ON ACZÉL'S INEQUALITY FOR REAL NUMBERS

On Aczél's Inequality for Real Numbers

This is the Published version of the following publication

Dragomir, Sever S and Cho, Yeol Je (2000) On Aczél's Inequality for Real Numbers. RGMIA research report collection, 3 (1).

The publisher’s official version can be found at

Note that access to this version may require subscription.

Downloaded from VU Research Repository https://vuir.vu.edu.au/17283/

S. S. Dragomir and Y. J. Cho

Abstract. In this note, we point out some new inequalities of Aczel’s type for real numbers.

I. Introduction

In 1956, J. Acz´el has proved the following interesting inequality ([2, p.

57], [3, p. 117 ]):

Theorem A. Let a = (a₁, a₂,· · · , a_n) and b = (b₁, b₂,· · · , b_n) be two se- quences of real numbers such that

0< a²₁−a²₂− · · · −a²_n or 0< b²₁−b²₂− · · · −b²_n. Then

(1.1) (a²₁−a²₂− · · · −a²_n)(b²₁−b²₂− · · · −b²_n)

≤(a₁b₁−a₂b₂ − · · · −a_nb_n)²,

with the equality if and only if the sequences a and b are proportional.

For various generalizations of Theorem A, see the recent book ([3, p. 117]) where further references are given.

Now, in this note, we give another proof than that embodied in [2, p. 57]

for a weighted variant of (1.1).

Assume that

Xn

i=1

p_ia²_i ≤a² and Xn

i=1

p_ib²_i ≤b²,

(1991) AMS Subject Classification: 28D15 Key Words and Phrases: Acz´el’s inequality.

Typeset byAMS-TEX 1

2 S. S. DRAGOMIR AND Y. J. CHO

where ai, bi, a, b ∈ R and 0 ≤ pi for i = 1,2,· · · , n. Then we have the following inequality:

(1.2) 

a²− Xn

i=1

p_ia²_i‘

b²− Xn

i=1

p_ib²_i‘

≤ ab−

Xn

i=1

p_ia_ib_i‘2

.

Indeed, by a simple calculation, we have

(a²−c²)(b²−d²)≤(|ab| − |cd|)² for all a, b, c, d∈R. Thus we have

(1.3)

a²− Xn

i=1

p_ia²_i‘

b²− Xn

i=1

p_ib²_i‘

≤

|ab| −Xⁿ

i=1

p_ia²_i‘1/2Xⁿ

i=1

p_ib²_i‘1/2‘2

.

By Cauchy-Buniakowski-Schwarz’s inequality, we have

ŒŒ

ΠXn

i=1

p_ia_ib_iŒŒŒ≤Xⁿ

i=1

p_ia²_i‘1/2Xⁿ

i=1

p_ib²_i‘1/2

and so

0≤ |ab| −Xⁿ

i=1

p_ia²_i‘1/2Xⁿ

i=1

p_ib²_i‘1/2

≤ |ab| −ŒŒŒ Xn

i=1

piaibi

ŒŒ

Œ

=ŒŒŒ|ab| −ŒŒŒ Xn

i=1

p_ia_ib_iŒŒŒŒŒŒ

≤ŒŒŒab− Xn

i=1

p_ia_ib_iŒŒŒ.

Thus, we have

(1.4) 

|ab| −Xⁿ

i=1

p_ia²_i‘1/2Xⁿ

i=1

p_ib²_i‘1/2‘2

≤ ab−

Xn

i=1

p_ia_ib_i‘2

.

Therefore, from (1.3) and (1.4), we have the inequality (1.2). This completes the proof.

II. The Results

We will start with the following theorem which give a refinement of the following variant of Acz´el’s inequality:

(2.1)  a²−

Xn

i=1

p_ia²_i‘1/2 b²−

Xn

i=1

p_ib²_i‘1/2

≤ŒŒŒab− Xn

i=1

p_ia_ib_iŒŒŒ, assuming that

Xn

i=1

p_ia²_i ≤a² and Xn

i=1

p_ib²_i ≤b²

and ai, bi, a, b∈R, 0≤pi for i= 1,2,· · · , n.

Theorem 2.1. Assume that a_i, b_i, p_i, a, b are as above and0≤q_i ≤p_i for all i= 1,2,· · · , n. Then we have the following inequality:

(2.2)

0≤ Xn

i=1

qi|aibi| −ŒŒŒ Xn

i=1

qiaibi

ŒŒ

Œ

≤hXⁿ

i=1

q_ia²_i Xn

i=1

q_ib²_ii1/2

−ŒŒŒ Xn

i=1

q_ia_ib_iŒŒŒ

≤ŒŒŒab− Xn

i=1

p_ia_ib_iŒŒŒ− a²−

Xn

i=1

p_ia²_i‘1/2 b²−

Xn

i=1

p_ib²_i‘1/2

.

Proof. From a²−Pn

i=1pia²_i ≥0 andb²−Pn

i=1pia²_i ≥0, it follows that a²−

Xn

i=1

(p_i−q_i)a²_i ≥a²− Xn

i=1

p_ia²_i ≥0,

b²− Xn

i=1

(pi−qi)b²_i ≥b²− Xn

i=1

pib²_i ≥0.

Now, for t_i =p_i−q_i ≥0, by (1.2), we have

a²− Xn

i=1

t_ia²_i‘

b²− Xn

i=1

t_ib²_i‘

≤ ab−

Xn

i=1

t_ia_ib_i‘2

,

4 S. S. DRAGOMIR AND Y. J. CHO

i.e.,

ha²− Xn

i=1

p_ia²_i‘ +

Xn

i=1

q_ia²_iih

b²− Xn

i=1

p_ib²_i‘ +

Xn

i=1

q_ib²_ii

≤h

ab− Xn

i=1

p_ia_ib_i‘ +

Xn

i=1

q_ia_ib_ii2

.

Applying the well-known Cauchy-Buniakowski-Schwarz’s inequality for real number, we have

(2.4) hXⁿ

i=1

q_ia²_i)^1/2Xⁿ

i=1

q_ib²_i‘1/2

+ a²−

Xn

i=1

p_ia²_i‘1/2 b²−

Xn

i=1

p_ib²_i‘1/2i2

≤Xⁿ

i=1

q_ia²_i +h

a²− Xn

i=1

p_ia²_i‘1/2i2‘Xⁿ

i=1

q_ib²_i +h

b²− Xn

i=1

p_ib²_i‘1/2i2‘ and, by the triangle inequality,

(2.5)

ŒŒ

Œ ab−

Xn

i=1

piaibi

‘+ Xn

i=1

qiaibi

ŒŒ

Œ

≤ŒŒŒab− Xn

i=1

p_ia_ib_iŒŒŒ+ŒŒŒ Xn

i=1

q_ia_ib_iŒŒŒ. Thus, from (2.3), (2.4) and (2.5), it follows that

Xⁿ

i=1

q_ia²_i‘1/2Xⁿ

i=1

q_ib²_i‘1/2

+ a²−

Xn

i=1

p_ia²_i‘1/2 b²−

Xn

i=1

p_ib²_i‘1/2

≤ŒŒŒab− Xn

i=1

piaibi

ŒŒ

Œ+ŒŒŒ Xn

i=1

qiaibi

ŒŒ

Œ,

which implies that 0≤

Xn

i=1

q_i|a_ib_i| −ŒŒŒ Xn

i=1

q_ia_ib_iŒŒŒ

≤Xⁿ

i=1

q_ia²_i‘1/2Xⁿ

i=1

q_ib²_i‘1/2

−ŒŒŒ Xn

i=1

q_ia_ib_iŒŒŒ

≤ŒŒŒab− Xn

i=1

pia_ib_iŒŒŒ− a²−

Xn

i=1

p_ia²_i‘1/2 b²−

Xn

i=1

p_ib²_i‘1/2

. This completes the proof.

Corollary 2.2. With the above assumptions for ai, bi, a, b∈R and 0≤pi

for i= 1,2,· · · , n, we have the following inequality:

0≤ Xn

i=1

p_i|a_ib_i| −ŒŒŒ Xn

i=1

p_ia_ib_iŒŒŒ

≤hXⁿ

i=1

pia²_i Xn

i=1

pib²_ii1/2

−ŒŒŒ Xn

i=1

piaibi

ŒŒ

Œ

≤ŒŒŒab− Xn

i=1

p_ia_ib_iŒŒŒ− a²−

Xn

i=1

p_ia²_i‘1/2 b²−

Xn

i=1

p_ib²_i)^1/2.

Another result of Acz´el’s type is as follows:

Theorem 2.3. Assume that a, b, a_i, b_i ∈ R and 0≤ p_i for i = 1,2,· · · , n are such that

Xn

i=1

p_ia²_i ≤a² and Xn

i=1

p_ib²_i ≤b².

Then we have the following inequality:

h|a| −Xⁿ

i=1

p_ia²_i‘1/2i1/2h

|b| −Xⁿ

i=1

p_ib²_i‘1/2i1/2

≤ |ab|^1/2−ŒŒŒ Xn

i=1

p_ia_ib_iŒŒŒ^1/2.

Proof. We will start with the following elementary inequality:

(2.7) p

(x−y)(z−u)≤√

xz−√yu,

where x ≥ y ≥ 0 and z ≥ u ≥ 0. Indeed, the inequality (2.7) is equivalent with

(x−y)(z−u)≤(√

xz−√yu)² =xz−2√xzyu+yu, i.e.,

xz+yu−yz−xu≤xz−2√xzyu+yu, which is equivalent with

2√xzyu≤yz+xu

6 S. S. DRAGOMIR AND Y. J. CHO

for x, y, z, u≥0, which is obvious.

Now, putting

x=|a|, y=Xⁿ

i=1

p_ia²_i‘1/2

, z =|b|, u=Xⁿ

i=1

p_ib²_i)^1/2,

then, by the inequality (2.7), we have

(2.8)

h|a| −Xⁿ

i=1

p_ia²_i‘1/2i1/2h

|b| −Xⁿ

i=1

p_ib²_i‘1/2i1/2

≤ |ab|^1/2−Xⁿ

i=1

pia²_i‘1/4Xⁿ

i=1

pib²_i‘1/4

.

By Cauchy-Buniakowski-Schwarz’s inequality, we have

ŒŒ

ΠXn

i=1

p_ia_ib_iŒŒŒ^1/2 ≤hXⁿ

i=1

p_ia²_i Xn

i=1

p_ib²_ii1/4

and so, by (2.8), we have the desired inequality (2.6). This completes the proof.

Corollary 2.4. Let a, b, a_i, b_i ∈R for i= 1,2,· · · , n be such that Xn

i=1

a²_i ≤a² and Xn

i=1

b²_i ≤b².

Then we have the following inequality:

h|a| −Xⁿ

i=1

a²_i‘1/2i1/2h

|b| −Xⁿ

i=1

b²_i‘1/2i1/2

≤ |ab|^1/2−ŒŒŒ Xn

i=1

a_ib_iŒŒŒ^1/2.

Remark. The inequality (2.9) was proved in [1] as a particular case of an inequality holding in inner product spaces.

References

1. S. S. Dragomir,A generalization of J. Acz´el’s inequality in inner product spaces, Acta Math. Hungarica 65(2)(1994), 141–148.

2. D. S. Mitrinovi´c,Analytic Inequalities, Springer-Verlag, 1970.

3. D. S. Mitrinovi´c, J. E. Peˇcari´c and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Acad. Publ., 1993.

Department of Mathematics, Victoria University of Technology, Mel- bourne, Victoria 8001, Australia

Department of Mathematics, Gyeongsang National University, Chinju 660- 701, Korea