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Applied Mathematics and Computation 231 (2014) 463–477

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Another look at some new Cauchy–Schwarz type inner productinequalities

0096-3003/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.11.047

⇑ Corresponding author at: University of Pittsburgh, Department of Mathematics, Pittsburgh, PA 15260, USA.E-mail addresses: [email protected], [email protected] (C. Lupu), [email protected] (D. Schwarz).

Cezar Lupu a,b,⇑, Dan Schwarz c

a University of Pittsburgh, Department of Mathematics, Pittsburgh, PA 15260, USAb University of Craiova, Department of Mathematics, Str. Alexandru Ioan Cuza 13, RO–200585 Craiova, Romaniac Intuitext Softwin Group, D. Pompeiu St. 10A, RO–020337 Bucharest, Romania

a r t i c l e i n f o

Keywords:Inner product spaceCauchy–Schwarz inequalityGram determinantBuzano’s inequalityIntegral inequalitiesDiscrete inequalitiesLagrange multipliers

a b s t r a c t

In this paper we take a refreshing look at a Cauchy–Schwarz type inner product inequali-ties. We also provide a Cauchy–Schwarz based proof and a refinement of Buzano’s inequal-ity in inner product spaces. Some elementary applications such as trace inequalities forunitary matrices and discrete or integral inequalities are given.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

The theory of inner product inequalities plays a central role in many branches of mathematics like Linear Operators, Par-tial Differential Equations, Approximation & Optimization Theory, Information Theory or Statistics and many other fields.Such inequalities were the pioneering work of mathematicians like Cauchy, Minkovski, Holder, Hilbert, Hardy, Kantorovichand many others.

On the other hand, it is worth mentioning the contributions of Bellman, Boas, van der Corput, Ostrowski, Selberg, Enfloand Bombieri who have applied successfully in obtaining applications for Fourier and Mellin transforms, oscillatory integrals,approximation of polynomials or large sieve. All these results as well as their applications are covered in the monograph [4].

In [2], Buzano gave the following extension of the celebrated Cauchy–Schwarz’s inequality in a real or complex innerproduct space ðH; h�; �iÞ,

jha; cihc; bij 6 12ðjjajjjjbjj þ jha; bijÞ � jjcjj2 ð1Þ

for all a; b; c 2 H.Clearly, when a ¼ b, the above inequality becomes the celebrated Cauchy–Schwarz’s inequality,

jha; cij2 6 jjajj2jjcjj2 ð2Þ

for all a; c 2 H.As pointed out in [5], the original proof of Buzano is a little bit complicated. Buzano’s inequality also mentioned in [8]

as an interesting generalization of the Cauchy–Schwarz’s inequality and its proof requires some facts about orthogonal

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464 C. Lupu, D. Schwarz / Applied Mathematics and Computation 231 (2014) 463–477

decomposition of a complete inner product space. Moreover, Fuji and Kubo [5] gave a simple proof by using the orthogonalprojection on a subspace of an inner product space H and Cauchy–Schwarz’s inequality.

The equality case in (1) holds if

c ¼a ajjajj þ

ha;bijha;bij � b

jjbjj

� �; when ha; bi – 0;

a ajjajj þ b b

jjbjj

� �; when ha; bi ¼ 0;

8><>:

where a; b 2 K 2 fR;Cg. A detailed proof of Buzano’s inequality can also be found in [15], pp. 60–61.For real inner product spaces, Richard [14] obtained the following stronger inequality

ha; cihc; bi � 12ha; bijjcjj2

�� 6 1

2jjajjjjbjjjjcjj2; ð3Þ

a; b; c 2 H.Dragomir [4] showed that the above inequality is true with coefficients 1

jaj instead of 12, where a is a non-zero number with

j1� aj ¼ 1. Also, Dragomir [4] obtained a refinement of Buzano’s inequality,

jha; cihc; bij 6 jha; cihc; bi � 12ha; bijjcjj2j þ 1

2ha; bijjcjj2 6 1

2ðjjajjjjbjj þ jha; bijÞÞjjcjj2; ð4Þ

a; b; c 2 H.In [6], Gavrea generalized Buzano’s inequality to nonegative real exponents. In fact, he proved that

jha; xija � jhx; bijb 6 ta21t

b22jjxjj

aþb;

where t1 and t2 are expressed in terms of the constants a and b and vectors a; b.

2. A Cauchy–Schwarz setting to prove Buzano’s inequality

In this section we prove Buzano’s inequality using a Krein type inequality which is deduced only from Cauchy–Schwarz’sinequality. As mentioned in [9], for any given two vectors a; b 2 Cn � f0g, one can define

cos Uab ¼Reha; bijjajjjjbjj

and

cos Wab ¼jha; bijjjajjjjbjj :

In 1969, Krein proved the following inequality,

Uac 6 Uab þUbc

for any a; b; c 2 Cn.A proof of the above result can be found in [7,9]. Moreover, the main idea from the proof of Krein’s inequality is hidden

in the following Krein-type inequality,

Theorem 2.1. For any vectors a; b; c 2 ðH; h�; �iÞ we have the following inequality

jjajj2jhb; cij2 þ jjbjj2jhc; aij2 þ jjcjj2jha; bij2 6 jjajj2jjbjj2jjcjj2 þ 2jha; bihb; cihc; aij:
First proof. We will give a proof using only Cauchy–Schwarz’s inequality. Without loss of generality, we assume jjcjj– 0,
otherwise the inequality is trivial. Now, for any complex scalar k, by the Cauchy–Schwarz’s inequality, we have

jja� kbjj2jjcjj2 P jha� kb; cij2

for all a; b; c 2 H. By expanding, this inequality is successively equivalent to

ðjjajj2 � 2kjha; bij þ k2jjbjj2Þjjcjj2 P jhc; aij2 � 2kjhc; aijjhb; cij þ k2jhc; aij2;

ðjjbjj2jjcjj2 � jhb; cij2Þk2 þ 2kðjha; bijjjcjj2 � jhc; aijjhb; cijÞ þ jjajj2jjcjj2 � jhc; aij2 P 0

for all complex scalars k. The last inequality can be viewed as a quadratic polynomial in k with its dominant coefficient non-negative (by Cauchy–Schwarz’s inequality), and thus, the discriminant D must be negative,

Dk ¼ 4ðjha; bijjjcjj2 � jhc; aijjhb; cijÞ2� 4ðjjbjj2jjcjj2 � jhb; cij2Þðjjajj2jjcjj2 � jhc; aij2Þ 6 0;

which is equivalent to

C. Lupu, D. Schwarz / Applied Mathematics and Computation 231 (2014) 463–477 465

jha; bij2jjcjj4 � 2jha; bijjhb; cijjhc; aijjjcjj2 þ jhc; aij2jhb; cij2 6 jjajj2jjbjj2jjcjj4 � jhc; aij2jjbjj2jjcjj2

and the desired inequality follows.Second proof. Consider the following matrix

Dða; bÞ ¼jjcjj2 jhc; aji

jhb; cij jha; bij

24

35:

We will prove that

jdet Dða; bÞj2 6 jdet Dða; aÞj � jdet Dðb; bÞj:

Denote PðtÞ ¼ detðDða; aÞÞt2 � 2 detðDða; bÞÞt þ detðDðb; bÞÞ. A simple calculation yields

PðtÞ ¼ det Dðat � b; at � bÞ ¼ detjjcjj2 hta� b; ci

hc; ta� bi hta� b; ta� bi

" # !¼ jjcjj2jjat � bjj2 � jhat � b; cij2 P 0;

by Cauchy–Schwarz’s inequality. This implies that the quadratic polynomial is nonnegative and thus, its discriminant mustbe negative,

DPðtÞ ¼ jdet Dða; bÞj2 � jdet Dða; aÞj � jdet Dðb; bÞj 6 0;


ðjjcjj2jha; bij � jhc; aijjhc; bijÞ26 ðjjcjj2jjajj2 � jhc; aij2Þðjjcjj2jjbjj2 � jhb; cij2Þ

and by expanding our inequality follows. h

Remark. One can consider the Gram matrix

Gða; b; cÞ ¼

jjajj2 jha; bij jha; cij

jha; bij jjbjj2 jhb; cij

jhc; aij jhb; cij jjcjj2

26664

37775:

It is well-known that Gða; b; cÞ is positive semidefinite and thus det Gða; b; cÞ is nonnegative, and finally this implies

det Gða; b; cÞ ¼ jjajj2jjbjj2jjcjj2 þ 2jha; bijjhb; cijjhc; aij � ðjjajj2jhb; cij2 þ jjbjj2jhc; aij2 þ jjcjj2jha; bij2Þ

and the inequality follows. This idea appears also in [1].As a consequence we have the following interesting.

Corollary 2.2. For any vectors a; b; c 2 ðH; h�; �iÞ we have the following inequality

jjajj2jjbjj2 � jha; bij2 P jjajj2jhb;1ij2 þ jjbjj2jha;1ij2 � 2jha;1ijjhb;1ijjha; bij:

Proof. Take c to be the unit vector in Theorem 1.2. h

The above corollary served as application of several refinements of Heisenberg type inequalities related to uncertaintyprinciple in the paper [13].

As a consequence of Theorem 1.2 we have the general integral version.

Corollary 2.3. Let ðH;R;lÞ be a positive measure space and let f ; g;h 2 L2ðH;R;lÞ the Hilbert space of complex-valued 2� lintegrable functions defined on H. Then
Z
Hf 2ðtÞdlðtÞ

� � ZH

gðtÞhðtÞdlðtÞ��

�� 2

þZ

Hg2ðtÞdlðtÞ

� � ZH

hðtÞf ðtÞdlðtÞ��

�� 2

þZ

Hh2ðtÞdlðtÞ

� � ZH

f ðtÞgðtÞdlðtÞ��

�� 2

6

ZH

f 2ðtÞdlðtÞ� � Z

Hg2ðtÞdlðtÞ

� � ZH

g2ðtÞdlðtÞ� �

þ 2Z

Hf ðtÞgðtÞdlðtÞ

� � ZH

f ðtÞgðtÞdlðtÞ� � Z


� ��:

It might be useful to see that out of Theorem 3.2 one can get the following discrete version.

Corollary 2.4. If pi P 0; i ¼ 1;2; . . . ;n and ai; bi; ci are complex numbers, then


Xn

i¼1

pijaij2 ! Xn

i¼1

pibici

��

!2

þXn

i¼1

pijbij2 ! Xn

i¼1

piciai

��

!2

þXn

i¼1

pijaij2 ! Xn

i¼1

pibici

��

!2

6

Xn

i¼1

pijaij2 ! Xn

i¼1

pijbij2 ! Xn

i¼1

pijcij2 !

þ 2Xn

i¼1

piaibi

��Xn

i¼1

pibici

��Xn

i¼1

piciai

��:

In the case when H is the space of real-valued continuous functions defined on the interval ½a; b� endowed with the innerproduct hf ; gi :¼

R ba f ðxÞgðxÞdx, we have.

Corollary 2.5. Let f ; g;h be continuous real-valued functions on ½a; b�. Then, we have the following inequality

Z b

af 2ðtÞdt

! Z b

agðtÞhðtÞdt

!2

þZ b

ag2ðtÞdt

! Z b

ahðtÞf ðtÞdt

!2

þZ b

ah2ðtÞdt

! Z b

af ðtÞgðtÞdt

!2

6

Z b

af 2ðtÞdt

! Z b

ag2ðtÞdt

! Z b

ah2ðtÞdt

!þ 2

Z b

af ðtÞgðtÞdt

��Z b

agðtÞhðtÞdt

��Z b

ahðtÞf ðtÞdt

��

!:

The discrete real version of Corollary 2.4 is given by.

Corollary 2.6. Given real numbers ai; bi; ci;1 6 i 6 n, then

Xn

i¼1

a2i

! Xn

i¼1

bici

!2

þXn

i¼1

b2i

! Xn

i¼1

aici

!2

þXn

i¼1

a2i

! Xn

i¼1

bici

!2

6

Xn

i¼1

a2i

! Xn

i¼1

b2i

! Xn

i¼1

c2i

!þ 2

Xn

i¼1

aibi

��Xn

i¼1

bici

��Xn

i¼1

ciai

��:

In other words, if f ; g;h are random variables of the space L2ðX;lÞ, where l is a probability measure. Define the follwow-ing covariance between f ; g and h by

covhðf ; gÞ :¼ Eðf gÞEðh2Þ � EðfhÞEðghÞ;

where Eðf Þ ¼R

X fdl is the expectation of f. The variance of f is given by

varhðf Þ ¼ covhðf ; f Þ ¼ Eðjf j2ÞEðh2Þ � jEðfhÞj2:

Thus, after some computations, Theorem 1.2 can be rewritten as the following variance–covariance inequality:

jcovhðf ; gÞj2 6 varhðf ÞvarhðgÞ:

In the case when h ¼ 1 we obtain the classical variance–covariance inequality which is used in Statistics.Now, we are able to prove Buzano’s inequality which is given by.

Theorem 2.7. For any a; b; c 2 ðH; h�; �iÞ we have the inequality

jha; cihb; cij 6 12ðjjajjjjbjj þ jha; bijÞjjcjj2:

Proof. Theorem 2.1 can be rewritten as

jha; bij2

jjajj2jjbjj2þ jhb; cij

2

jjbjj2jjcjj2þ jhc; aij

2

jjcjj2jjajj26 1þ 2jha; bijjhb; cijjhc; aij

jjajj2jjbjj2jjcjj2;


ajjajj ;

bjjbjj

� ��2

þ bjjbjj ;

cjjcjj

� ��2

þ cjjcjj ;

ajjajj

� ��2

6 1þ 2ajjajj ;

bjjbjj

� �� bjjbjj ;

cjjcjj

� �� cjjcjj ;

ajjajj

� ��:

Denote u ¼ ajjajj ;v ¼ b

jjbjj and x ¼ cjjcjj. Our inequality rewrites as

jhu; vij2 þ jhx;uij2 þ jhx;vij2 6 1þ 2jhu; vihx; uihx; vij;

which is equivalent further with

jhx;uij2 þ jhx;vij2 6 1þ jhu; vij � jhu;vijð1þ jhu;vij � 2jhx;uijjhx;vijÞ:

By the arithmetic–geometric mean, we obtain


jhx;uij2 þ jhx;vij2 P 2jhx;uijjhx;vij

and we have

2jhx;uijjhx; vij 6 1þ jhu; vij � jhu;vijð1þ jhu; vij � 2jhx;uijjhx;vijÞ;

or
ð1� jhu; vijÞð1þ jhu; vij � 2jhx;uijjhx;vijÞP 0:
Then either jhu;vij ¼ 1 which by Cauchy–Schwarz’s inequality, we have 2 ¼ 1þ jhu;vijP 2jhx;uijjhx;vij or 1 > jhu;vijwhich implies

1þ jhu;vij � 2jhx;uijjhx; vijÞP 0;

or

jhx;uijjhx;vijÞ 6 12ð1þ jhu; vijÞ

which is exactly what we needed to prove. h

Remark. In fact, one can prove even more:

jhx;uij2 þ jhx;vij2 6 1þ jhu; vij:

Indeed, at some point in the proof of Theorem 2.2 we derived the following inequality

jhx;uij2 þ jhx;vij2 6 1þ jhu; vij � jhu;vijð1þ jhu; vij � 2jhx;uijjhx;vijÞ:

By Theorem 2.2 we have

1þ jhu;vij � jhu;vijð1þ jhu;vij � 2jhx;uijjhx; vijÞ 6 1þ jhu; vij:

Putting u ¼ ajjajj ; v ¼ b

jjbjj and x ¼ cjjcjj we obtain the following inequality

ðjjbjjjha; cijÞ2 þ ðjjajjjhb; cijÞ2 6 jjajjjjbjjðjjajjjjbjj þ jha; bijÞjjcjj2

for all a; b; c 2 H.Other proofs and applications of this last inequality are given in the section that will follow and extensions and general-

izations to different settings will be given elsewhere.

3. Main result

Now, we are ready to state and prove the main result of this paper. But first, the following result will be crucial in provingour refinement of Buzano’s inequality.

Theorem 3.1. Given vectors u;v 2 H, with jjujj ¼ jjv jj ¼ 1 (thus u;v 2 Sn�1), then

supjjxjj¼1ðjhu; xij2 þ jhv ; xij2Þ ¼ 1þ jhu;vij:

First proof. (Trigonometric proof) Consider the unique vectorial representation x ¼ x0 þ x00 with x0 2 spanðu;vÞ andx00 ? spanðu; vÞ. Then 1 ¼ jjxjj2 ¼ hx0 þ x00; x0 þ x00i ¼ jjx0jj2 þ 2hx0; x00i þ jjx00jj2 ¼ jjx0jj2 þ jjx00jj2 (Pythagoras’ relation), sojjx0jj 6 1. Now, jhu; xij ¼ jhu; x0ij 6 jhu; yij where y ¼ 0 if x0 ¼ 0 and y ¼ x0

jjx0 jj if x0 – 0 (thus jjyjj ¼ 1). Similarlyjhv; xij ¼ jhv ; x0ij 6 jhv; yij. The maximum asked is therefore obtained when x 2 spanðu;vÞ.

Thus our vectorial problem has been transferred in the 2-dimensional space, with unit vectors u;v and x. The endpoints ofthe vectors ±u and ±v partition the circle S1 into four arcs, each of measure at most p (and possibly 0, when u = ±v); the end-point of x will fall into one of them. Let x be the measure of that arc, and a; b, with aþ b ¼ x, the measures of the arcs be-tween the endpoint of x and the ends of that arc. Then hu; xi2 þ hv ; xi2 ¼ cos2 aþ cos2 b, by the well-known geometricinterpretation of the dot-product (independently of the dimension of the space). But thenhu; xi2 þ hv ; xi2 ¼ cos2 aþ cos2 b ¼ 1þ 1

2 ðcos 2aþ cos 2bÞ ¼ 1þ cosðaþ bÞ cosða� bÞ 6 1þ j cos xj ¼ 1þ jhu;vij, with equal-ity in the obvious cases:

� when a ¼ b ¼ x=2, therefore when jhu; xij ¼ jhv ; xij, therefore x ¼ ð�u� vÞ=jj � u� v jj, where the signs are such that0 6 x < p=2;� when x ¼ p=2, therefore when hu;vi ¼ 0, i.e., u ? v , for all x 2 spanðu;vÞ.

Second Proof. (Quadratic forms) For jjxjj ¼ jjujj ¼ jjv jj ¼ 1 we have

0 6 jjkxþ luþ mv jj2 ¼ hkxþ luþ mv; kxþ luþ mvi ¼ k2 þ l2 þ m2 þ 2klhx;ui þ 2kmhx;vi þ 2lmhu;vi;


a quadratic form which takes non-negative values for any real parameters k;l; m, thus corresponding to a positive semide-fined matrix

1 jhx;uij jhx;vijjhx;uij 1 jhu;vijjhx;vij jhu;vij 1

264

375

The principal minors of order 1 are thus non-negative, which yields the semi-positivity of the norm; the principal minors oforder 2 are non-negative, which yields the Cauchy–Schwarz’s inequality, e.g., 1 P jhu;vij2; finally, also the determinant ofthe matrix is non-negative

D ¼ 1� ðjhu; vij2 þ jhx; uij2 þ jhx;vij2Þ þ 2jhu;vihx;uihx;vijP 0;

which can be arranged as jhx;uij2 þ jhx;vij2 6 1þ jhu;vij � jhu; vijð1þ jhu;vij � 2jhx;uihx;vijÞ. But jhx;uij2 þ jhx;vij2 P2jhx;uihx;vij, which plugged into the relation yields that ð1� jhu;vijÞð1þ jhu;vij � 2jhx;uihx; vijÞP 0. Then either1 ¼ jhu;vij, when 1þ jhu;vij ¼ 2 P 2jhx;uihx;vij (by Cauchy–Schwarz), or 1 > jhu;vij, which implies 1þ jhu;vijP2jhx;uihx;vij. Thus always hx;ui2 þ hx;vi2 6 1þ jhu;vij.

Third proof. (Lagrange multipliers) Define

Lðx; kÞ ¼ hu; xi2 þ hv; xi2 � kðjjxjj2 � 1Þ

and consider the system

@L@xi¼ 2uihu; xi þ 2v ihv ; xi � 2kxi ¼ 0

for 1 6 i 6 n, and

@L@k¼ jjxjj2 � 1 ¼ 0:

Now

0 ¼ 12

Xn

i¼1

xi@L@xi¼ hu; xi

Xn

i¼1

uixi þ hv ; xiXn

i¼1

v ixi � kXn

i¼1

x2i ¼ hu; xi

2 þ hv ; xi2 � kjjxjj2 ¼ hu; xi2 þ hv; xi2 � k;

so k needs be, computed at the critical point(s), the very expression we are considering (of no relevance, in fact). On the otherhand,

0 ¼ 12

Xn

i¼1

ui@L@xi¼ hu; xi

Xn

i¼1

u2i þ hv ; xi

Xn

i¼1

v iui � kXn

i¼1

xiui ¼ hu; xijjujj2 þ hv ; xihu;vi � khu; xi

¼ hu; xi þ hu; vihv ; xi � khu; xi

and similarly for v, thus reaching the system of two equations (in the variables hu; xi and hv ; xi)

ð1� kÞhu; xi þ hu; vihv; xi ¼ 0;hu;vihu; xi þ ð1� kÞhv; xi ¼ 0:

The determinant D of the matrix of this particular system is D ¼ ð1� kÞ2 � hu;vi2, and, if not null, the only solution is thetrivial hu; xi ¼ hv ; xi ¼ 0, when our expression reaches an evident minimum, equal to zero (therefore since then k ¼ 0, thismeans hu;vi – � 1, which translates in u – � v , and x ? spanðu;vÞ). We are therefore interested in D ¼ 0 (for all other crit-ical points), leading to k ¼ 1� hu;vi, thus k ¼ 1þ jhu;vij at a maximum point, and k ¼ 1� jhu;vij at a critical point. There aresome particular cases.

When u ¼ �v , thus hu;vi ¼ �1, the situation is simple. We have a maximum of 2 when x ¼ �u, and a minimum of 0 whenx ? u.

When u ? v , thus hu;vi ¼ 0, then k (thus the expression) is 1 at the maximum points, for all x 2 spanðu;vÞ, and k (thus theexpression) is 0 at the minimum points, for all x ? spanðu;vÞ. h.

The main result of this paper is given by

Theorem 3.2. Let ðH; h�; �iÞ be a real or complex inner product space and a; b; c 2 H. Then

ðjjajjjhb; cijÞ2 þ ðjjbjjjha; cijÞ2 6 jjajjjjbjjðjjajjjjbjj þ jha; bijÞjjcjj2:

Proof. For any not-null a; b; c, let us take u ¼ ajjajj ;v ¼ b

jjbjj ; x ¼ cjjcjj in Theorem 2.1 and, therefore hu; xi2 ¼ 1

jjajj�jjcjj2ha; ci2;

hv ; xi2 ¼ 1jjbjj�jjcjj2

hb; ci2; hu;vi ¼ ha;bijjajj�jjbjj, so the proven relation becomes


1jjajj jha; cij

2 þ 1jjbjj jhb; cij

26 1þ jha; bij

jjajj � jjbjj

� �jjcjj2;

equivalent with the required inequality, also true for a; b or c equal to zero. h

Remark. We can continue by Arithmetic–Geometric mean to obtain

jhu; xi � hv ; xij 6 12jhu; xij2 þ jhv; xij2� �

612

1þ jhu; vijð Þ:

A simple computation now yields

jha; ci � hb; cij 6 12ðjjajj � jjbjj þ jha; bijÞjjcjj2;

also true for a; b or c equal to zero, which is exactly Buzano’s inequality.As a consequence of Theorem 3.2 we have the general integral version.

Corollary 3.3. Let ðH;R;lÞ be a positive measure space and let f ; g;h 2 L2ðH;R;lÞ the Hilbert space of complex-valued 2� lintegrable functions defined on H. Then
Z
Hf 2ðtÞdlðtÞ

� � ZH

gðtÞhðtÞdlðtÞ��

�� 2

þZ

Hg2ðtÞdlðtÞ

� � ZH

hðtÞf ðtÞdlðtÞ��

�� 2

6

ZH

f 2ðtÞdlðtÞ� �1=2 Z

Hg2ðtÞdlðtÞ

� �1=2

�Z

Hf 2ðtÞdlðtÞ

� �1=2 ZH

g2ðtÞdlðtÞ� �1=2

þZ


��

!ZH

h2ðtÞdlðtÞ:

It might be useful to see that out of Theorem 3.2 one can get the following discrete version.

Corollary 3.4. If pi P 0; i ¼ 1;2; . . . ;n and ai; bi; ci are complex numbers, then

Xn

i¼1

pijaij2 ! Xn

i¼1

pibici

��

!2

þXn

i¼1

pijbij2 ! Xn

i¼1

piciai

��

!2

6

Xn

i¼1

pijaij2 !1=2 Xn

i¼1

pijbij2 !1=2 Xn

i¼1

pijcij2 !

�Xn

i¼1

pijaij2 !1=2 Xn

i¼1

pijbij2 !1=2

þXn

i¼1

piaibi

��

0@

1A:

In the case when H is the space of real-valued continuous functions defined on the interval ½a; b� endowed with the innerproduct hf ; gi :¼

R ba f ðxÞgðxÞdx, we have.

Corollary 3.5. Let f ; g;h be continuous real-valued functions on ½a; b�. Then, we have the following inequality

Z b

af 2ðtÞdt

! Z b

agðtÞhðtÞdt

!2

þZ b

ag2ðtÞdt

! Z b

ahðtÞf ðtÞdt

!2

6

Z b

af 2ðtÞdt

!1=2 Z b

ag2ðtÞdt

!1=2 Z b

ah2ðtÞdt �

Z b

af 2ðtÞdt

!1=2 Z b

ag2ðtÞdt

!1=2

þZ b

af ðtÞgðtÞdt

��

0@

1A:

The discrete real version of Corollary 2.4 is given by.

Corollary 3.6. Given real numbers ai; bi; ci;1 6 i 6 n, then

Xn

i¼1

a2i

! Xn

i¼1

bici

!2

þXn

i¼1

b2i

! Xn

i¼1

aici

!2

6

Xn

i¼1

a2i

! Xn

i¼1

b2i

! Xn

i¼1

c2i

!þXn

i¼1

aibi

��Xn

i¼1

a2i

!1=2 Xn

i¼1

b2i

!1=2 Xn

i¼1

c2i

!:

4. Some elementary applications

In [16] Wang and Zhang proved an inequality for unitary matrices. Now, we are able to give other trace inequalities forunitary matrices. A similar inequality was obtained by Zhang [17] but with a slightly different method.

Theorem 4.1. For a complex n� n matrix X denote mðXÞ the algebraic mean of its eigenvalues. For any two unitary n� n complexmatrices U;V, we have


jmðUÞj2 þ jmðVÞj2 þ jmðUVÞj2 6 1þ 2jmðUÞjjmðVÞjjmðUVÞj

and

jmðUÞj2 þ jmðVÞj2 6 1þ jmðUVÞj:

Proof. Let c ¼ 1ffiffinp I; a ¼ 1ffiffi

np U and b ¼ 1ffiffi

np V where U and V are unitary matrices. Clearly, jjajj ¼ jjbjj ¼ jjcjj ¼ 1 and moreover

ha; bi ¼ 1n

trðUV�Þ ¼ 1n

trðUVÞ;

hb; ci ¼ 1n

trðUÞ; hc; ai ¼ 1n

trðVÞ:

Now, by Theorems 2.1 and 3.2 we have

1n2 ðjtrðUÞj2 þ jtrðVÞj2 þ jtrðUVÞj2Þ 6 1þ 2

n3 ðjtrðUÞjjtrðVÞjjtrðUVÞjÞ

and

1njtrðUÞj

� �2

þ 1njtrðVÞj

� �2

6 1þ 1njtrðUVÞj:

Taking into account that mðXÞ ¼ 1n trðXÞ, we obtain exactly our inequalities. h

Remark. In general, for the Hilbert–Schmidt inner product, hX;Yi :¼ trðX�YÞ, for any complex square matrices A;B and C, wehave the following inequalities

trðAA�ÞjtrðBC�Þj2 þ trðBB�ÞjtrðCA�Þj2 þ trðCC�ÞjtrðCA�Þj2 6 trðAA�ÞtrðBB�ÞtrðCC�Þ þ 2jtrðAB�ÞtrðBC�ÞtrðCA�Þj

and

trðAA�Þ � jtrðB�CÞj2 þ trðBB�Þ � jtrðC�AÞj2 6 tr1=2ðAA�Þ � tr1=2ðBB�ÞtrðCC�Þ tr1=2ðAA�Þ � tr1=2ðBB�Þ þ jtrðA�BÞj� �

:

Theorem 4.2. For reals ai; bi; ci; i ¼ 1;2; . . . ;n such thatPn

i¼1aibi ¼ 0, we have

Xn

i¼1

a2i

! Xn

i¼1

c2i

! ! Xn

i¼1

b2i

! Xn

i¼1

c2i

! !P 4

Xn

i¼1

aici

!2 Xn

i¼1

bici

!2

and

Xn

i¼1

a2i

! Xn

i¼1

c2i

! ! Xn

i¼1

ciai

!2

þXn

i¼1

b2i

! Xn

i¼1

c2i

! ! Xn

i¼1

bici

!2

P 4Xn

i¼1

aici

!2 Xn

i¼1

bici

!2

:

Proof. By Arithmetic–Geometric mean inequality, we have

Xn

i¼1

a2i

! Xn

i¼1

bici

!2

þXn

i¼1

b2i

! Xn

i¼1

ciai

!2

P 2Xn

i¼1

bici

�� Xn

i¼1

ciai

��

Xn

i¼1

a2i

!1=2

�Xn

i¼1

b2i

!1=2

:

By Corollary 3.6 and taking into account thatPn

i¼1aibi ¼ 0, we obtain

Xn

i¼1

a2i

! Xn

i¼1

b2i

! Xn

i¼1

c2i

!P 2

Xn

i¼1

bici

��Xn

i¼1

ciai

��

Xn

i¼1

a2i

!1=2 Xn

i¼1

b2i

!1=2

;

which is equivalent with the desired inequality. As for the second inequality, applying again arithmetic–geometric meaninequality, we have

Xn

i¼1

a2i

! Xn

i¼1

c2i

! Xn

i¼1

aici

!2

þXn

i¼1

b2i

! Xn

i¼1

c2i

! Xn

i¼1

bici

!2

P 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

i¼1

a2i

! Xn

i¼1

b2i

! Xn

i¼1

c2i

!2vuut Xn

i¼1

ciai

! Xn

i¼1

bici

!

and by using the first inequality we finally obtain the second one. h


Theorem 4.3. For a matrix X with real entries, let sðXÞ be the sum of its entries. If A;B and C are n� n real matrices, then

sðCCTÞðsðAATÞsðBBTÞ � sðABTÞsðBATÞÞP sðAATÞsðBCTÞsðCBTÞ þ sðBBTÞsðACTÞsðCT AÞ � sðACTÞsðCBTÞsðBATÞ

� sðBCTÞsðCATÞsðABTÞ

and

sðAT AÞsðBT CÞsðBCTÞ þ sðBT BÞsðCT AÞsðAT CÞ 66 s1=2ðAT AÞs1=2ðBT BÞsðCT CÞðs1=2ðAT AÞs1=2ðBT BÞ þ sðAT BÞÞ:

Proof. For a square matrix A 2MnðRÞ; A ¼ ðaijÞ16i;j6n we consider sðAÞ ¼P

16i;j6naij, the sum of all the elements of the matrix.We denote AT the transpose of the matrix A.

We want to express sðABTÞ in terms of the elements of A and B. We have

A � BT ¼

a11 a12 . . . a1n

a21 a22 . . . a2n

..

. ... . .

. ...

an1 an2 . . . ann

0BBBB@

1CCCCA �

b11 b21 . . . bn1

b12 b22 . . . bn2

..

. ... . .

. ...

b1n b2n . . . bnn

0BBBB@

1CCCCA:

If we look at the element a1i, this element appears in the product matrix multiplied only by the elements of the i-th line of BT ,and this means that the sum of all the terms that contain a1i is a1iðb1i þ b2i þ . . .þ bniÞ. We repeat the same argument for allthe elements for the i-th column of A, and we get that the sum of all the terms which contain one of them isða1i þ a2i þ . . .þ aniÞðb1i þ b2i þ . . .þ bniÞ. Writing this for all the columns of A, we get that

sðABTÞ ¼Xn

i¼1

ða1i þ a2i þ . . .þ aniÞðb1i þ b2i þ . . .þ bniÞ:

Thus, we have xi ¼Pn

i¼1a1i; yi ¼Pn

i¼1b1i and zi ¼Pn

i¼1c1i.With xi and yi; zi considered above, we have

Xn

i¼1

x2i ¼ sðAATÞ;

Xn

i¼1

y2i ¼ sðBBTÞ

andPn

i¼1z2i ¼ sðCCTÞ. On the other hand,Pn

i¼1xiyi ¼ sðABTÞ ¼ sðAT BÞ;Pn

i¼1yizi ¼ sðBCTÞ ¼ sðCT BÞ andPn

i¼1zixi ¼ sðACTÞ ¼ sðCT AÞ. Now, by Corollaries 2.6 and 3.6 weobtain the desired inequalities. h

Remark. If we put C ¼ I in the first inequality we recover [12].

Theorem 4.4. If m; n are positive integers and A; B and C are m� n real matrices, then

ðdetðABTÞÞ2

detðCCTÞ þ ðdetðBCTÞÞ2 detðAATÞ þ ðdetðCATÞÞ2

detðBBTÞ

6 detðAATÞdetðBBTÞdetðCCTÞ þ 2jdetðABTÞdetðBCTÞdetðCT AÞj:

and

detðAT AÞdetðBT CÞdetðBCTÞ þ detðBT BÞdetðCT AÞdetðAT CÞ 6 ðdetðAT AÞÞ1=2ðdetðBT BÞÞ1=2

detðCT CÞ

� ððdetðAT AÞÞ1=2ðdetðBT BÞÞ1=2 þ detðAT BÞÞ:

Proof. If m > n then ABT;BCT ;CAT and AAT

; BBT ;CCT are m�m matrices with rank at most m, so our inequality is obvious. So,we may assume wothout loss of generality that m 6 n. Consider S ¼ ðj1; j2; . . . ; jmÞ : 1 6 j1 < � � � < jm 6 nf g. Now, for a m by nmatrix X with real entries and w ¼ ðj1; j2; . . . ; jmÞ 2 S denote by Xw the m�m matirx whose columns are columns j1; j2; . . . ; jm

of X. For X;Y two m� n matrices with real entries, by the Cauchy-Binet formula, we have

detðXYTÞ ¼Xw2S

det Xw det Yw;

n� �

n� �

with respect to the inner product h�; �i : R m � R m ! R. Now, if a ¼ ðdet AwÞw2S ; b ¼ ðdet BwÞw2S and c ¼ ðdet CwÞw2S ,then by applying Theorems 2.1 and 3.2, we obtain


Xw2S

det Aw det Bw

!2 Xw2Sðdet CwÞ2

!þ

Xw2S

det Bw det Cw

!2 Xw2Sðdet AwÞ2

!þ

Xw2S

det Cw det Aw

!2 Xw2Sðdet BwÞ2

!

6

Xw2Sðdet CwÞ2

! Xw2Sðdet CwÞ2

! Xw2Sðdet CwÞ2

!þ 2

Xw2S

det Aw det Bw

! Xw2S

det Aw det Bw

! Xw2S

det Aw det Bw

!��

and

Xw2Sðdet AwÞ2

! Xw2S

det Bw det Cw

!2

þXw2Sðdet BwÞ2

! Xw2S

det Cw det Aw

!2

6

Xw2Sðdet AwÞ2

!��

1=2 Xw2Sðdet BwÞ2

!��1=2

�Xw2Sðdet CwÞ2

!�

Xw2Sðdet AwÞ2

!��

1=2 Xw2Sðdet BwÞ

!��þ

Xw2S

det Aw det Bw

!��

0@

1A

and the conclusion follows immediately. h

An interesting particular case was one of the hardest problems given at Romanian selection test for International Math-ematical Olympiad in 2007.

Theorem 4.5. For n P 2 positive integer, ai; bi reals, i ¼ 1;2; . . . ;n, such that

Xn

i¼1

a2i ¼

Xn

i¼1

b2i ¼ 1;

Xn

i¼1

aibi ¼ 0;

we have

Xn

i¼1

ai

!2

þXn

i¼1

bi

!2

6 n:

Proof. Take ci ¼ 1; i ¼ 1;2; . . . ; n in Corollary 3.6. h

The integral analog of Theorem 4.2 is given by.

Theorem 4.6. Let f ; g;h be contionuous real-valued functions on ½0;1� satisfying the conditionR½0;1� fg ¼ 0. Then, we have

Z½0;1�

f 2Z½0;1�

g2Z½0;1�

h2

!2

P 4Z½0;1�

hf

!2 Z½0;1�

gh

!2

and

Z½0;1�

f 2Z½0;1�

h2

! Z½0;1�

hf

!2

þZ½0;1�

g2Z½0;1�

h2

! Z½0;1�

gh

!2

P 4Z½0;1�

hf

!2 Z½0;1�

gh

!2

:

Proof. Let us consider by Cð½0;1�Þ be the space of continuous functions endowed with the inner product

hf ; gi :¼Z½0;1�

f ðxÞgðxÞdx

and the associated norm jjf jj ¼R½0;1� f

2ðxÞdx� �1=2

. Our inequalities can be rewritten as

jjf jj2jjhjj2 � jjgjj2jjhjj2 P 4jhg;hij2 � jhh; f ij2 ð5Þ

and

jjf jj2jjhjj2jhf ;hij2 þ jjgjj2jjhjj2jhg; hij2 P 4jhg;hij2 � jhh; f ij2; ð6Þ

by the Corollary 3.4, we have

ðjjf jjhg;hiÞ2 þ ðjjgjjhf ;hiÞ2 6 jjf jjjjgjjðjjf jjjjgjj þ jhf ; gijÞjjhjj2

for any f ; g;h 2 Cð½0;1�Þ. On the other hand, by Arithmetic–Geometric mean inequality we obtain

ðjjf jjhg;hiÞ2 þ ðjjgjjhf ;hiÞ2 P 2jjf jjjjgjj � jhg; hijjhh; f ij:


Combining this inequality with the one obtained from the Corollary 2.5 and taking into account that hf ; gi ¼ 0, we get

jjf jj2jjgjj2jjhjj2 P 2jjf jjjjgjj � jhg;hijjhh; f ij;

which is finally equivalent to

jjf jj2jjgjj2jjhjj4 P 4jhg;hij2jhh; f ij2

and thus the first inequality is proved. For the second inequality, we shall use the first inequality. Hence, again by Arithme-tic–Geometric mean inequality we have

jjf jj2jjhjj2jhf ;hij2 þ jjgjj2jjhjj2jhg; hij2 P 2jjf jjjjhjj � jhh; f ij � jjgjjjjhjj � jhg;hij:

Now, we only need to see that

2jjf jjjjhjj � jhh; f ij � jjgjjjjhjj � jhg; hijP 4jhg;hij2jhh; f ij2:

This last inequality finally reduces to

jjf jjjjgjjjjhjj2 P 2jhg; hijjhh; f ij

and this exactly inequality (5) written in the form

jjf jj2jjgjj2jjhjj4 P 4jhg;hij2jhh; f ij2: �

Remark. For h 1, we obtain [10].

Theorem 4.7. [[11]] Let n be a positive integer, and write a vector x 2 Rn as ðx1; x2; . . . ; xnÞ. For x; y; a; b 2 Rn let

½x; y�a;b ¼X

16i;j6n

xiyj minðai; bjÞ:

Then, for x; y; z; a; b; c 2 Rnwith nonnegative entries,

½x; x�a;a � ½y; z�2b;c þ ½y; y�b;b � ½z; x�

2c;a þ ½z; z�c;c � ½x; y�

2a;b 6 ½x; x�a;a � ½y; y�b;b � ½z; z�c;c þ 2½x; y�a;b � ½y; z�b;c � ½z; x�c;a:

and

½x; x�a;a � ½y; z�2b;c þ ½y; y�b;b � ½z; x�

2c;a 6 ½x; x�

1=2a;a � ½y; y�

1=2b;b � ½z; z�c;c � ½x; x�

1=2a;a � ½y; y�

1=2b;b þ ½x; y�a;b

� �:

Proof. Our inequalities can be restated as

X16i;j6n

xixj minðai; ajÞ !

�X

16i;j6n

yizj minðbi; cjÞ !2

þX

16i;j6n

yiyj minðbi; bjÞ !

�X

16i;j6n

zixj minðci; ajÞ !2

þX

16i;j6n

zizj minðci; cjÞ !

�X

16i;j6n

yixj minðbi; cjÞ !2

6

X16i;j6n


�X

16i;j6n


�X

16i;j6n


þ 2X

16i;j6n

xiyj minðai; bjÞ !

�X

16i;j6n

yizj minðbi; cjÞ !

�X

16i;j6n

zixj minðci; ajÞ !

and

X16i;j6n


�X

16i;j6n

yizj minðbi; cjÞ !2

þX

16i;j6n


�X

16i;j6n

zixj minðci; ajÞ !2

6

X16i;j6n

xixj minðai; ajÞ !1=2 X

16i;j6n

yiyj minðbi; bjÞ !1=2

�X

16i;j6n

xixj minðai; ajÞ !1=2 X

16i;j6n

yiyj minðbi; bjÞ !1=2

0@

þX

16i;j6n

xiyj minðai; bjÞ!�

X16i;j6n


:

Let vA be the characteristic function of an arbitrary set A. Consider the functions f ; g;h : ½0;1Þ ! R defined by

f ðxÞ ¼Xn

i¼1

xiv½0;ai �ðxÞ;


gðxÞ ¼Xn

i¼1

yiv½0;bi �ðxÞ;

hðxÞ ¼Xn

i¼1

ziv½0;ci �ðxÞ:

We have
Z 1
0f 2ðxÞdx ¼

X16i;j6n

xixj

Z 1

0v½0;ai �ðxÞv½0;bi �ðxÞdx ¼

X16i;j6n

xixj minðai; ajÞ:

Analogously, we obtain
Z 1
0g2ðxÞdx ¼

X16i;j6n

yiyj minðbi; bjÞ;Z 1

0h2ðxÞdx ¼

X16i;j6n

zizj minðci; cjÞ;

as well as
Z 1
0f ðxÞgðxÞdx ¼

X16i;j6n

xiyj minðai; bjÞ;

Z 1

0gðxÞhðxÞdx ¼

X16i;j6n

yizj minðbi; cjÞ

and
Z 1
0gðxÞhðxÞdx ¼

X16i;j6n

zixj minðci; ajÞ:

Finally, by applying Theorem 2.1 and Corollary 3.3 with the inner product given by hf ; gi :¼R1

0 f ðxÞgðxÞdx, with f ; gdefined as above, we derive the inequalities
Z 1
0f 2ðxÞdx

Z 1

0gðxÞhðxÞdx

� �2

þZ 1

0g2ðxÞdx

Z 1

0hðxÞf ðxÞdx

� �2

þZ 1

0h2ðxÞdx

Z 1

0f ðxÞgðxÞdx

� �2

6

Z 1

0f 2ðxÞdx

Z 1

0g2ðxÞdx

Z 1

0h2ðxÞdxþ 2

Z 1

0hðxÞf ðxÞdx

Z 1

0hðxÞf ðxÞdx

Z 1

0hðxÞf ðxÞdx

��

and
Z 1
0f 2ðxÞdx

Z 1

0gðxÞhðxÞdx

� �2

þZ 1

0g2ðxÞdx

Z 1

0hðxÞf ðxÞdx

� �2

6

Z 1

0f 2ðxÞdx

� �1=2 Z 1

0g2ðxÞdx

� �1=2 Z 1

0h2ðxÞdx

� ��

Z 1

0f 2ðxÞdx

� �1=2 Z 1

0g2ðxÞdx

� �1=2

þZ 1

0f ðxÞgðxÞdx

��

!

and, now the conclusion is obvious. h

Remark. These kind of min–max inequalities for vectors arise in coding of messages. A list of such inequalities is provided in[3].

Theorem 4.8. For a matrix A 2 MnðRÞ, let fA :MnðRÞ ! R,

fAðXÞ ¼Xn

i;j¼1

bijxij;

where B ¼ ½bij�16i;j6n ¼ AAT . Then, we have

fAðXXTÞ � f 2A ðYZTÞ þ fAðYYTÞ � f 2

A ðZXTÞ þ fAðZZTÞ � f 2A ðXYTÞ 6 fAðXXTÞ � fAðYYTÞ � fAðZZTÞ þ 2f AðXYTÞ � fAðYZTÞ � fAðZXTÞ

and

fAðXXTÞ � f 2A ðYZTÞ þ fAðYYTÞ � f 2

A ðZXTÞ 6 f 1=2A ðXXTÞ � f 1=2

A ðYYTÞ � fAðZZTÞ f 1=2A ðXXTÞ � f 1=2

A ðYYTÞ þ fAðXYTÞ� �

:


Proof. We define the inner product

hX;Yi :¼ fAðXYTÞ ¼X

16i;j;k6n

bi;jxi;kyj;k ¼Xn

k¼1

columnkðXÞT BcolumnkðYÞ:

Moreover, since B is positive semidefinite, we have

fAðXXTÞ ¼Xn

k¼1

columnkðXÞT BcolumnkðXÞ

and clearly, fAðXXTÞ ¼ 0 implies that columnkðXÞ ¼ 0 for every k, that is X ¼ 0. Now, by Theorems 2.1 and 3.2, we derive ourinequalities. h

Theorem 4.9. Let A ¼ ½aij�16i;j6n 2 MnðRÞ, be a symmetric matrix with aii >Pn

j¼1;j–ijaijj and xi; yi; zi; i ¼ 1;2; . . . ;n are nonnegativereal numbers. Then, we have

X16i;j6n

aijxixj

! X16i;j6n

aijyizj

!2

þX

16i;j6n

aijyiyj

! X16i;j6n

aijzixj

!2

þX

16i;j6n

aijzizj

! X16i;j6n

aijxiyj

!2

6

X16i;j6n

aijxixj

! X16i;j6n

aijyiyj

! X16i;j6n

aijzizj

!þ 2

X16i;j6n

aijxiyj

��X

16i;j6n

aijyizj

��X

16i;j6n

aijzixj

��

and

X16i;j6n

aijxixj

! X16i;j6n

aijyizj

!2

þX

16i;j6n

aijyiyj

! X16i;j6n

aijzixj

!2

6

X16i;j6n

aijxixj

!1=2 X16i;j6n

aijyiyj

!1=2 X16i;j6n

aijzizj

!�

X16i;j6n

aijxixj

!1=2 X16i;j6n

aijyiyj

!1=2

þX

16i;j6n

aijxiyj

��

0@

1A:

Proof. For vectors x; y 2 Rn, define the function f : Rn � Rn ! R,

f ðx; yÞ :¼X

16i;j6n

aijxiyj:

Clearly, f ðax; byÞ ¼ a � bf ðx; yÞ; f ðx; yÞ ¼ f ðy; xÞ and f ðxþ y; zÞ ¼ f ðx; zÞ þ f ðy; zÞ. The only property that we need to verify isf ðx; xÞ > 0 for x – 0. We have

f ðx; xÞ ¼X

16i;j6n

aijxiyj ¼Xn

i¼1

aiix2i þ 2

Xi<j

aijxixj >Xn

i¼1

jaijj� �

x2ij þ 2

Xi<j

aijxixj ¼Xi<j

jaijjðx2i þ x2

j Þ þ 2Xi<j

aijxixj

P 2Xi<j

jaijjjxijjxjj þ 2Xi<j

aijxixj ¼ 2Xi<j

ðjaijjjxijjxjj þ aijxixjÞP 0:

Thus, f ðx; yÞ is an inner product on Rn � Rn. Now, by applying Theorems 2.1 and 3.2, we obtain our result. h

Remark. More generally, one can prove that if A is a positive definite symmetric matrix with real entries, then the functionf : Rn � Rn ! R,

f ðx; yÞ ¼X

16i;j6n

aijxiyj;

defines an inner product on Rn. Thus, Theorem 4.8 holds in a slightly general setting.One can consider some classical examples of positive definite and symmetric matrices. For instance, the Hilbert matrix

A1 ¼ ðaijÞ16i;j6n ¼ 1iþj�1

h i16i;j6n

which is positive definite. This is equivalent with showing that for positive xi; i ¼ 1;2; . . . ;n

we haveP

16i;j6nxixj

iþj�1 > 0. For instance, one ca easily prove that

X16i;j6n

xixj

iþ j� 1¼Z 1

0

Xn

i¼1

xiti�1

!2

dt > 0;


when f1ðtÞ ¼Pn

i¼1xiti�1� �2

is not the zero function. The same for Cauchy matrix, A2 ¼ ðaijÞ16i;j6n ¼ 1iþj

h i16i;j6n

, one can provethat

X16i;j6n

xixj

iþ j¼Z 1

0

Xn

i¼1

xie�it

!2

dt > 0;

when f2ðtÞ ¼Pn

i¼1xie�it� �2 is not the zero function.

On the other hand, on the axis of real numbers we consider the intervals Ii; i ¼ 1;2; . . . ;n and denote by aij be the length of

the interval Ii \ Ij, for i; j ¼ 1;2; . . . ; n. One can prove that the matrix A3 ¼ ðaijÞ16i;j6n ¼ lengthðIi \ IjÞ� �

16i;j6n is positive definite.

Indeed, denote I ¼Sn

i¼1Ii and for J I consider the characteristic function of the set J;vJ : I ! f0;1g. If J is an interval, then the

length of J is given by lengthðJÞ ¼R

I vJðxÞdx ¼R

I v2J dx. On the other hand,

aij ¼Z

IvIiðxÞvIj

ðxÞdx

and thus,

X16i;j6n

aijxixj ¼Z

I

X16i;j6n

vIiðxÞvIj

ðxÞxixj

!dx ¼

ZI

Xn

i¼1

xivIiðxÞ

!2

dx > 0

for positive xi; i ¼ 1;2; . . . ;n.The same idea works when A4 ¼ ðaijÞ16i;j6n ¼ areaðDi \ DjÞ, where Di; i ¼ 1;2; . . . ;n are discs in the Euclidian plane. Indeed,

we have the characteristic function of the set Di;vDi: R2 ! f0;1g, and moreover

areaðDiÞ ¼Z Z

R2vDiðx; yÞdxdy ¼

Z ZR2

v2Diðx; yÞdxdy

and

areaðDi \ DjÞ ¼Z Z

R2vDiðx; yÞvDj

ðx; yÞdxdy:

Now, we have

X16i;j6n

aijxixj ¼Z Z

R2

X16i;j6n

xivDiðx; yÞxjvDj

ðx; yÞdxdy ¼Z Z

R2

Xn

i¼1

xivDiðx; yÞ

!2

dxdy > 0:

Last but not least, we end the note with an elementary inequality with its remniscents in the theory of improper integrals.

Theorem 4.10. For all nonnegative reals a; b; c we have the inequalities

ab

ðaþ bÞ2þ bc

ðbþ cÞ2þ ca

ðc þ aÞ26

14þ 4abcðaþ bÞðbþ cÞðc þ aÞ

and

bc

ðbþ cÞ2þ ca

ðc þ aÞ26

18abc

þ 14cðaþ bÞ :

Proof. The key is the following identity:

1aþ b

¼Z 1

0e�ðaþbÞxdx:

Define the inner product hf ; gi :¼R1

0 f ðxÞgðxÞ, where f ðxÞ ¼ e�ax and gðxÞ ¼ e�bx. A short computation shows thatR10 e�2axdx ¼ 1

2a. So by Theorem 2.1 we have

1

ðbþ cÞ2aþ 1

ðc þ aÞ2bþ 1

ðaþ bÞ2c6

18abc

þ 2ðaþ bÞðbþ cÞðc þ aÞ ;

which is equivalent to the first inequality.


Now, by Corolarry 3.3, we obtain

1

aðbþ cÞ2� 1

bðc þ aÞ26

1

4abc2

12abþ 1

aþ b

� �;

which is our second inequality. h

Acknowledgements

The first author would like to thank Politehnica University of Bucharest for hospitality during May–June 2012 when partof this work was initiated. The authors would also like to thank Beniamin Bogos�el for crucial observations which improvedthe initial version of this paper.

References

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