Approaching the Power Logarithmic and Difference Means by Iterative Algorithms Involving the Power Binomial Mean

(1)

Volume 2011, Article ID 687825,12pages doi:10.1155/2011/687825

Research Article

Approaching the Power Logarithmic and Difference

Means by Iterative Algorithms Involving the Power

Binomial Mean

Mustapha Ra¨ıssouli

Applied Functional Analysis Team, Department of Mathematics and Computer Science, Faculty of Science, Moulay Isma¨ıl University, P.O. Box 11201, Mekn`ees, Morocco

Correspondence should be addressed to Mustapha Ra¨ıssouli,raissouli [email protected]

Received 4 December 2010; Accepted 9 February 2011

Academic Editor: B. N. Mandal

Copyrightq2011 Mustapha Ra¨ıssouli. 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.

Introducing the notion of cross means we give iterative algorithms involving the power binomial mean and converging to the power logarithmic and diﬀerence means. At the end, we address a list of open problems derived from our present work.

1. Introduction

Throughout this paper, we understand by mean a binary mapmbetween positive real num-bers satisfying the following statements:

ima, a a, for alla >0normalization axiom;

iima, b mb, a, for alla, b >0symmetry axiom;

iiimta, tb tma, b, for alla, b, t >0homogeneity axiom;

ivma, bis an increasing function inaand inb monotonicity axiom;

vma, bis a continuous function ofaandbcontinuity axiom.

(2)

m1a, b ≤ m2a, bfor alla, b > 0. Two trivial means area, b → mina, b anda, b →

maxa, b, and every meanmsatisfies

mina, b≤ma, b≤maxa, b, 1.1

for alla, b >0. We denote min and max the two trivial means which we call lower and upper means, respectively. The standard examples of means satisfying the above requirements are recalled in the following:

iArithmetic mean,Aa, b ab/2,

iiGeometric mean,Ga, b √ab,

iiiHarmonic mean,Ha, b 2ab/ab,

ivLogarithmic mean,La, b a−b/lna−lnb,_{a /}b,La, a a,

vIdentricor exponentialmean,Ia, b 1/ebb_/aa1/b−a_,_{a /}_b_,_I_{a, a} _a_,

viQuadratic mean,Ka, b a2_b2_/_2.

As well known, these means satisfy the following inequalities:

min≤H≤G≤L≤I≤A≤K≤max. 1.2

A meanmis called strict mean ifma, b is strictly increasing inaand in b. Also, every strict meanmsatisfies thatma, b a ⇒ a b. It is easy to see that the lower and upper means are not strict, whileA,G,H,L,I,Kare strict means.

There are many families of means, called power means, which extend the above standard ones. For instance, letpbe a real number; we cite

ipower binomial mean defined by

Bpa, b

apbp

2

1/p

. 1.3

It is understood that

B−∞:_plim_{→ −∞}Bpmin, B−1H, B0G, B1 A, B∞max. 1.4

Notice that

B1/2a, b

1 2Aa, b

1

2Ga, b:Hea, b 1.5

is called the Heron mean,

iipower logarithmic mean given by

Lpa, b

ap1₋_bp1

p1a−b

1/p

1

b−a b

a tpdt

1/p

(3)

The particular special values ofpare understood as

L−∞min, L−2G, L−1L, L0I, L1A, L∞max. 1.7

Further, the following inequalities are well known:

Bp≤Lp≤A forp≤1, A≤Lp≤Bp forp≥1, 1.8

iiipower diﬀerence mean defined as follows:

Dpa, b p p1

ap1₋_bp1

ap₋_bp , a /b, Dpa, a a. 1.9

This includes some of the most familiar means in the sense

D−∞min, D−2H, D−1 G 2

L , D−1/2G, D0L, D1A, D∞max.

1.10

It is not hard to see that the above power meansBp,Lp, andDpare strict means for all real numbersp−∞< p <∞.

The remainder of this paper will be organized as follows: after this section,Section 2

contains some new basic notions and results about a class of means, termed cross means.

Section 3 is devoted to introduce two adjacent recursive sequences, depending only of Bp

and converging to the power logarithmic meanLp.Section 4displays briefly an analogue of the above section for the power diﬀerence meanDp. Finally,Section 5is focused to address a list of open problems derived from our present work and put as future research for the interested readers.

2. Cross Means

In this section, we will introduce the tensor product of two binary means from which we derive the definition of a class of special means termed cross means.

Definition 2.1. Letm1 andm2 be two binary means. The tensor product ofm1andm2 is the

2-binary map, denotedm1⊗m2, defined by

∀a, b, c, d >0 m1⊗m2a, b, c, d m1m2a, b, m2c, d. 2.1

It is simple to verify thatm1⊗m2andm2⊗m1are, in general, diﬀerent. Further, the

mapm1⊗m2satisfies all axioms of a 2-binary mean except the symmetry axiomii. That is,

the tensor product of two binary means is not, in general, a 2-binary mean. For a meanm, we writem⊗2 _: _m_⊗_m_{. To not lengthen this section, we omit the study of the elementary}

properties of m1 ⊗m2 not needed later. However, our goal here is to derive the following

(4)

Definition 2.2. A binary meanmwill be called cross mean ifm⊗2_{is a 2-binary mean, that is,}

∀a, b, c, d >0 m⊗2a, b, c, d m⊗2a, c, b, d. 2.2

By the symmetry axiom ii for m, relation 2.2 is equivalent to one of the three following equalities:

∀a, b, c, d >0 m⊗2a, b, c, d m⊗2b, c, a, d,

∀a, b, c, d >0 m⊗2a, b, c, d m⊗2a, d, b, c,

∀a, b, c, d >0 m⊗2a, b, c, d m⊗2b, d, a, c.

2.3

It is not hard to see that the two trivial means min and max are cross means. Other examples of cross means are given in the following.

Theorem 2.3. For all real numbersp, the power binomial meanBpis a cross mean.

Proof. According toDefinition 2.2, with the explicit form ofBp, the desired result follows from

an elementary computation. We left the routine detail here.

Corollary 2.4. The arithmetic, geometric, and harmonic means are cross means.

Proof. Theorem 2.3can be formulated as follows:

∀a, b, c, d >0 BpBpa, b, Bpc, dBpBpa, c, Bpb, d, 2.4

from which, settingp1 andp−1, we obtain the announced result for the arithmetic and harmonic means, respectively. Lettingp → 0 in the latter formulae, and using1.4with an argument of continuity, we deduce the result for the geometric mean.

From the above theorem, we immediately deduce that the quadratic and Heron means

K andHe are also cross means. However, the logarithmic and identric means are not cross means. The following counterexample shows this latter situation.

Example 2.5. According to the above definitions, simple computations yield the following

results:

L⊗21,2,3,4 ln8/3

ln 2ln3/4lnln 2/ln3/4/

ln9/4

ln 2ln 3lnln 3/ln 2L

⊗2₁_,₃_,₂_,₄_,

I⊗21,2,3,4 4

e

9 4

5/4

/

9

4e I

⊗2₁

,3,2,4.

2.5

The above example, with 1.7 and 1.10, shows that the power logarithmic and diﬀerence means are not always cross means. In the next sections, we will approximateLp

(5)

3. Approximation of the Power Logarithmic Mean

Lp

As already pointed before, our aim in this section is to approximate the noncross meanLpby iterative scheme involving the cross meanBp. For all positive real numbersa,band all fixed real numbersp, define the following iterative algorithms:

Υp,n1a, b Bp

Υp,n

a,ab

2

,Υp,n

ab

2 , b

,

Υp,0a, b Bpa, b,

3.1

Θp,n1a, b Bp

Θp,n

a,ab

2

,Θp,n

ab

2 , b

,

Θp,0a, b Aa, b: ab

2 .

3.2

By a mathematical induction, it is easy to see thatΥp,nandΘp,nare means for alln≥0.

In particular, the symmetry axiom forΥp,nandΘp,nholds, that is,

∀n≥0 Υp,na, b Υp,nb, a, Θp,na, b Θp,nb, a. 3.3

In terms of tensor product, the above recursive relation defining the sequence

Υp,na, b_ncan be written as follows:

Υp,n1a, b Bp⊗Υp,n

a,ab

2 ,

ab

2 , b

, 3.4

with analogous form forΘp,na, b_n. However, for the sake of simplicity for the reader we

omit these tensor writings and we use the recursive forms 3.1 and 3.2 throughout the following.

In what follows, we will study the convergence of the above algorithms. We start with the next result giving a link between the two sequencesΥp,na, b_nandΘp,na, b_n.

Proposition 3.1. With the above, the sequencesΥp,na, b_nandΘp,na, b_nsatisfy the following

relationship:

Υp,n1a, b Bp

Υp,na, b,Θp,na, b

, 3.5

for alla, b >0 and everyn≥0.

Proof. Forn0, relations3.1give

Υp,1a, b Bp

Bp

a,ab

2

, Bp

ab

2 , b

(6)

According toTheorem 2.3, with the symmetry axiom ofBp, we obtain

Υp,1a, b Bp

Bpa, b, Bp

ab

2 ,

ab

2

Bp

Bpa, b,ab

2

. 3.7

This, with3.1and3.2, yields

Υp,1a, b Bp

Υp,0a, b,Θp,0a, b

. 3.8

By a mathematical induction, the desired result follows with the same arguments as previ-ously mentioned. The proof is complete.

Proposition 3.2. Assume thatp≤1, then, the following inequalities:

Bpa, b≤ · · · ≤Υp,n−1a, b≤Υp,na, b≤Θp,na, b≤Θp,n−1a, b≤ · · · ≤Aa, b 3.9

hold for alla, b >0 and everyn≥0.

Ifp≥1, the above inequalities are reversed, with equalities forp1.

Proof. Letp ≤ 1. The mapx → x1/p _{is convex on} ₀_,_∞_{and so}_Bp_{a, b} _≤ _A_{a, b}_{, that is,}

Υp,0a, b≤ Θp,0a, bfor alla, b >0. Using3.1and3.2, we easily show by mathematical

induction that, for alla, b >0,

Υp,na, b≤Θp,na, b, 3.10

for everyn≥0. This, withProposition 3.1and the monotonicity axiom ofBp, implies that

Υp,n1a, b≥Bp

Υp,na, b,Υp,na, b

Υp,na, b, 3.11

for eachn≥0, that is,Υp,na, b_n is an increasing sequence. Now, let us show the decrease

monotonicity ofΘp,na, b_n. By3.2, we have

Θp,1a, b Bp

A

a,ab

2

, A

ab

2 , b

, 3.12

which, withBpa, b≤Aa, b, yields

Θp,1a, b≤A

A

a,ab

2

, A

ab

2 , b

, 3.13

and, with the fact thatAis a cross mean, we obtain

Θp,1a, b≤A

Aa, b, A

ab

2 ,

ab

2

(7)

for all a, b > 0. This, with 3.2 and a simple mathematical induction, gives the decrease monotonicity ofΘp,na, b_n. The proof of inequalities3.9is complete. Forp≥ 1, the map x→x1/p_, _{p /}₀_{, is concave and all inequalities in the above case are reversed. The proof is}

completed.

Theorem 3.3. The sequencesΥp,na, bnandΘp,na, bnboth converge to the same limitLpa, b,

power logarithmic mean ofaandb, with the following estimations:

∀n≥0 Bpa, b≤ · · · ≤Υp,na, b≤Lpa, b≤Θp,na, b≤ · · · ≤Aa, b 3.15

ifp≤1, with reversed inequalities ifp≥1 and equalities ifp1.

Proof. By Proposition 3.2 the sequences Υp,na, b_n and Θp,na, b_n are monotone and

bounded then they converge. Calling mpa, b and Mpa, b their limits, respectively, we deduce fromProposition 3.1, with an argument of continuity, that

mpa, b Bpmpa, b, Mpa, b. 3.16

This, with the fact thatBpis a strict mean for all real numbersp, yieldsmpa, b Mpa, b, that is, Υp,na, b_n and Θp,na, b_n converge with the same limit. Let us prove that this

common limit is exactlyLpa, b. It is suﬃcient to show thatLpa, bis an intermediary mean between the two meansΥp,na, bandΘp,na, b, for alln≥0. First, using the integral explicit

form ofLpa, b, it is easy to verify the following relationship:

Lpa, b Bp

Lp

a,ab

2

, Lp

ab

2 , b

. 3.17

For n 0, inequalities 1.8 imply that Lpa, b is between Υp,0a, b : Bpa, b and Θp,0a, b:Aa, b. Assuming thatp ≤1 and using1.8again with the recursive relations 3.1and3.2, we easily prove with a simple mathematical induction that

Υp,na, b≤Lpa, b≤Θp,na, b, 3.18

for alla, b >0 and everyn≥0. Lettingn → ∞in inequalities3.18, we deduce that

mpa, b≤Lpa, b≤Mpa, b, 3.19

with reversed inequalities ifp ≥ 1. This, with the fact that mpa, b Mpa, b, yields the desired results. The proof of the theorem is complete.

We notice that inequalities3.15give some iterative refinements of1.8. Further, the above theorem has many consequences as recited in the two following corollaries.

Corollary 3.4. The sequences Υ0,na, bn and Θ0,na, bn converge to the same limit Ia, b,

identric mean ofaandb, with the following relationship:

(8)

Proof. Settingp0 in the above theorem, with the sake of convenience,

Ia, b lim

p→0Lpa, b, 3.21

we obtain the desired result.

We notice that forp 0, the inequalities3.15imply thatGa, b≤Ia, b≤ Aa, b, which is the known arithmetic-identric-geometric mean inequality, and the relationship

3.20can be directly verified from the explicit form ofIa, b.

Now, settingp −1 in the previous theorem, we immediately deduce the following result whose proof is similar to that of the above corollary.

Corollary 3.5. The sequences Υ−1,na, bn and Θ−1,na, bn both converge to the same limit

La, b, logarithmic mean ofaandb, with the relationship

La, b HLa, Aa, b, LAa, b, b. 3.22

We left to the reader the routine task of formulating, from the above corollaries with

3.1and3.2, the relevant iterative algorithms converging, respectively, to the identric and logarithmic means ofaandb.

4. Approximation of the Power Difference Mean

Dp

We preserve the same notations as in the previous sections. The present section is devoted to approximate the noncross meanDpin terms of the cross meanBp. For this, we define the following schemes:

Φp,n1a, b A

Φp,n

a, Bpa, b,Φp,n

Bpa, b, b,

Φp,0a, b Aa, b, Ψp,n1a, b A

Ψp,n

a, Bpa, b,Ψp,n

Bpa, b, b,

Ψp,0a, b Bpa, b.

4.1

Similarly to the above section,Φp,nandΨp,nare binary means for alln≥0, and

Φp,n1a, b A⊗Φp,n

a, Bpa, b, Bpa, b, b, 4.2

with analogous relation for Ψp,na, b_n. The study of the convergence of the sequences

Φp,na, b_nandΨp,na, b_n, together with related properties and common limit, is similar

(9)

Theorem 4.1. With the above, the following assertions are met.

iFor alln≥0,a, b >0, andpreal number,

Φp,n1a, b A

Φp,na, b,Ψp,na, b

. 4.3

iiForp≥1, the inequalities

Aa, b≤ · · · ≤Φp,n−1a, b≤Φp,na, b≤Ψp,na, b≤Ψp,n−1a, b≤ · · · ≤Bpa, b 4.4

hold and, ifp≤1 the above inequalities are reversed, with equalities forp1.

iiiThe sequencesΦp,na, b_n and Ψp,na, b_n both converge to the same limit Dpa, b,

power diﬀerence mean ofaandb, with the following relationship:

Dpa, b ADpa, Bpa, b, DpBpa, b, b. 4.5

Forp0in the sensep → 0, we recall thatsee1.4and1.10B0a, b Ga, b

√

abandD0a, b La, b. In this case, the above sequences become, respectively,

Φ0,n1a, b

Φ0,n

a,√ab Φ0,n

_√

ab, b

2 ,

Φ0,0a, b Aa, b: ab

2 ,

4.6

Ψ0,n1a, b

Ψ0,n

a,√ab Ψ0,n

_√

ab, b

2 ,

Ψ0,0a, b Ga, b:

ab.

4.7

With this, we may state the next result.

Corollary 4.2. The sequences Φ0,na, b_n and Ψ0,na, b_n defined by 4.6 and 4.7 both

converge to the same limitLa, blogarithmic mean ofaandb, with the following formulae:

La, b

∞

n1

a1/2n

b1/2n

2

. 4.8

Proof. The first part of the corollary follows from the above theorem with the fact that

D0a, b La, b. Let us prove the second part. Since Φ0,n is a mean for alln ≥ 0, the

homogeneity axiom with4.6yields

Φ0,n1a, b

√

a√b

2 Φ0,n

_√

(10)

for all n ≥ 0, with similar recursive relation forΨ0,n_n. By mathematical induction, with

Φ0,0a, b ab/2 andΨ0,0a, b

√

ab, we easily deduce that

Φ0,na, b

a1/2n

b1/2n

2

n

i1

a1/2i

b1/2i

2

,

Ψ0,na, b ab1/2

n1n

i1

a1/2i

b1/2i

2

,

4.10

for everyn≥0. This, when combined with the first part, gives the desired result so completes the proof.

The explicit formulae of La, b, in terms of infinite product, obtained in the above corollary is not obvious to establish directly. However, for p 0, inequalities 4.4 give

Ga, b ≤ La, b ≤ Aa, b the known arithmetic-logarithmic-geometric mean inequality, while relationship4.5implies that

La, b ALa, Ga, b, LGa, b, b, 4.11

which can be directly verified from the explicit form ofLa, b.

We end this section by stating the following remark showing the interest of the above algorithms and the generality of our approach.

Remark 4.3. In the two above sections, we have obtained the following.

iThe logarithmic mean La, b of a and b, containing logarithm, has been approached by two iterative algorithms and explicit formulae involving only the elementary operations sum, product, inverse, and square root of positive real numbers. Such algorithms are simple and practical in the theoretical context as in the numerical purpose.

iiThe identric mean Ia, b of a and b, having a transcendent expression, is here approached by algorithms of algebraic type, that is, containing only the sum, product, and square root of positive real numbers. Such algorithms are useful for the theoretical study and simple for the numerical computation.

5. Motivation and Some Open Problems

As we have already seen, the power binomial mean Bp is a cross mean while the power logarithmic and diﬀerence meansLp and Dp are not always cross means. Approximations ofLpandDpby simple iterative algorithms involvingBphave been discussed. In particular, relationships3.20,3.22, and4.11are derived from the related algorithms and appear to be new in their brief forms. From this, we may naturally arise the following.

Problem 1. 1Determine the set of all real numbersp, such thatLpresp.,Dpis a cross mean.

(11)

There are many other scalar means which we have not recalled above. For instance, let

r,sbe two given real numbers anda, b >0, we recall the following.

iThe Stolarsky meanEr,sa, bof orderr, sofaandbis given by,1,2 ,

Er,sa, b

r s

bs₋_as br−_ar

1/s−r

, Er,sa, a a. 5.1

This includes some of the most familiar cases in the sense

Er,ra, b exp

−1

r

ar_ln_a₋_br_ln_b ar₋_br

, Er,0a, b

1

r

br ₋_ar

lnb−lna

1/r

5.2

if_{r /}0, withE0,0a, b Ga, b.

iiThe Gini meanGr,sa, bof orderr, sofaandbis defined by,3 ,

Gr,sa, b

as_bs ar_br

1/s−r

. 5.3

The meanEr,s extends the power binomial, logarithmic, and diﬀerence means by virtue of the following relations:

Ep,2pBp, E1,p1Lp, Ep,p1 Dp, 5.4

for all real numbersp. So,Er,sis not always a cross mean. Our second open problem can be recited as follows.

Problem 2. 1Determine the set of all couplesr, sof real numbers, such thatEr,sis a cross

mean.

2Is it possible to approximateEr,sby an iterative algorithm involving only the power binomial cross mean?

Analogue of the second point of the above problem for the Gini meanGr,sis without any greatest interest sinceGr,scan be explicitly written in terms of the power binomial cross mean as follows:

Gr,sa, b

Bss Brr

1/s−r

. 5.5

Clearly,G0,pBpfor all real numbersp. However, it is not hard to verify by a counterexample

that the Gini meanGr,sis not always a cross mean. So, to determine the set of all couplesr, s

such thatGr,sis a cross mean is not obvious and appears to be interesting.

(12)

Problem 3. Prove or disprove that every mean m can be, explicitly or approximately, defined

in terms of cross means.

The extension of scalar means from the case that the variables are positive real numbers to the case that the variables are positive operatorsresp., convex functionalshas extensive several developments and interesting applications, see4,5 and the related references cited therein. So, it is natural to put the following.

Problem 4. What should be the reasonable analogues of the above notions and results for

means with operatorresp., functionalvariables?

References

1 K. B. Stolarsky, “Generalizations of the logarithmic mean,” Mathematics Magazine, vol. 48, no. 2, pp. 87–92, 1975.

2 K. B. Stolarsky, “The power and generalized logarithmic means,” The American Mathematical Monthly, vol. 87, no. 7, pp. 545–548, 1980.

3 C. Gini, “Di una formula comprensiva delle medie,” Metron, vol. 13, pp. 3–22, 1938.

4 M. Ra¨ıssouli, “Discrete operator and functional means can be reduced to the continuous arithmetic mean,” International Journal of Open Problems in Computer Science and Mathematics, vol. 3, no. 2, pp. 186– 199, 2010.

(13)

Submit your manuscripts at

http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

Journal of

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied MathematicsJournal of

http://www.hindawi.com Volume 2014 Mathematical PhysicsAdvances in

Complex Analysis

Journal of

Optimization

Journal of

Combinatorics

International Journal of Hindawi Publishing Corporation

Journal of

Function Spaces

Abstract and Applied Analysis Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporation http://www.hindawi.com Volume 2014

The Scientific

World Journal

Discrete Dynamics in Nature and Society Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation

Discrete Mathematics

Journal of

Approaching the Power Logarithmic and Difference Means by Iterative Algorithms Involving the Power Binomial Mean

Research Article