

*Research Article*

**On Generalized Flett’s Mean Value Theorem**

**Jana Moln ´arov ´a**

*Institute of Mathematics, Faculty of Science, Pavol Jozef ˇSaf´arik University in Koˇsice, Jesenn´a 5,*
*040 01 Koˇsice, Slovakia*

Correspondence should be addressed to Jana Moln´arov´a,jana.molnarova88@gmail.com

Received 27 March 2012; Accepted 20 September 2012

Academic Editor: Adolfo Ballester-Bolinches

Copyrightq2012 Jana Moln´arov´a. 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.

We present a new proof of generalized Flett’s mean value theorem due to Pawlikowskafrom 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*. Iff* : a, b → R*is continuous on* a, b *and*
*diﬀerentiable on*a, b*andfa fb, then there exist a numberη*∈a, b*such thatf*η *0.*

A geometric interpretation of Theorem 1.1 states that if the curve *y* *f*x has a
tangent at each point in a, b and *fa * *fb*, then there exists a point*η* ∈ a, b such
that the tangent atη, fηis parallel to the*x-axis. One may ask a natural question: What*

*if we remove the boundary conditionfa * *fb*? The answer is well known as the Lagrange
mean value theorem. For the sake of brevity put

*b*

*a*K

*f*n* _{, g}*n

_{}

*f*nb−

*f*na

*g*n_{b}_{−}* _{g}*n

_{a}

*, n*∈N∪ {0}, 1.1

for functions*f, g*defined ona, bfor which the expression has a sense. If*g*nb−*g*na

*x*
*y*

*a* *η* *b*

*y*=*f(a) +f*′_{(η)(x}_{−}_{a)}

**Figure 1: Geometric interpretation of Flett’s mean value theorem.**

**Theorem 1.2**Lagrange’s mean value theorem*. Iff* :a, b → R*is continuous on*a, b*and*
*diﬀerentiable on*a, b*, then there exist a numberη*∈a, b*such thatf*η *b _{a}*Kf

*.*

Clearly, Theorem1.2reduces to Theorem1.1if*fa fb*. Geometrically, Theorem1.2

states that given a line joining two points on the graph of a diﬀerentiable function*f*, namely,

a, faandb, fb, then there exists a point*η*∈a, bsuch that the tangent atη, fηis
parallel to the given line.

In connection with Theorem1.1*, the following question may arise: Are there changes if in*

*Theorem1.1the hypothesisfa fbrefers to higher-order derivatives? Flett, see*1, first proved
in 1958 the following answer to this question for*n* 1 which gives a variant of Lagrange’s
mean value theorem with the Rolle-type condition.

**Theorem 1.3**Flett’s mean value theorem*. Iff*:a, b → R*is a diﬀerentiable function on*a, b
*andf*a *f*b*, then there exists a numberη*∈a, b*such that*

*f _{η}*

_{}

*η*

*a*K

*f.* 1.2

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 *y* *fx* has a tangent at each point ina, b
and if the tangents ata, faandb, fbare parallel, then there exists a point*η* ∈a, b
such that the tangent atη, fηpasses through the pointa, fa; see Figure1.

Similarly as in the case of Rolle’s theorem, we may ask about possibility to remove the
boundary assumption*f*a *f*bin Theorem1.3. As far as we know, the first result of that
kind is given in the book3.

**Theorem 1.4**Riedel-Sahoo*. Iff* :a, b → R*is a diﬀerentiable function on*a, b*, then there*
*exist a numberη*∈a, b*such that*

*f _{η}*

_{}

*η*

*a*K

*fb _{a}*K

*f*

_{·}

*η*−

*a*

2 *.* 1.3

An interesting idea is presented in paper5where 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, in6–8.

In recent years there has been renewed interest in Flett’s mean value theorem; see a survey in9. Among the many other extensions and generalizations of Theorem1.3, see, for example,10–13, we focus on that of Iwona Pawlikowska14solving the question of Zsolt Pales raised at the 35th International Symposium on Functional Equations held in Graz in 1997.

**Theorem 1.5**Pawlikowska*. Letfben-times diﬀerentiable on*a, b*andf*na *f*nb*. Then*
*there existsη*∈a, b*such that*

*fη*−*fa n*

*i1*

−1*i1*

*i*!

*η*−*aif*i_{η}_{.}_{}_{1.4}_{}

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

*Tnf, x*0

x *fx*0

*f*_{x}

0

1! x−*x*0 · · ·

*f*n_{x}

0

*n*! x−*x*0

*n _{,}*

_{}

_{1.5}

_{}

Pawlikowska’s theorem has the following very easy form*f*a *Tn*f, ηa.

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

*Gf*x
⎧
⎨
⎩

*g*n−1_{x, x}_{∈}_{a, b}_{,}

1

*nf*na, x*a,*

1.6

where*gx * *x _{a}*Kffor

*x*∈ a, band using Theorem1.1. In what follows we provide a diﬀerent proof of Theorem1.5which uses only iterations of an appropriate auxiliary function and Theorem1.3. In Section3 we give a general version of Trahan condition, compare4

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

*ϕk*x *xf*n−k1a
*k*

*i0*

−1*i1*

*i*! k−*ix*−*a*

*i _{f}*n−ki

_{x, k}

_{}

_{1}

_{,}_{2}

_{, . . . , n.}_{}

_{2.1}

_{}

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 the

Indeed, for*k*1 we have*ϕ*1x −fn−1x *xf*naand*ϕ*_{1}x −fnx *f*na.

Clearly,*ϕ*_{1}a 0*ϕ*_{1}b, so applying Flett’s mean value theorem for*ϕ*1ona, bthere exists

*u*1∈a, b, such that*ϕ*_{1}u1u1−*a ϕ*1u1−*ϕ*1a; that is,

*f*n−1_{u}

1−*f*n−1a u1−*af*nu1. 2.2

Then for*ϕ*2x −2*f*n−2x x−*af*n−1x *xf*n−1a, we get

*ϕ*

2x −fn−1x x−*af*nx *f*n−1a 2.3

and*ϕ*_{2}a 0 *ϕ*_{2}u1by2.2. So, by Flett’s mean value theorem for*ϕ*2 ona, u1, there

exists*u*2∈a, u1⊂a, bsuch that*ϕ*_{2}u2u2−*a ϕ*2u2−*ϕ*2a, which is equivalent to

*f*n−2_{u}

2−*f*n−2a u2−*af*n−1u2−

1

2u2−*a*

2* _{f}*n

_{u}

2. 2.4

Continuing this way after*n*−1 steps,*n*≥2, there exists*un−1*∈a, bsuch that

*f*_{u}

*n−1*−*f*a

*n−1*

*i1*

−1*i1*

*i*! u*n−1*−*a*

*i _{f}*i1

_{u}

*n−1*. 2.5

Considering the function*ϕn*, we get

*ϕ*

*nx* −f*x* *fa*

*n−1*

*i1*

−1*i1*

*i*! *x*−*a*

*i _{f}*i

_{}

_{x}_{}

*f*_{a }*n−1*
*i0*

−1*i1*

*i*! x−*a*

*i _{f}*i1

_{x.}

2.6

Clearly,*ϕ _{n}*a 0

*ϕ*u

_{n}*n−1*by2.5. Then by Flett’s mean value theorem for

*ϕn*ona, u

*n−1*,

there exists*η*∈a, u*n−1*⊂a, bsuch that

*ϕ*

*nηη*−*aϕnη*−*ϕn*a. 2.7

Since

*ϕ*

*nηη*−*aη*−*af*a

*n*

*i1*

−1*i*

i−1!

*η*−*a*i*f*i_{η}_{,}

*ϕnη*−*ϕn*a *η*−*af*a−*nfη*−*fa*
*n*

*i1*

−1*i1*

*i*! n−*i*

*η*−*aif*i_{η}_{,}

the equality2.7yields

−n*fη*−*fan*

*i1*

−1*i*

i−1!

*η*−*aif*i_{η}_{1}_{}*n*−*i*

*i*

*nn*

*i1*

−1*i*

*i*!

*η*−*a*i*f*i_{η}_{,}

2.9

which corresponds to1.4.

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

*ψk*x *ϕk*x −1
*k1*

*k*1!x−*a*

*k1*_{·}*b*

*a*K

*f*n_{,}_{}_{2.10}_{}

then the analogous way as in the proof of Theorem1.5yields the following result also given in14including Riedel-Sahoo’s Theorem1.4as a special casen1.

* Theorem 2.1. Iff* :a, b → R

*isn-times diﬀerentiable on*a, b

*, then there existsη*∈a, b

*such*

*that*

*f*a *Tnf, η*a

*a*−*ηn1*
*n*1! ·

*b*

*a*K

*f*n_{.}_{}_{2.11}_{}

Note that the case *n* 1 is used to extend Flett’s mean value theorem for
holomorphic functions; see10. An easy generalization of Pawlikowska’s theorem involving
two functions is the following one.

**Theorem 2.2. Let**f,g*ben-times diﬀerentiable on*a, b*andg*na*/g*n_{b}_{. Then there exists}

*η*∈a, b*such that*

*fa*−*Tnf, η*a
*b*

*a*K

*f*n* _{, g}*n

_{·}

_{ga}_{−}

_{T}*ng, η*a*.* 2.12

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

*hx f*x− *b _{a}*K

*f*n

*n*

_{, g}_{·}

*,gx,* *x*∈ *a, b.* 2.13

**3. A Trahan-Type Condition**

In4Trahan gave a suﬃcient condition for the existence of a point*η*∈a, bsatisfying1.2

under the assumptions of diﬀerentiability of*f*ona, band inequality

*f*_{b}_{−} *b*

*a*K

*f*·*f*_{a}_{−} *b*

*a*K

*f*≥0*.* 3.1

Modifying Trahan’s original proof using Pawlikowska’s auxiliary function*Gf*, we are

able to state the following condition for validity1.4.

* Theorem 3.1. Letfben-times diﬀerentiable on*a, b

*and*

*f*n_{aa}_{−}_{b}n

*n*! *Mf*

*f*n_{ba}_{−}_{b}n

*n*! *Mf*

≥0*,* 3.2

*whereMf* *Tn−1*f, ba−*f*a*. Then there existsη*∈a, b*satisfying*1.4*.*

*Proof. SinceGf*is continuous ona, band diﬀerentiable ona, bwith

*G*

*f*x *g*nx −

1*nn*!

x−*an1*

*fx*−*fa n*

*i1*

−1*i*

*i*! x−*a*

*i _{f}*i

_{x}

*,* 3.3

for*x*∈a, b,14, page 282, then

*Gf*b−*Gf*a*Gf*b

*g*n−1_{b}_{−}1

*nf*na

*g*n_{b}

− *n*!n−1!
*b*−*a*2n1

*f*n_{aa}_{−}_{b}n

*n*! *Tn−1*

*f, b*a−*f*a

·

*f*n_{ba}_{−}_{b}n

*n*! *Tn−1*

*f, b*a−*fa*

≤0*.*

3.4

According to4, Lemma 1, there exists*η* ∈a, bsuch that*G _{f}*η 0 which corresponds to

1.4.

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

*Proof of Theorem2.2. For* *x* ∈ a, b put *ϕx * *x _{a}*Kf and

*ψx*

*x*Kg. Define the

_{a}auxiliary function*F*as follows:

*Fx *

⎧ ⎨ ⎩

*ϕ*n−1_{x}_{−} *b*

*a*K*f*n*, g*n·*ψ*n−1x, x∈a, b,

1

*n*

*f*n_{a}_{−} *b*

*a*K

Clearly,*F*is continuous ona, b, diﬀerentiable ona, b, and for*x*∈a, bthere holds

*F*_{x }* _{ϕ}*n

_{x}

_{−}

*b*

*a*K

*f*n* _{, g}*n

_{·ψ}n

_{x.}

_{}

_{3.6}

_{}

Then

*F*_{bFb}_{−}_{Fa }_{−} *n*

b−*a*Fb−*Fa*2≤0*,* 3.7

and by4, Lemma 1there exists*η*∈a, bsuch that*F*η 0; that is,

*fa*−*Tnf, η*a
*b*

*a*K

*f*n* _{, g}*n

_{·}

_{ga}_{−}

_{T}_{n}_{g, η}_{a}

_{,}_{}

_{3.8}

_{}

which completes the proof.

**Acknowledgment**

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

**References**

1 *T. M. Flett, “A mean value theorem,” The Mathematical Gazette, vol. 42, pp. 38–39, 1958.*

2 *T.-L. R˘adulescu, V. D. R˘adulescu, and T. Andreescu, Problems in Real Analysis: Advanced Calculus on the*

*Real Axis, Springer, New York, NY, USA, 2009.*

3 *P. K. Sahoo and T. Riedel, Mean Value Theorems and Functional Equations, World Scientific, River Edge,*
NJ, USA, 1998.

4 *D. H. Trahan, “A new type of mean value theorem,” Mathematics Magazine, vol. 39, pp. 264–268, 1966.*
5 *J. Tong, “On Flett’s mean value theorem,” International Journal of Mathematical Education in Science &*

*Technology, vol. 35, pp. 936–941, 2004.*

6 J. Haluˇska and O. Hutn´ı*k, “Some inequalities involving integral means,” Tatra Mountains Mathematical*

*Publications, vol. 35, pp. 131–146, 2007.*

7 O. Hutn´ık, “On Hadamard type inequalities for generalized weighted quasi-arithmetic means,”

*Journal of Inequalities in Pure and Applied Mathematics, vol. 7, no. 3, article 96, 2006.*

8 O. Hutn´ı*k, “Some integral inequalities of H ¨older and Minkowski type,” Colloquium Mathematicum,*
vol. 108, no. 2, pp. 247–261, 2007.

9 O. Hutn´ık and J. Moln´arov´a, “On Flett’s mean value theorem,” In press.

10 R. M. Davitt, R. C. Powers, T. Riedel, and P. K. Sahoo, “Flett’s mean value theorem for holomorphic
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11 M. Das, T. Riedel, and P. K. Sahoo, “Flett’s mean value theorem for approximately diﬀerentiable
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*International Journal of Mathematics and Mathematical Sciences, vol. 27, no. 11, pp. 689–694, 2001.*

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