On Generalized Flett's Mean Value Theorem

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,[email protected]

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 → Ris continuous on a, b and differentiable ona, bandfa fb, then there exist a numberη∈a, bsuch thatfη 0.

A geometric interpretation of Theorem 1.1 states that if the curve y fx has a tangent at each point in a, b and fa fb, then there exists a pointη ∈ a, b such that the tangent atη, fηis parallel to thex-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

aK

fn_{, g}n fnb−fna

gn_b−gn_a, n∈N∪ {0}, 1.1

for functionsf, gdefined ona, bfor which the expression has a sense. Ifgnb−gna

x y

a η b

y=f(a) +f′_(η)(x−a)

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

Theorem 1.2Lagrange’s mean value theorem. Iff :a, b → Ris continuous ona, band differentiable ona, b, then there exist a numberη∈a, bsuch thatfη b_aKf.

Clearly, Theorem1.2reduces to Theorem1.1iffa fb. Geometrically, Theorem1.2

states that given a line joining two points on the graph of a differentiable functionf, namely,

a, faandb, fb, then there exists a pointη∈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, see1, first proved in 1958 the following answer to this question forn 1 which gives a variant of Lagrange’s mean value theorem with the Rolle-type condition.

Theorem 1.3Flett’s mean value theorem. Iff:a, b → Ris a differentiable function ona, b andfa fb, then there exists a numberη∈a, bsuch that

f_η η

aK

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 pointη ∈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 assumptionfa fbin Theorem1.3. As far as we know, the first result of that kind is given in the book3.

Theorem 1.4Riedel-Sahoo. Iff :a, b → Ris a differentiable function ona, b, then there exist a numberη∈a, bsuch that

f_η η

aK

fb_aKf_· 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 1.5Pawlikowska. Letfben-times differentiable ona, bandfna fnb. Then there existsη∈a, bsuch that

fη−fa n

i1

−1i1

i!

η−aifi_{η. 1.4}

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

Tnf, x0

x fx0

f_x

0

1! x−x0 · · ·

fn_x

0

n! x−x0

n_{, 1.5}

Pawlikowska’s theorem has the following very easy formfa Tnf, ηa.

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

Gfx ⎧ ⎨ ⎩

gn−1_{x, x∈a, b,}

1

nfna, xa,

1.6

wheregx x_aKfforx ∈ 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

ϕkx xfn−k1a k

i0

−1i1

i! k−ix−a

i_fn−ki_{x, k1,2, . . . , n. 2.1}

Running through all indicesk1,2, . . . , 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, fork1 we haveϕ1x −fn−1x xfnaandϕ₁x −fnx fna.

Clearly,ϕ₁a 0ϕ₁b, so applying Flett’s mean value theorem forϕ1ona, bthere exists

u1∈a, b, such thatϕ₁u1u1−a ϕ1u1−ϕ1a; that is,

fn−1_u

1−fn−1a u1−afnu1. 2.2

Then forϕ2x −2fn−2x x−afn−1x xfn−1a, we get

ϕ

2x −fn−1x x−afnx fn−1a 2.3

andϕ₂a 0 ϕ₂u1by2.2. So, by Flett’s mean value theorem forϕ2 ona, u1, there

existsu2∈a, u1⊂a, bsuch thatϕ₂u2u2−a ϕ2u2−ϕ2a, which is equivalent to

fn−2_u

2−fn−2a u2−afn−1u2−

1

2u2−a

2_fn_u

2. 2.4

Continuing this way aftern−1 steps,n≥2, there existsun−1∈a, bsuch that

f_u

n−1−fa

n−1

i1

−1i1

i! un−1−a

i_fi1_u

n−1. 2.5

Considering the functionϕn, we get

ϕ

nx −fx fa

n−1

i1

−1i1

i! x−a

i_fi_x

f_a n−1 i0

−1i1

i! x−a

i_fi1_x.

2.6

Clearly,ϕ_na 0ϕ_nun−1by2.5. Then by Flett’s mean value theorem forϕnona, un−1,

there existsη∈a, un−1⊂a, bsuch that

ϕ

nηη−aϕnη−ϕna. 2.7

Since

ϕ

nηη−aη−afa

n

i1

−1i

i−1!

η−aifi_η,

ϕnη−ϕna η−afa−nfη−fa n

i1

−1i1

i! n−i

η−aifi_η,

the equality2.7yields

−nfη−fan

i1

−1i

i−1!

η−aifi_η1n−i

i

nn

i1

−1i

i!

η−aifi_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

ψkx ϕkx −1 k1

k1!x−a

k1_·b

aK

fn_{, 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 → Risn-times differentiable ona, b, then there existsη∈a, bsuch that

fa Tnf, ηa

a−ηn1 n1! ·

b

aK

fn_{. 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. Letf,g ben-times differentiable ona, bandgna/gn_{b. Then there exists}

η∈a, bsuch that

fa−Tnf, ηa b

aK

fn_{, g}n_·ga−T

ng, ηa. 2.12

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

hx fx− b_aKfn_{, g}n_·

,gx, x∈ a, b. 2.13

3. A Trahan-Type Condition

In4Trahan gave a sufficient condition for the existence of a pointη∈a, bsatisfying1.2

under the assumptions of differentiability offona, band inequality

f_b− b

aK

f·f_a− b

aK

f≥0. 3.1

Modifying Trahan’s original proof using Pawlikowska’s auxiliary functionGf, we are

able to state the following condition for validity1.4.

Theorem 3.1. Letfben-times differentiable ona, band

fn_aa−bn

n! Mf

fn_ba−bn

n! Mf

≥0, 3.2

whereMf Tn−1f, ba−fa. Then there existsη∈a, bsatisfying1.4.

Proof. SinceGfis continuous ona, band differentiable ona, bwith

G

fx gnx −

1nn!

x−an1

fx−fa n

i1

−1i

i! x−a

i_fi_x

, 3.3

forx∈a, b,14, page 282, then

Gfb−GfaGfb

gn−1_b−1

nfna

gn_b

− n!n−1! b−a2n1

fn_aa−bn

n! Tn−1

f, ba−fa

·

fn_ba−bn

n! Tn−1

f, ba−fa

≤0.

3.4

According to4, Lemma 1, there existsη ∈a, bsuch thatG_fη 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_aKf and ψx x_aKg. Define the

auxiliary functionFas follows:

Fx

⎧ ⎨ ⎩

ϕn−1_x− b

aKfn, gn·ψn−1x, x∈a, b,

1

n

fn_a− b

aK

Clearly,Fis continuous ona, b, differentiable ona, b, and forx∈a, bthere holds

F_{x ϕ}n_x− b

aK

fn_{, g}n_·ψn_{x. 3.6}

Then

F_{bFb−Fa −} n

b−aFb−Fa2≤0, 3.7

and by4, Lemma 1there existsη∈a, bsuch thatFη 0; that is,

fa−Tnf, ηa b

aK

fn_{, g}n_{·ga−Tng, ηa, 3.8}

which completes the proof.

Acknowledgment

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

References

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

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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 &

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