KY FAN INEQUALITY AND BOUNDS FOR DIFFERENCES OF MEANS

PII. S0161171203207171 http://ijmms.hindawi.com

© Hindawi Publishing Corp.

KY FAN INEQUALITY AND BOUNDS FOR DIFFERENCES OF MEANS

PENG GAO Received 23 July 2002

We prove an equivalent relation between Ky Fan-type inequalities and certain bounds for the differences of means. We also generalize a result of Alzer et al.

(2001).

2000 Mathematics Subject Classification: 26D15, 26D20.

1. Introduction. Let Pn,r(x) be the generalized weighted power means:

Pn,r(x)=(n

i=1ωix^r_i)^1/r, whereωi>0, 1≤i≤nwithn

i=1ωi=1 andx= (x1,x2,...,x_n). Here,P_n,0(x)=n

i=1x_i^ωidenotes the limit ofP_n,r(x)asr→0⁺, which can be proved by noting that ifp(r )=ln(n

i=1ω_ix_i^r), thenp(0)= ln(n

i=1x_i^ωi)=ln(Pn,0(x)). We writePn,r forPn,r(x)when there is no risk of confusion.

In this paper, we assume that 0< x1≤x2≤ ··· ≤x_n. With any givenx, we associatex=(1−x1,1−x2,...,1−xn)and writeAn=Pn,1,Gn =Pn,0, and H_n=P_n,−1. When 1−x_i≥0 for alli, we defineA_n=P_n,1(x)and similarly for G_nandH_n. We also letσ_n=n

i=1ω_i[x_i−A_n]².

The following counterpart of the arithmetic mean-geometric mean inequal- ity, due to Ky Fan, was first published by Beckenbach and Bellman [7].

Theorem1.1. Forx_i∈(0,1/2], A_n Gn ≤An

G_n (1.1)

with equality holding if and only ifx1= ··· =x_n.

In this paper, we consider the validity of the following additive Ky Fan-type inequalities (withx1< xn<1):

x1

1−x1

<P_n,r −P_n,s P_n,r−P_n,s < xn

1−x_n. (1.2)

Note that by a change of variablesxi→1−xi, the left-hand side inequality is equivalent to the right-hand side inequality in (1.2). We can deduce (see [9]) Theorem 1.1from the caser=1,s=0, andxn≤1/2 in (1.2), which is a result

of Alzer [5]. Gao [9] later proved the validity of (1.2) forr=1,−1≤s <1, and xn≤1/2.

What is worth mentioning is a nice result of Mercer [12] who showed that the validity ofr=1 ands=0 in (1.2) is a consequence of a result of Cartwright and Field [8] who established the validity ofr=1 ands=0 for the following bounds for the differences between power means (r > s):

r−s 2x1

σn≥Pn,r−Pn,s≥r−s

2xnσn, (1.3)

where the constant(r−s)/2 is the best possible (see [10]).

We point out that inequalities (1.2) and (1.3) do not hold for allr > s. We refer the reader to the survey article [2] and the references therein for an account of Ky Fan’s inequality, and to [4,5,10,11] for other interesting refinements and extensions of (1.3).

Mercer’s result reveals a close relation between (1.3) and (1.2), and it is our main goal in the paper to prove that the validities of (1.3) and (1.2) are equiva- lent for fixedrands. As a consequence of this result, we give a characterization of the validity of (1.3) forr=1 ors=1. A solution of an open problem from [11] is also given.

Among the numerous sharpenings of Ky Fan’s inequality in the literature, we have the following inequalities connecting the three classical means (with ω_i=1/nhere):

H_n Hn

n−1A_n An ≤G_n

Gn

n

≤A_n An

n−1H_n

Hn. (1.4)

The right-hand side inequality of (1.4) is due to W. L. Wang and P. F. Wang [14] and the left-hand side inequality was recently proved by Alzer et al. [6].

It is natural to ask whether we can extend the above inequality to the weighted case, and using the same idea as in [6], we show that this is indeed true inSection 5.

2. The main theorem

Theorem2.1. For fixedr > s, the following inequalities are equivalent:(i)in- equality (1.2) forxn≤1/2;(ii)inequality (1.2);(iii)inequality (1.3).

Proof. (iii)⇒(ii) follows from a similar argument as given in [12], (ii)⇒(i) is trivial, so it suffices to show that (i)⇒(iii).

Fix r > s assuming that (1.2) holds for xn≤1/2. Without loss of gener- ality, we can assume that x1< xn. For a given x=(x1,x2,...,xn), let y = (x1,x2,...,x_n). We can choose small so thatx_n≤1/2. Now, applying the right-hand side inequality (1.2) fory, we get

x_nP_n,r(x)−P_n,s(x)>1−x_n ²

P_n,r(y)−P_n,s(y). (2.1)

Letf ()=Pn,r(y)−Pn,s(y), thenf(0)=0 andf(0)=(r−s)σn. Thus, by lettingtend to 0, it is easy to verify that the limit of the expression on the right-hand side of (2.1) is(r−s)σ_n/2. We can consider the left-hand side of (1.2) by a similar argument and this completes the proof.

3. An application ofTheorem 2.1

Lemma3.1. If inequality (1.3) holds forr > s, then0≤r+s≤3.

Proof. Letn=2, and writeω1=1−q,ω2=q,x1=1, andx2=1+twith t≥ −1. Let

D(t;r ,s,q)=r−s 2

2 i=1

w_i x_i−A2

2−P_2,r+P_2,s. (3.1)

Fort≥0,D(t;r ,s,q)≥0 implies the validity of the left-hand side inequality of (1.3) while for−1≤t≤0,D(t;r ,s,q)≤0 implies the validity of the right- hand side inequality of (1.3).

Using the Taylor series expansion ofD(t;r ,s,q)aroundt=0, it is readily seen that D(0;r ,s,q)=D⁽¹⁾(0;r ,s,q)=D⁽²⁾(0;r ,s,q)=0. Thus, by the La- grangian remainder term of the Taylor expansion,

D(t;r ,s,q)=D⁽³⁾(θt;r ,s,q)

3! t³ (3.2)

with 0< θ <1.

Since

t→0lim⁺D⁽³⁾(θt;r ,s,q)=D⁽³⁾(0;r ,s,q), (3.3)

a necessary condition for (1.3) to hold isD⁽³⁾(0;r ,s,q)≥0 for 0≤q≤1. The calculation yields

D⁽³⁾(0;r ,s,q)=(r−s)q(q−1)(3−2r−2s)q−(3−r−s). (3.4) It is easy to check that this is equivalent to 0≤r+s≤3.

Theorem3.2. Letr > s. Ifr=1, inequality (1.3) holds if and only if−1≤ s <1. Ifs=1, inequality (1.3) holds if and only if1< r≤2.

Proof. A result of Gao [9] shows the validity of (1.2) forr=1,−1≤s <1, x_n≤1/2, and a similar result of his [10] shows the validity of (1.2) fors=1, 1< r≤2,xn≤1/2. Thus, it follows fromTheorem 2.1that (1.3) holds forr= 1,−1≤s <1, ands=1, 1< r≤2. This proves the “if” part of the statement, and the “only if” part follows from the previous lemma.

We note here that a special case ofTheorem 3.2answers an open problem of Mercer [11], namely, we have shown that

1 x1

σ_n≥A_n−H_n≥ 1

xnσ_n. (3.5)

4. Two lemmas

Lemma4.1. Letx,b,u, andvbe real numbers with0< x≤b,u≥1,v≥0, andu+v≥2, thenf (u,v,x,b)≤0, where

f (u,v,x,b)=u+v−1

ux+vb+ 1

x²(u/x+v/b)−1

x− u+v−2

b²(u+v)²v(x−b) (4.1) with equality holding if and only ifx=borv=0oru=v=1.

Proof. Letx < b,u >1, andv >1. We have f (u,v,x,b)=v(b−x)

− (u−1)b+(v−1)x

x(bv+ux)(bu+vx)+(u−1)+(v−1) b²(u+v)²

< v(b−x) xb²(u+v)²

(u−1)+(v−1)

x−(u−1)b−(v−1)x

= −v(u−1)(b−x)² xb²(u+v)² <0

(4.2)

sinceb²(u+v)²> (bv+ux)(bu+vx). Thus, we conclude thatf (u,v,x,b)≤ 0 for 0< x≤b,u≥1,v≥0, andu+v≥2.

Lemma4.2. Letx, a,b,u, v, ands be real numbers with0< x≤a≤b, u≥1,v≥1,u+v≥3, and0≤s≤v, then

u+v−1

ux+sa+(v−s)b+ 1 x²

u/x+s/a+(v−s)/b−1 x

− u+v−2 b²(u+v)²

s(x−a)+(v−s)(x−b)

≤0

(4.3)

with equality holding if and only if one of the following cases is true:(1)x=a= b;(2)s=0andx=b;(3)s=vandx=a.

Proof. LetM= {(s,a)∈R²|0≤s≤v, x≤a≤b}. Furthermore, we define H(s,a)as the expression on the left-hand side of (4.3), where(s,a)∈M. It suffices to show thatH(s,a) <0. We denote the absolute minimum ofH by m=(s0,a0). Ifmis an interior point ofM, then we obtain

0=1 s

∂H

∂a− 1 a−b

∂H

∂s _(s,a)=(s

0,a₀)= b−a

x⁴a²b

u/x+s/a+(v−s)/b2>0.

(4.4)

Hence,mis a boundary point ofM, so we get m∈

s0,x ,

s0,b ,

0,a0

, v,a0

. (4.5)

UsingLemma 4.1, we obtain H

s0,x

=f

u+s0,v−s0,x,b

≤0, H

s0,b

=H 0,a0

=f (u,v,x,b)≤0,

H v,a0

=f

u,v,x,a0

−v(u+v−2)a0−xb²−a²₀ a²₀b²(u+v)² ≤0.

(4.6)

Thus, we get that if(s,a)∈M, thenH(s,a)≤0. The conditions for equality can be easily checked usingLemma 4.1.

5. A sharpening of Ky Fan’s inequality. In this section, we prove the fol- lowing theorem.

Theorem5.1. For0< x1≤ ··· ≤x_n,q=min{ω_i}, 1−2q

2x₁² σ_n≥(1−q)lnA_n+qlnH_n−lnG_n≥1−2q

2x²n σ_n, (5.1) 1−2q

2x₁² σ_n≥lnG_n−qlnA_n−(1−q)lnH_n≥1−2q

2x²_n σ_n (5.2) with equality holding if and only ifq=1/2orx1= ··· =xn.

Proof. The proof uses the ideas in [6]. We prove the right-hand side in- equality of (5.1); the proofs for other inequalities are similar. Fix 0< x=x1, x_n=bwithx1< x_n,n≥2; we define

fn xn,q

=(1−q)lnAn+qlnHn−lnGn−1−2q

2x²n σn, (5.3) where we regardAn,Gn, andHnas functions ofxn=(x1,...,xn).

We then have gn

x2,...,x_n−1 := 1

ω1

∂fn

∂x1=1−q A_n +qHn

x²₁ − 1

x1−1−2q xn²

x1−An

. (5.4)

We want to show thatgn≤0. Let D= {(x2,...,x_n−1)∈Rⁿ⁻²|0< x≤x2≤

··· ≤x_n−1≤b}. Leta=(a2,...,a_n−1)∈Dbe the point in which the absolute minimum ofg_nis reached. Next, we show that

a=(x,...,x,a,...,a,b,...,b) withx < a < b, (5.5) where the numbers x, a, and b appear r, s, and ttimes, respectively, with r ,s,t≥0 andr+s+t=n−2.

Suppose not, this implies that two components ofahave different values and are interior points of D. We denote these values byak and al. Partial differentiation leads to

B

a²_i+C=0 (5.6)

fori=k,l, where

B=qH_n²

x²₁, C= −1−q

A²n +1−2q

xn² . (5.7)

SincezB/z²+C is strictly monotonic for z >0, then (5.6) yieldsa_k=a_l. This contradicts our assumption thata_k≠a_l. Thus, (5.5) is valid and it suffices to show thatgn≤0 for the casen=2,3.

Whenn=2, by setting x1=x,x2=b,ω1/q=u, andω2/q=v, we can identifyg2as (4.1), and the result follows fromLemma 4.1.

When n= 3, by setting x1 =x, x2 = a, x3 = b, ω1/q = u, ω2/q = s, and ω3/q =v−s, we can identify g3 as (4.3), and the result follows from Lemma 4.2.

Thus, we have shown thatgn=(1/ω1)∂fn/∂x1≤0 with equality holding if and only ifn=1 orn=2,q=1/2. By lettingx1tend tox2, we have

f_n x_n,q

≥f_n−1 x_n−1,q

≥f_n−1

x_n−1,q, (5.8)

wherex_n−1=(x2,...,xn)with weightsω1+ω2,...,ω_n−1,ωnandq=min{ω1

+ω2,...,ωn}. Here, we have used the following inequality, which is a conse- quence of (3.5) (see [9]):

lnAn−lnHn≥ 1

x²nσn. (5.9)

It then follows by induction thatfn≥f_n−1≥ ··· ≥f2=0 whenq=1/2 in f2or elsef_n≥f_n−1≥ ··· ≥f1=0, and this completes the proof.

We note that the above theorem gives a sharpening of Sierpi´nski’s inequality [13], originally stated for the unweighted case (ωi=1/n) as

H_nⁿ⁻¹A_n≤G_n≤Aⁿ⁻¹_n H_n. (5.10)

The following corollary gives refinements of (1.4).

Corollary5.2. For0< x1≤ ··· ≤xn<1,q=min{ωi}, A⁽¹n ^−q)Hn^q

Gn

(1−x1)²/x₁²

≥A¹n^−qHn^q

Gn ≥

A⁽¹n ^−q)Hn^q

Gn

(1−xn)²/x²n

, G_n

A^qnH^(1−q)n

(1−x1)²/x₁²

≥ Gn

A^qnHn^1−q ≥

G_n A^qnHn^(1−q)

(1−xn)²/x²n

,

(5.11)

with equality holding if and only ifx1=x2= ··· =xnorq=1/2.

Proof. This is a direct consequence ofTheorem 5.1, following from a sim- ilar argument as in [12].

6. Concluding remarks. We note that if forxn≤1/2, we have x1

1−x1

β

<P_n,r −P_n,s Pn,r−Pn,s <

xn

1−xn

α

, (6.1)

thenβ≥1 andα≤1; otherwise, by lettingtend to 0 in (2.1), we get contra- dictions.

It was conjectured that an additive companion of (1.4) is true (see [1]) n

Gn−G_n

≤(n−1)

An−A_n

+Hn−H_n. (6.2)

In [3], Alzer asked if the above conjecture is true and whether there exists a weighted version. Based on what we have got in this paper, it is natural to give the following conjecture of the weighed version of (6.2).

Conjecture6.1. For0< x1≤ ··· ≤x_n≤1/2andq=min{ω_i}, G_n−G_n≤(1−q)A_n−A_n

+qH_n−H_n. (6.3)

Recently, Alzer et al. [6] asked the following question: what is the largest numberα=α(n)and what is the smallest numberβ=β(n)such that

α

A_n−A_n

+(1−α)

H_n−H_n

≤G_n−G_n≤β

A_n−A_n

+(1−β)

H_n−H_n (6.4) for allx_i∈(0,1/2] (i=1,...,n)?

We note here thatα≤0 since the left-hand side inequality above can be written as

αAn+(1−α)Hn−Gn≤αA_n+(1−α)H_n−G_n. (6.5)

By a similar argument as in the proof ofTheorem 2.1, replacing(x1,...,xn) by(x1,...,xn)and lettingtend to 0 in (6.5), we find that (6.5) implies that

αA_n+(1−α)H_n−G_n≤0 (6.6)

for anyx. If we further letx1tend to 0 in (6.6), we get

αA_n≤0 (6.7)

which implies thatα≤0.

Acknowledgment. The author is grateful to the referees for their helpful comments and suggestions.

References

[1] H. Alzer,An inequality for arithmetic and harmonic means, Aequationes Math.

46(1993), no. 3, 257–263.

[2] ,The inequality of Ky Fan and related results, Acta Appl. Math.38(1995), no. 3, 305–354.

[3] ,On Ky Fan’s inequality and its additive analogue, J. Math. Anal. Appl.204 (1996), no. 1, 291–297.

[4] ,A new refinement of the arithmetic mean–geometric mean inequality, Rocky Mountain J. Math.27(1997), no. 3, 663–667.

[5] ,On an additive analogue of Ky Fan’s inequality, Indag. Math. (N.S.) 8 (1997), no. 1, 1–6.

[6] H. Alzer, S. Ruscheweyh, and L. Salinas,On Ky Fan-type inequalities, Aequationes Math.62(2001), no. 3, 310–320.

[7] E. F. Beckenbach and R. Bellman,Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Bd. 30, Springer-Verlag, Berlin, 1961.

[8] D. I. Cartwright and M. J. Field,A refinement of the arithmetic mean-geometric mean inequality, Proc. Amer. Math. Soc.71(1978), no. 1, 36–38.

[9] P. Gao,A generalization of Ky Fan’s inequality, Int. J. Math. Math. Sci.28(2001), no. 7, 419–425.

[10] ,Certain bounds for the differences of means, RGMIA Res. Rep. Coll. 5 (2002), no. 3, Article 7.

[11] A. McD. Mercer,Bounds forA–G, A–H, G–H, and a family of inequalities of Ky Fan’s type, using a general method, J. Math. Anal. Appl.243(2000), no. 1, 163–173.

[12] P. R. Mercer,A note on Alzer’s refinement of an additive Ky Fan inequality, Math.

Inequal. Appl.3(2000), no. 1, 147–148.

[13] W. Sierpi´nski,On an inequality for arithmetic, geometric and harmonic means, Warsch. Sitzungsber.2(1909), 354–358 (Polish).

[14] W. L. Wang and P. F. Wang,A class of inequalities for the symmetric functions, Acta Math. Sinica27(1984), no. 4, 485–497 (Chinese).

Peng Gao: Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

E-mail address:[email protected]

Mathematical Problems in Engineering

Special Issue on

Time-Dependent Billiards

Call for Papers

This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi accelera- tion (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.

This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).

We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers dis- cussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.

To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.

Authors should follow the Mathematical Problems in Engineering manuscript format described at

submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at

mts.hindawi.com/

according to the following timetable:

Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009

Guest Editors

Edson Denis Leonel,

Departamento de Estatística, Matemática Aplicada e Computação, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Avenida 24A, 1515 Bela Vista, 13506-700 Rio Claro, SP, Brazil ; [email protected]

Alexander Loskutov,

Physics Faculty, Moscow State University, Vorob’evy Gory, Moscow 119992, Russia;

[email protected]

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