Many companion, reverse and related results both for real and complex numbers are also presented

http://jipam.vu.edu.au/

Volume 4, Issue 3, Article 63, 2003

A SURVEY ON CAUCHY-BUNYAKOVSKY-SCHWARZ TYPE DISCRETE INEQUALITIES

S.S. DRAGOMIR

SCHOOL OFCOMPUTERSCIENCE ANDMATHEMATICS

VICTORIAUNIVERSITY, PO BOX14428, MCMC 8001, MELBOURNE,

VICTORIA, AUSTRALIA. sever.dragomir@vu.edu.au

Received 22 January, 2003; accepted 14 May, 2003 Communicated by P.S. Bullen

ABSTRACT. The main purpose of this survey is to identify and highlight the discrete inequalities that are connected with(CBS)−inequality and provide refinements and reverse results as well as to study some functional properties of certain mappings that can be naturally associated with this inequality such as superadditivity, supermultiplicity, the strong versions of these and the corresponding monotonicity properties. Many companion, reverse and related results both for real and complex numbers are also presented.

Key words and phrases: Cauchy-Bunyakovsky-Schwarz inequality, Discrete inequalities.

2000 Mathematics Subject Classification. 26D15, 26D10.

CONTENTS

1. Introduction 4

2. (CBS)– Type Inequalities 5

2.1. (CBS)−Inequality for Real Numbers 5

2.2. (CBS)−Inequality for Complex Numbers 6

2.3. An Additive Generalisation 7

2.4. A Related Additive Inequality 8

2.5. A Parameter Additive Inequality 10

2.6. A Generalisation Provided by Young’s Inequality 11

2.7. Further Generalisations via Young’s Inequality 12

2.8. A Generalisation InvolvingJ−Convex Functions 16

2.9. A Functional Generalisation 18

2.10. A Generalisation for Power Series 19

2.11. A Generalisation of Callebaut’s Inequality 21

2.12. Wagner’s Inequality for Real Numbers 22

2.13. Wagner’s inequality for Complex Numbers 24

ISSN (electronic): 1443-5756

c 2003 Victoria University. All rights reserved.

010-03

References 26

3. Refinements of the(CBS)−Inequality 28

3.1. A Refinement in Terms of Moduli 28

3.2. A Refinement for a Sequence Whose Norm is One 29

3.3. A Second Refinement in Terms of Moduli 31

3.4. A Refinement for a Sequence Less than the Weights 33

3.5. A Conditional Inequality Providing a Refinement 35

3.6. A Refinement for Non-Constant Sequences 37

3.7. De Bruijn’s Inequality 40

3.8. McLaughlin’s Inequality 41

3.9. A Refinement due to Daykin-Eliezer-Carlitz 42

3.10. A Refinement via Dunkl-Williams’ Inequality 43

3.11. Some Refinements due to Alzer and Zheng 45

References 50

4. Functional Properties 51

4.1. A Monotonicity Property 51

4.2. A Superadditivity Property in Terms of Weights 52

4.3. The Superadditivity as an Index Set Mapping 54

4.4. Strong Superadditivity in Terms of Weights 55

4.5. Strong Superadditivity as an Index Set Mapping 57

4.6. Another Superadditivity Property 59

4.7. The Case of Index Set Mapping 61

4.8. Supermultiplicity in Terms of Weights 63

4.9. Supermultiplicity as an Index Set Mapping 67

References 71

5. Reverse Inequalities 72

5.1. The Cassels’ Inequality 72

5.2. The Pólya-Szegö Inequality 73

5.3. The Greub-Rheinboldt Inequality 75

5.4. A Cassels’ Type Inequality for Complex Numbers 75

5.5. A Reverse Inequality for Real Numbers 77

5.6. A Reverse Inequality for Complex Numbers 79

5.7. Shisha-Mond Type Inequalities 82

5.8. Zagier Type Inequalities 83

5.9. A Reverse Inequality in Terms of thesup−Norm 85

5.10. A Reverse Inequality in Terms of the1−Norm 87

5.11. A Reverse Inequality in Terms of thep−Norm 90

5.12. A Reverse Inequality Via an Andrica-Badea Result 92

5.13. A Refinement of Cassels’ Inequality 94

5.14. Two Reverse Results Via Diaz-Metcalf Results 97

5.15. Some Reverse Results Via the ˇCebyšev Functional 99

5.16. Another Reverse Result via a Grüss Type Result 105

References 107

6. Related Inequalities 109

6.1. Ostrowski’s Inequality for Real Sequences 109

6.2. Ostrowski’s Inequality for Complex Sequences 110

6.3. Another Ostrowski’s Inequality 111

6.4. Fan and Todd Inequalities 113

6.5. Some Results for Asynchronous Sequences 114

6.6. An Inequality viaA−G−HMean Inequality 115

6.7. A Related Result via Jensen’s Inequality for Power Functions 116

6.8. Inequalities Derived from the Double Sums Case 117

6.9. A Functional Generalisation for Double Sums 118

6.10. A(CBS)−Type Result for Lipschitzian Functions 119

6.11. An Inequality via Jensen’s Discrete Inequality 121

6.12. An Inequality via Lah-Ribari´c Inequality 122

6.13. An Inequality via Dragomir-Ionescu Inequality 123

6.14. An Inequality via a Refinement of Jensen’s Inequality 124

6.15. Another Refinement via Jensen’s Inequality 127

6.16. An Inequality via Slater’s Result 129

6.17. An Inequality via an Andrica-Ra¸sa Result 131

6.18. An Inequality via Jensen’s Result for Double Sums 132

6.19. Some Inequalities for the ˇCebyšev Functional 134

6.20. Other Inequalities for the ˇCebyšev Functional 136

6.21. Bounds for the ˇCebyšev Functional 137

References 140

Index 141

1. INTRODUCTION

The Cauchy-Bunyakovsky-Schwarz inequality, or for short, the(CBS)−inequality, plays an important role in different branches of Modern Mathematics including Hilbert Spaces Theory, Probability & Statistics, Classical Real and Complex Analysis, Numerical Analysis, Qualitative Theory of Differential Equations and their applications.

The main purpose of this survey is to identify and highlight the discrete inequalities that are connected with(CBS)−inequality and provide refinements and reverse results as well as to study some functional properties of certain mappings that can be naturally associated with this inequality such as superadditivity, supermultiplicity, the strong versions of these and the corresponding monotonicity properties. Many companions and related results both for real and complex numbers are also presented.

The first section is devoted to a number of(CBS)−type inequalities that provides not only natural generalizations but also several extensions for different classes of analytic functions of a real variable. A generalization of the Wagner inequality for complex numbers is obtained.

Several results discovered by the author in the late eighties and published in different journals of lesser circulation are also surveyed.

The second section contains different refinements of the (CBS)− inequality including de Bruijn’s inequality, McLaughlin’s inequality, the Daykin-Eliezer-Carlitz result in the version presented by Mitrinovi´c-Peˇcari´c and Fink as well as the refinements of a particular version obtained by Alzer and Zheng. A number of new results obtained by the author, which are connected with the above ones, are also presented.

Section 4 is devoted to the study of functional properties of different mappings naturally associated to the(CBS)−inequality. Properties such as superadditivity, strong superadditivity, monotonicity and supermultiplicity and the corresponding inequalities are mentioned.

In the next section, Section 5, reverse results for the(CBS)− inequality are surveyed. The results of Cassels, Pólya-Szegö, Greub-Rheinbold, Shisha-Mond and Zagier are presented with their original proofs. New results and versions for complex numbers are also obtained. Reverse results in terms of p−norms of the forward difference recently discovered by the author and some refinements of Cassels and Pólya-Szegö results obtained via Andrica-Badea inequality are mentioned. Some new facts derived from Grüss type inequalities are also pointed out.

Section 6 is devoted to various inequalities related to the (CBS)− inequality. The two in- equalities obtained by Ostrowski and Fan-Todd results are presented. New inequalities obtained via Jensen type inequality for convex functions are derived, some inequalities for the ˇCeby¸sev functionals are pointed out. Versions for complex numbers that generalize Ostrowski results are also emphasised.

It was one of the main aims of the survey to provide complete proofs for the results consid- ered. We also note that in most cases only the original references are mentioned. Each section concludes with a list of the references utilized and thus may be read independently.

Being self contained, the survey may be used by both postgraduate students and researchers interested in Theory of Inequalities & Applications as well as by Mathematicians and other Scientists dealing with numerical computations, bounds and estimates where the(CBS)−in- equality may be used as a powerful tool.

The author intends to continue this survey with another one devoted to the functional and integral versions of the(CBS)−inequality. The corresponding results holding in inner-product and normed spaces will be considered as well.

2. (CBS)– TYPE INEQUALITIES

2.1. (CBS)−Inequality for Real Numbers. The following inequality is known in the lit- erature as Cauchy’s or Cauchy-Schwarz’s or Cauchy-Bunyakovsky-Schwarz’s inequality. For simplicity, we shall refer to it throughout this work as the(CBS)−inequality.

Theorem 2.1. If¯a= (a₁, . . . , a_n)andb¯= (b₁, . . . , b_n)are sequences of real numbers, then (2.1)

n

X

k=1

a_kb_k

!2

≤

n

X

k=1

a²_k

n

X

k=1

b²_k

with equality if and only if the sequences¯aandb¯ are proportional, i.e., there is ar ∈ Rsuch thata_k=rb_kfor eachk ∈ {1, . . . , n}.

Proof. (1) Consider the quadratic polynomialP :R→R,

(2.2) P (t) =

n

X

k=1

(a_kt−b_k)². It is obvious that

P(t) =

n

X

k=1

a²_k

! t²−2

n

X

k=1

a_kb_k

! t+

n

X

k=1

b²_k for anyt∈R.

SinceP (t)≥0for anyt∈Rit follows that the discriminant∆ofP is negative, i.e., 0≥ 1

4∆ =

n

X

k=1

a_kb_k

!2

−

n

X

k=1

a²_k

n

X

k=1

b²_k and the inequality (2.1) is proved.

(2) If we use Lagrange’s identity

n

X

i=1

a²_i

n

X

i=1

b²_i −

n

X

i=1

a_ib_i

!2

= 1 2

n

X

i,j=1

(a_ib_j −a_jb_i)² (2.3)

= X

1≤i<j≤n

(a_ib_j−a_jb_i)² then (2.1) obviously holds.

The equality holds in (2.1) iff

(a_ib_j−a_jb_i)² = 0

for anyi, j ∈ {1, . . . , n}which is equivalent with the fact that¯aand¯bare proportional.

Remark 2.2. The inequality (2.1) apparently was firstly mentioned in the work [2] of A.L.

Cauchy in 1821. The integral form was obtained in 1859 by V.Y. Bunyakovsky [1]. The cor- responding version for inner-product spaces obtained by H.A. Schwartz is mainly known as Schwarz’s inequality. For a short history of this inequality see [3]. In what follows we use the spelling adopted in the paper [3]. For other spellings of Bunyakovsky’s name, see MathSciNet.

2.2. (CBS)−Inequality for Complex Numbers. The following version of the(CBS)−inequality for complex numbers holds [4, p. 84].

Theorem 2.3. If ¯a = (a₁, . . . , a_n) and ¯b = (b₁, . . . , b_n)are sequences of complex numbers, then

(2.4)

n

X

k=1

a_kb_k

2

≤

n

X

k=1

|a_k|²

n

X

k=1

|b_k|²,

with equality if and only if there is a complex number c ∈ C such that a_k = c¯b_k for any k ∈ {1, . . . , n}.

Proof. (1) For any complex numberλ ∈Cone has the equality

n

X

k=1

a_k−λ¯b_k

2 =

n

X

k=1

a_k−λ¯b_k

¯

a_k−¯λb_k (2.5)

=

n

X

k=1

|a_k|²+|λ|²

n

X

k=1

|b_k|²−2 Re ¯λ

n

X

k=1

a_kb_k

! . If in (2.5) we chooseλ₀ ∈C,

λ₀ :=

Pn k=1a_kb_k Pn

k=1|b_k|², ¯b6=0 then we get the identity

(2.6) 0≤

n

X

k=1

a_k−λ₀¯b_k

2 =

n

X

k=1

|a_k|²−|Pn

k=1a_kb_k|² Pn

k=1|b_k|² , which proves (2.4).

By virtue of (2.6), we conclude that equality holds in (2.4) if and only if a_k = λ₀¯b_k for anyk∈ {1, . . . , n}.

(2) Using Binet-Cauchy’s identity for complex numbers

n

X

i=1

x_iy_i

n

X

i=1

z_it_i−

n

X

i=1

x_it_i

n

X

i=1

z_iy_i (2.7)

= 1 2

n

X

i,j=1

(x_iz_j −x_jz_i) (y_it_j −y_jt_i)

= X

1≤i<j≤n

(xizj−xjzi) (yitj−yjti)

for the choicesx_i = ¯a_i, z_i =b_i, y_i =a_i, t_i = ¯b_i, i={1, . . . , n}, we get

n

X

i=1

|a_i|²

n

X

i=1

|b_i|²−

n

X

i=1

a_ib_i

2

= 1 2

n

X

i,j=1

|¯a_ib_j −a¯_jb_i|² (2.8)

= X

1≤i<j≤n

|¯a_ib_j−¯a_jb_i|². Now the inequality (2.4) is a simple consequence of (2.8).

The case of equality is obvious by the identity (2.8) as well.

Remark 2.4. By the(CBS)−inequality for real numbers and the generalised triangle inequal- ity for complex numbers

n

X

i=1

|z_i| ≥

n

X

i=1

z_i

, z_i ∈C, i∈ {1, . . . , n}

we also have

n

X

k=1

a_kb_k

2

≤

n

X

k=1

|a_kb_k|

!2

≤

n

X

k=1

|a_k|²

n

X

k=1

|b_k|².

Remark 2.5. The Lagrange identity for complex numbers stated in [4, p. 85] is wrong. It should be corrected as in (2.8).

2.3. An Additive Generalisation. The following generalisation of the (CBS)−inequality was obtained in [5, p. 5].

Theorem 2.6. If¯a = (a₁, . . . , a_n),b¯ = (b₁, . . . , b_n),¯c = (c₁, . . . , c_n)andd¯ = (d₁, . . . , d_n) are sequences of real numbers andp¯ = (p₁, . . . , p_n),¯q= (q₁, . . . , q_n)are nonnegative, then (2.9)

n

X

i=1

pia²_i

n

X

i=1

qib²_i +

n

X

i=1

pic²_i

n

X

i=1

qid²_i ≥2

n

X

i=1

piaici n

X

i=1

qibidi.

Ifp¯andq¯are sequences of positive numbers, then the equality holds in (2.9) iffa_ib_j =c_id_j for anyi, j ∈ {1, . . . , n}.

Proof. We will follow the proof from [5].

From the elementary inequality

(2.10) a²+b² ≥2abfor anya, b∈R

with equality iffa=b,we have

(2.11) a²_ib²_j +c²_id²_j ≥2a_ic_ib_jd_j for any i, j ∈ {1, . . . , n}.

Multiplying (2.11) byp_iq_j ≥ 0, i, j ∈ {1, . . . , n} and summing overi andj from1to n,we deduce (2.9).

If p_i, q_j > 0 (i= 1, . . . , n), then the equality holds in (2.9) iff a_ib_j = c_id_j for anyi, j ∈

{1, . . . , n}.

Remark 2.7. The conditiona_ib_j =c_id_j forc_i 6= 0, b_j 6= 0 (i, j = 1, . . . , n)is equivalent with

ai

ci = ^d_b^j

j (i, j = 1, . . . , n),i.e.,¯a,¯candb,¯ d¯are proportional with the same constantk.

Remark 2.8. If in (2.9) we choose pi = qi = 1 (i= 1, . . . , n), ci = bi, and di = ai

(i= 1, . . . , n),then we recapture the(CBS)−inequality.

The following corollary holds [5, p. 6].

Corollary 2.9. If¯a,¯b,¯cand¯dare nonnegative, then

(2.12) 1

2

" _n X

i=1

a³_ic_i

n

X

i=1

b³_id_i+

n

X

i=1

c³_ia_i

n

X

i=1

d³_ib_i

#

≥

n

X

i=1

a²_ic²_i

n

X

i=1

b²_id²_i,

(2.13) 1

2

" _n X

i=1

a²_ib_id_i·

n

X

i=1

b²_ia_ic_i+

n

X

i=1

c²_ib_id_i·

n

X

i=1

d²_ia_ic_i

#

≥

n

X

i=1

a_ib_ic_id_i

!2

. Another result is embodied in the following corollary [5, p. 6].

Corollary 2.10. If¯a,¯b,¯cand¯dare sequences of positive and real numbers, then:

(2.14) 1

2

" _n X

i=1

a³_i c_i

n

X

i=1

b³_i d_i +

n

X

i=1

a_ic_i

n

X

i=1

b_id_i

#

≥

n

X

i=1

a²_i

n

X

i=1

b²_i,

(2.15) 1

2

" _n X

i=1

a²_ib_i c_i

n

X

i=1

b²_ia_i d_i +

n

X

i=1

b_ic_i

n

X

i=1

a_id_i

#

≥

n

X

i=1

a_ib_i

!2

. Finally, we also have [5, p. 6].

Corollary 2.11. If¯a,andb¯are positive, then 1

2





n

X

i=1

a³_i b_i

n

X

i=1

b³_i a_i −

n

X

i=1

a_ib_i

!2

≥

n

X

i=1

a²_i

n

X

i=1

b²_i −

n

X

i=1

a_ib_i

!2

≥0.

The following version for complex numbers also holds.

Theorem 2.12. Let¯a= (a₁, . . . , a_n),¯b= (b₁, . . . , b_n),¯c= (c₁, . . . , c_n)and¯d= (d₁, . . . , d_n) be sequences of complex numbers and p¯ = (p₁, . . . , p_n), q¯ = (q₁, . . . , q_n) are nonnegative.

Then one has the inequality

(2.16)

n

X

i=1

p_i|a_i|²

n

X

i=1

q_i|b_i|²+

n

X

i=1

p_i|c_i|²

n

X

i=1

q_i|d_i|² ≥2 Re

" _n X

i=1

p_ia_ic¯_i

n

X

i=1

q_ib_id¯_i

# . The case of equality for¯p,¯qpositive holds iffa_ib_j =c_id_j for anyi, j ∈ {1, . . . , n}. Proof. From the elementary inequality for complex numbers

|a|²+|b|² ≥2 Re a¯b

, a, b ∈C, with equality iffa=b,we have

(2.17) |a_i|²|b_j|²+|c_i|²|d_j|² ≥2 Re

a_ic¯_ib_jdj¯

for anyi, j ∈ {1, . . . , n}.Multiplying (2.17) byp_iq_j ≥ 0and summing overiandj from1to n,we deduce (2.16).

The case of equality is obvious and we omit the details.

Remark 2.13. Similar particular cases may be stated but we omit the details.

2.4. A Related Additive Inequality. The following inequality was obtained in [5, Theorem 1.1].

Theorem 2.14. If ¯a = (a₁, . . . , a_n), ¯b = (b₁, . . . , b_n) are sequences of real numbers and

¯c= (c₁, . . . , c_n),¯d= (d₁, . . . , d_n)are nonnegative, then (2.18)

n

X

i=1

d_i

n

X

i=1

c_ia²_i +

n

X

i=1

c_i

n

X

i=1

d_ib²_i ≥2

n

X

i=1

c_ia_i

n

X

i=1

d_ib_i.

If c_i and d_i (i= 1, . . . , n) are positive, then equality holds in (2.18) iff ¯a = b¯ = ¯k where

¯k= (k, k, . . . , k)is a constant sequence.

Proof. We will follow the proof from [5].

From the elementary inequality

(2.19) a²+b² ≥2ab for any a, b∈R

with equality iffa=b;we have

(2.20) a²_i +b²_j ≥2aibj for any i, j ∈ {1, . . . , n}.

Multiplying (2.20) by c_id_j ≥ 0, i, j ∈ {1, . . . , n}and summing overi from1tonand over j from1ton,we deduce (2.18).

If ci, dj > 0 (i= 1, . . . , n), then the equality holds in (2.18) iff ai = bj for any i, j ∈ {1, . . . , n}which is equivalent with the fact thatai =bi =k for anyi∈ {1, . . . , n}.

The following corollary holds [5, p. 4].

Corollary 2.15. If¯aandb¯are nonnegative sequences, then

(2.21) 1

2

" _n X

i=1

a³_i

n

X

i=1

b_i+

n

X

i=1

a_i

n

X

i=1

b³_i

#

≥

n

X

i=1

a²_i

n

X

i=1

b²_i;

(2.22) 1

2

" _n X

i=1

ai n

X

i=1

a²_ibi+

n

X

i=1

bi n

X

i=1

b²_iai

#

≥

n

X

i=1

aibi

!2

. Another corollary that may be obtained is [5, p. 4 – 5].

Corollary 2.16. If¯aandb¯are sequences of positive real numbers, then

(2.23)

n

X

i=1

a²_i +b²_i 2a_ib_i ≥

Pn i=1

1 ai

Pn i=1

1 bi

Pn i=1

1 aibi

,

(2.24)

n

X

i=1

a_i

n

X

i=1

1 b_i +

n

X

i=1

1 a_i

n

X

i=1

b_i ≥2n², and

(2.25) n

n

X

i=1

a²_i +b²_i 2a²_ib²_i ≥

n

X

i=1

1 a_i

n

X

i=1

1 b_i. The following version for complex numbers also holds.

Theorem 2.17. If¯a= (a₁, . . . , a_n),¯b= (b₁, . . . , b_n)are sequences of complex numbers, then forp¯ = (p₁, . . . , p_n)and¯q= (q₁, . . . , q_n)two sequences of nonnegative real numbers, one has the inequality

(2.26)

n

X

i=1

q_i

n

X

i=1

p_i|a_i|²+

n

X

i=1

p_i

n

X

i=1

q_i|b_i|² ≥2 Re

" _n X

i=1

p_ia_i

n

X

i=1

q_i¯b_i

# . For¯p,¯qpositive sequences, the equality holds in (2.26) iff¯a=b¯ =k¯= (k, . . . , k).

The proof goes in a similar way with the one in Theorem 2.14 on making use of the following elementary inequality holding for complex numbers

(2.27) |a|²+|b|² ≥2 Re

a¯b

, a, b ∈C; with equality iffa=b.

2.5. A Parameter Additive Inequality. The following inequality was obtained in [5, Theorem 4.1].

Theorem 2.18. Let ¯a = (a₁, . . . , a_n), b¯ = (b₁, . . . , b_n) be sequences of real numbers and

¯c= (c₁, . . . , c_n),¯d= (d₁, . . . , d_n)be nonnegative. Ifα, β > 0andγ ∈ Rsuch thatγ² ≤αβ, then

(2.28) α

n

X

i=1

di n

X

i=1

a²_ici+β

n

X

i=1

ci n

X

i=1

b²_idi ≥2γ

n

X

i=1

ciai n

X

i=1

dibi. Proof. We will follow the proof from [5].

Sinceα, β >0andγ² ≤αβ,it follows that for anyx, y ∈Rone has

(2.29) αx²+βy² ≥2γxy.

Choosing in (2.29)x=a_i, y =b_j (i, j = 1, . . . , n),we get

(2.30) αa²_i +βb²_j ≥2γa_ib_j for any i, j ∈ {1, . . . , n}.

If we multiply (2.30) by c_id_j ≥ 0 and sum over i and j from 1 to n, we deduce the desired

inequality (2.28).

The following corollary holds.

Corollary 2.19. If¯aand¯bare nonnegative sequences andα, β, γare as in Theorem 2.18, then

(2.31) α

n

X

i=1

b_i

n

X

i=1

a³_i +β

n

X

i=1

a_i

n

X

i=1

b³_i ≥2γ

n

X

i=1

a²_i

n

X

i=1

b²_i,

(2.32) α

n

X

i=1

a_i

n

X

i=1

a²_ib_i+β

n

X

i=1

b_i

n

X

i=1

b²_ia_i ≥2γ

n

X

i=1

a_ib_i

!2

. The following particular case is important [5, p. 8].

Theorem 2.20. Let¯a, ¯bbe sequences of real numbers. Ifp¯is a sequence of nonnegative real numbers withPn

i=1p_i >0,then:

(2.33)

n

X

i=1

p_ia²_i

n

X

i=1

p_ib²_i ≥ Pn

i=1p_ia_ib_iPn

i=1p_ia_iPn i=1p_ib_i Pn

i=1p_i .

In particular, (2.34)

n

X

i=1

a²_i

n

X

i=1

b²_i ≥ 1 n

n

X

i=1

a_ib_i

n

X

i=1

a_i

n

X

i=1

b_i. Proof. We will follow the proof from [5, p. 8].

If we choose in Theorem 2.18, c_i = d_i = p_i (i= 1, . . . , n) and α = Pn

i=1p_ib²_i, β = Pn

i=1p_ia²_i, γ = Pn

i=1p_ia_ib_i, we observe, by the (CBS)−inequality with the weights p_i (i= 1, . . . , n)one hasγ² ≤αβ, and then by (2.28) we deduce (2.33).

Remark 2.21. If we assume that¯aand¯bare asynchronous, i.e.,

(a_i−a_j) (b_i−b_j)≤0for any i, j ∈ {1, . . . , n}, then by ˇCebyšev’s inequality

(2.35)

n

X

i=1

p_ia_i

n

X

i=1

p_ib_i ≥

n

X

i=1

p_i

n

X

i=1

p_ia_ib_i

respectively (2.36)

n

X

i=1

a_i

n

X

i=1

b_i ≥n

n

X

i=1

a_ib_i, we have the following refinements of the(CBS)−inequality

n

X

i=1

p_ia²_i

n

X

i=1

p_ib²_i ≥ Pn

i=1p_ia_ib_iPn

i=1p_ia_iPn i=1p_ib_i Pn

i=1p_i (2.37)

≥

n

X

i=1

p_ia_ib_i

!2

providedPn

i=1p_ia_ib_i ≥0,respectively (2.38)

n

X

i=1

a²_i

n

X

i=1

b²_i ≥ 1 n

n

X

i=1

aibi n

X

i=1

ai n

X

i=1

bi ≥

n

X

i=1

aibi

!2

providedPn

i=1aibi ≥0.

2.6. A Generalisation Provided by Young’s Inequality. The following result was obtained in [5, Theorem 5.1].

Theorem 2.22. Let¯a= (a₁, . . . , a_n),¯b= (b₁, . . . , b_n),¯p= (p₁, . . . , p_n)and¯q= (q₁, . . . , q_n) be sequences of nonnegative real numbers andα, β > 1 with _α¹ + _β¹ = 1. Then one has the inequality

(2.39) α

n

X

i=1

qi n

X

i=1

pib^β_i +β

n

X

i=1

pi n

X

i=1

qia^α_i ≥αβ

n

X

i=1

pibi n

X

i=1

qiai.

If p¯ and ¯q are sequences of positive real numbers, then the equality holds in (2.39) iff there exists a constantk≥0such thata^α_i =b^β_i =k for eachi∈ {1, . . . , n}.

Proof. It is, by the Arithmetic-Geometric inequality [6, p. 15], well known that

(2.40) 1

αx+ 1

βy≥x^1αy^β1 for x, y ≥0, 1 α + 1

β = 1, α, β >1 with equality iffx=y.

Applying (2.40) forx=a^α_i, y =b^β_j (i, j = 1, . . . , n)we have (2.41) αb^β_j +βa^α_i ≥αβa_ib_j for anyi, j ∈ {1, . . . , n}

with equality iffa^α_i =b^β_j for anyi, j ∈ {1, . . . , n}.

If we multiply (2.41) byq_ip_j ≥ 0 (i, j ∈ {1, . . . , n}) and sum overi andj from1ton we deduce (2.39).

The case of equality is obvious by the above considerations.

The following corollary is a natural consequence of the above theorem.

Corollary 2.23. Let¯a,b, α¯ andβbe as in Theorem 2.22. Then

(2.42) 1

α

n

X

i=1

b_i

n

X

i=1

a^α+1_i + 1 β

n

X

i=1

a_i

n

X

i=1

b^β+1_i ≥

n

X

i=1

a²_i

n

X

i=1

b²_i;

(2.43) 1 α

n

X

i=1

a_i

n

X

i=1

b_ia^α_i + 1 β

n

X

i=1

b_i

n

X

i=1

a_ib^β_i ≥

n

X

i=1

a_ib_i

!2

.

The following result which provides a generalisation of the(CBS)−inequality may be ob- tained by Theorem 2.22 as well [5, Theorem 5.2].

Theorem 2.24. Let¯xandy¯be sequences of positive real numbers. Ifα, βare as above, then

(2.44) 1

α

n

X

i=1

x^α_iy^2−α_i + 1 β

n

X

i=1

x^β_iy_i^2−β

!

·

n

X

i=1

y_i² ≥

n

X

i=1

xiyi

!2

. The equality holds iff¯xand¯yare proportional.

Proof. Follows by Theorem 2.22 on choosingp_i =q_i =y_i², a_i = ^x_yⁱ

i, b_i = ^x_yⁱ

i, i∈ {1, . . . , n}. Remark 2.25. Forα=β= 2,we recapture the(CBS)−inequality.

Remark 2.26. Fora_i = |z_i|, b_i =|w_i|,withz_i, w_i ∈ C;i = 1, . . . , n,we may obtain similar inequalities for complex numbers. We omit the details.

2.7. Further Generalisations via Young’s Inequality. The following inequality is known in the literature as Young’s inequality

(2.45) px^q+qy^p ≥pqxy, x, y ≥0 and 1 p+ 1

q = 1, p >1 with equality iffx^q=y^p.

The following result generalising the (CBS)−inequality was obtained in [7, Theorem 2.1]

(see also [8, Theorem 1]).

Theorem 2.27. Let¯x= (x₁, . . . , x_n),¯y= (y₁, . . . , y_n)be sequences of complex numbers and

¯

p = (p₁, . . . , p_n),q¯ = (q₁, . . . , q_n)be two sequences of nonnegative real numbers. Ifp > 1,

1

p +¹_q = 1,then (2.46) 1

p

n

X

k=1

p_k|x_k|^p

n

X

k=1

q_k|y_k|^p+ 1 q

n

X

k=1

q_k|x_k|^q

n

X

k=1

p_k|y_k|^q≥

n

X

k=1

p_k|x_ky_k|

n

X

k=1

q_k|x_ky_k|. Proof. We shall follow the proof in [7].

Choosingx=|x_j| |y_i|, y=|x_i| |y_j|, i, j ∈ {1, . . . , n},we get from (2.45) (2.47) q|x_i|^p|y_j|^p+p|x_j|^q|y_i|^q ≥pq|x_iy_i| |x_jy_j|

for anyi, j ∈ {1, . . . , n}.

Multiplying withp_iq_j ≥ 0 and summing over i and j from 1 to n, we deduce the desired

result (2.46).

The following corollary is a natural consequence of the above theorem [7, Corollary 2.2] (see also [8, p. 105]).

Corollary 2.28. Ifx¯ andy¯ are as in Theorem 2.27 and m¯ = (m₁, . . . , m_n)is a sequence of nonnegative real numbers, then

(2.48) 1 p

n

X

k=1

m_k|x_k|^p

n

X

k=1

m_k|y_k|^p+ 1 q

n

X

k=1

m_k|x_k|^q

n

X

k=1

m_k|y_k|^q ≥

n

X

k=1

m_k|x_ky_k|

!2

, wherep >1, _p¹ +¹_q = 1.

Remark 2.29. If in (2.48) we assume that m_k = 1, k ∈ {1, . . . , n},then we obtain [7, p. 7]

(see also [8, p. 105])

(2.49) 1

p

n

X

k=1

|x_k|^p

n

X

k=1

|y_k|^p+1 q

n

X

k=1

|x_k|^q

n

X

k=1

|y_k|^q ≥

n

X

k=1

|x_ky_k|

!2

, which, in the particular casep=q= 2will provide the(CBS)−inequality.

The second generalisation of the(CBS)−inequality via Young’s inequality is incorporated in the following theorem [7, Theorem 2.4] (see also [8, Theorem 2]).

Theorem 2.30. Let¯x,¯y,¯p,q¯andp, q be as in Theorem 2.27. Then one has the inequality (2.50) 1

p

n

X

k=1

pk|xk|^p

n

X

k=1

qk|yk|^q+ 1 q

n

X

k=1

qk|xk|^q

n

X

k=1

pk|yk|^p

≥

n

X

k=1

p_k|x_k| |y_k|^p−1

n

X

k=1

q_k|x_k| |y_k|^q−1. Proof. We shall follow the proof in [7].

Choosing in (2.45),x= ^|x_|y^j|

j|, y = ^|x_|y^i|

i|,we get

(2.51) p

|xj|

|y_j| q

+q |xi|

|y_i| p

≥pq|xi| |xj|

|y_i| |y_j| for anyy_i 6= 0, i, j ∈ {1, . . . , n}.

It is easy to see that (2.51) is equivalent to

(2.52) q|x_i|^p|y_j|^q+p|y_i|^p|x_j|^q ≥pq|x_i| |y_i|^p−1|x_j| |y_j|^q−1 for anyi, j ∈ {1, . . . , n}.

Multiplying (2.52) byp_iq_j ≥0 (i, j ∈ {1, . . . , n})and summing overiandj from1ton,we

deduce the desired inequlality (2.50).

The following corollary holds [7, Corollary 2.5] (see also [8, p. 106]).

Corollary 2.31. Let¯x,y,¯ m¯ andp,¯ q¯be as in Corollary 2.28. Then

(2.53) 1 p

n

X

k=1

m_k|x_k|^p

n

X

k=1

m_k|y_k|^q+ 1 q

n

X

k=1

m_k|x_k|^q

n

X

k=1

m_k|y_k|^p

≥

n

X

k=1

m_k|x_k| |y_k|^p−1

n

X

k=1

m_k|x_k| |y_k|^q−1. Remark 2.32. If in (2.53) we assume thatm_k = 1, k ∈ {1, . . . , n},then we obtain [7, p. 8]

(see also [8, p. 106]) (2.54) 1

p

n

X

k=1

|x_k|^p

n

X

k=1

|y_k|^q+1 q

n

X

k=1

|x_k|^q

n

X

k=1

|y_k|^p ≥

n

X

k=1

|x_k| |y_k|^p−1

n

X

k=1

|x_k| |y_k|^q−1, which, in the particular casep=q= 2will provide the(CBS)−inequality.

The third result is embodied in the following theorem [7, Theorem 2.7] (see also [8, Theorem 3]).

Theorem 2.33. Let¯x,y,¯ ¯p,¯qandp, q be as in Theorem 2.27. Then one has the inequality (2.55) 1

p

n

X

k=1

pk|xk|^p

n

X

k=1

qk|yk|^q+ 1 q

n

X

k=1

qk|xk|^p

n

X

k=1

pk|yk|^q

≥

n

X

k=1

pk|xkyk|

n

X

k=1

pk|xk|^p−1|yk|^q−1. Proof. We shall follow the proof in [7].

If we choosex= _|y^|yi|

j| andy= ^|x_|x^i|

j| in (2.45) we get p

|y_i|

|y_j| q

+q |x_i|

|x_j| p

≥pq|x_i| |y_i|

|x_j| |y_j|, for anyx_i, y_j 6= 0, i, j ∈ {1, . . . , n},giving

(2.56) q|x_i|^p|y_j|^q+p|y_i|^q|x_j|^p ≥pq|x_iy_i| |x_j|^p−1|y_j|^q−1 for anyi, j ∈ {1, . . . , n}.

Multiplying (2.56) byp_iq_j ≥0 (i, j ∈ {1, . . . , n})and summing overiandj from1ton,we

deduce the desired inequality (2.55).

The following corollary is a natural consequence of the above theorem [8, p. 106].

Corollary 2.34. Let¯x,y,¯ m¯ andp,¯ q¯be as in Corollary 2.28. Then one has the inequality:

(2.57)

n

X

k=1

m_k|x_k|^p

n

X

k=1

m_k|y_k|^q ≥

n

X

k=1

m_k|x_ky_k|

n

X

k=1

m_k|x_k|^p−1|y_k|^q−1.

Remark 2.35. If in (2.57) we assume thatm_k = 1, k = {1, . . . , n},then we obtain [7, p. 8]

(see also [8, p. 10]) (2.58)

n

X

k=1

|x_k|^p

n

X

k=1

|y_k|^q ≥

n

X

k=1

|x_ky_k|

n

X

k=1

|x_k|^p−1|y_k|^q−1, which, in the particular casep=q= 2will provide the(CBS)−inequality.

The fourth generalisation of the(CBS)−inequality is embodied in the following theorem [7, Theorem 2.9] (see also [8, Theorem 4]).

Theorem 2.36. Let¯x,y,¯ ¯p,¯qandp, q be as in Theorem 2.27. Then one has the inequality (2.59) 1

q

n

X

k=1

p_k|x_k|²

n

X

k=1

q_k|y_k|^q+ 1 p

n

X

k=1

p_k|y_k|²

n

X

k=1

q_k|x_k|^p

≥

n

X

k=1

q_k|x_ky_k|

n

X

k=1

p_k|x_k|^2q |y_k|^2p. Proof. We shall follow the proof in [7].

Choosing in (2.45),x=|x_i|^2q |y_j|, y =|x_j| |y_i|^p2 ,we get

(2.60) p|x_i|²|y_j|^q+q|x_j|^p|y_i|² ≥pq|x_i|^2q |y_i|^2p|x_jy_j| for anyi, j ∈ {1, . . . , n}.

Multiply (2.60) byp_iq_j ≥ 0 (i, j ∈ {1, . . . , n})and summing overi andj from1to n,we

deduce the desired inequality (2.60).