CUBO A Mathematical Journal

Vol.19, No¯ 01, (53–77). March 2017

Inequalities for Chebyshev Functional in Banach Algebras

S. S. Dragomir1, M. V. Boldea2 and M. Megan3

1 Mathematics, School of Engineering & Science, Victoria University,

PO Box 14428, Melbourne City, MC 8001, Australia.

School of Computational & Applied Mathematics, University of the Witwatersrand, .

Private Bag 3, Johannesburg 2050, South Africa.

2 Mathematics and Statistics, Banat University of Agricultural Sciences and Veterinary Medicine

Timişoara,

119 Calea Aradului, 300645, Timişoara, România

3 Department of Mathematics, West University of Timişoara,

B-dul V. Pârvan 4, 1900-Timişoara, România

sever.dragomir@vu.edu.au, http://rgmia.org/dragomir

ABSTRACT

By utilizing some identities for double sums, some new inequalities for the Cheby-

shev functional in Banach algebras are given. Some examples for the exponential and

resolvent functions on Banach algebras are also provided.

RESUMEN

Usando algunas identidades para sumas dobles, encontramos algunas nuevas desigual-

dades para el funcional de Chebyshev en álgebras de Banach. También entregamos

algunos ejemplos para las funciones exponencial y resolvente en álgebras de Banach.

Keywords and Phrases: Banach algebras, Power series, Exponential function, Resolvent func-

tion, Norm inequalities.

2010 AMS Mathematics Subject Classification: 47A63; 47A99.



54 S. S. Dragomir, M. V. Boldea, M. Megan CUBO
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1 Introduction

In 1950, M. Biernacki, H. Pidek and C. Ryll-Nardzewski [2] established the following discrete

version of Grüss’ inequality, see also [39, Ch. X]:

Theorem 1.1. Let a = (a1, ..., an) , b = (b1, ..., bn) be two n-tuples of real numbers such that

r ≤ ai ≤ R and s ≤ bi ≤ S for i = 1, ..., n. Then one has the inequality:
∣
∣
∣
∣
∣
1

n

n∑

i=1

aibi −
1

n

n∑

i=1

ai ·
1

n

n∑

i=1

bi

∣
∣
∣
∣
∣

(1.1)

≤
1

n

[n

2

](

1 −
1

n

[n

2

])

(R − r) (S − s)

when [x] is the integer part of x, x ∈ R.

A weighted version of Grüss’ discrete inequality was proved by J.E. Pec̆arić in 1979, see for

instance [39, Ch. X]:

Theorem 1.2. Let a, b be two monotonic n-tuples and p a positive one. Then
∣
∣
∣
∣
∣

1

Pn

n∑

i=1

piaibi −
1

Pn

n∑

i=1

piai ·
1

Pn

n∑

i=1

pibi

∣
∣
∣
∣
∣

(1.2)

≤ |an − a1| |bn − b1| max
1≤k≤n−1

(
PkP̄k+1

P2n

)

where Pn :=
n∑

i=1

pi , P̄k+1 = Pn − Pk+1 .

In 1981 , A. Lupaş [39, Ch. X] proved some similar results for the first difference of a as

follows :

Theorem 1.3. Let a, b two monotonic n-tuples in the same sense and p a positive n-tuple. Then

min
1≤i≤n−1

|ai+1 − ai| min
1≤i≤n−1

|bi+1 − bi|

⎡

⎣ 1

Pn

n∑

i=1

i2pi −

(
1

Pn

n∑

i=1

ipi

)2
⎤

⎦ (1.3)

≤
1

Pn

n∑

i=1

piaibi −
1

Pn

n∑

i=1

piai ·
1

Pn

n∑

i=1

pibi

≤ max
1≤i≤n−1

|ai+1 − ai| max
1≤i≤n−1

|bi+1 − bi|

⎡

⎣ 1

Pn

n∑

i=1

i2pi −

(
1

Pn

n∑

i=1

ipi

)2
⎤

⎦ .

If there exists the numbers ā, ā1, r, r1, (rr1 > 0) such that ak = ā + kr and bk = ā1 + kr1, then in

(1.3) the equality holds.

For some generalizations of Gruss’ inequality for isotonic linear functionals defined on certain

spaces of mappings see Chapter X of the book [39] where further references are given .

For related results, see [1]-[21], [25]-[31] and [34]-[44].



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Inequalities for Čeby šev Functional in Banach Algebras 55

2 Some Facts on Banach Algebras

In order to extend the above results for Banach algebras, we need some preliminary facts as follows:

Let B be an algebra. An algebra norm on B is a map ∥·∥ : B→[0, ∞) such that (B, ∥·∥) is a

normed space, and, further:

∥ab∥ ≤ ∥a∥ ∥b∥

for any a, b ∈ B. The normed algebra (B, ∥·∥) is a Banach algebra if ∥·∥ is a complete norm.

We assume that the Banach algebra is unital, this means that B has an identity 1 and that

∥1∥ = 1.

Let B be a unital algebra. An element a ∈ B is invertible if there exists an element b ∈ B with

ab = ba = 1. The element b is unique; it is called the inverse of a and written a−1 or 1
a
. The set of

invertible elements of B is denoted by InvB. If a, b ∈InvB then ab ∈InvB and (ab)
−1

= b−1a−1.

For a unital Banach algebra we also have:

(i) If a ∈ B and limn→∞ ∥a
n∥

1/n
< 1, then 1 − a ∈InvB;

(ii) {a ∈ B: ∥1 − a∥ < 1} ⊂InvB;

(iii) InvB is an open subset of B;

(iv) The map InvB ∋ a &−→ a−1 ∈InvB is continuous.

For simplicity, we denote λ1, where λ ∈ C and 1 is the identity of B, by λ. The resolvent set

of a ∈ B is defined by

ρ(a) := {λ ∈ C : λ − a ∈ InvB} ;

the spectrum of a is σ(a) , the complement of ρ(a) in C, and the resolvent function of a is

Ra : ρ(a) →InvB, Ra (λ) := (λ − a)
−1

. For each λ,γ ∈ ρ(a) we have the identity

Ra (γ) − Ra (λ) = (λ − γ) Ra (λ) Ra (γ) .

We also have that σ(a) ⊂ {λ ∈ C : |λ| ≤ ∥a∥} . The spectral radius of a is defined as ν(a) =

sup {|λ| : λ ∈ σ(a)} .

If a, b are commuting elements in B, i.e. ab = ba, then

ν(ab) ≤ ν(a)ν(b) and ν(a + b) ≤ ν(a) + ν(b) .

Let f be an analytic functions on the open disk D (0, R) given by the power series f (λ) :=
∑

∞

j=0 αjλ
j (|λ| < R) . If ν(a) < R, then the series

∑
∞

j=0 αja
j
converges in the Banach algebra B

because
∑

∞

j=0 |αj|
∥
∥aj
∥
∥ < ∞, and we can define f (a) to be its sum. Clearly f (a) is well defined and



56 S. S. Dragomir, M. V. Boldea, M. Megan CUBO
19, 1 (2017)

there are many examples of important functions on a Banach algebra B that can be constructed

in this way. For instance, the exponential map on B denoted exp and defined as

exp a :=
∞∑

j=0

1

j!
aj for each a ∈ B.

If B is not commutative, then many of the familiar properties of the exponential function from the

scalar case do not hold. The following key formula is valid, however with the additional hypothesis

of commutativity for a and b from B

exp (a + b) = exp (a) exp (b) .

In a general Banach algebra B it is difficult to determine the elements in the range of the exponential

map exp (B) , i.e. the element which have a ”logarithm”. However, it is easy to see that if a is an

element in B such that ∥1 − a∥ < 1, then a is in exp (B) . That follows from the fact that if we set

b = −
∞∑

n=1

1

n
(1 − a)

n
,

then the series converges absolutely and, as in the scalar case, substituting this series into the

series expansion for exp (b) yields exp (b) = a.

It is known that if x and y are commuting, i.e. xy = yx, then the exponential function satisfies

the property

exp (x) exp (y) = exp (y) exp (x) = exp (x + y) .

Also, if x is invertible and a, b ∈ R with a < b then

∫
b

a

exp (tx) dt = x−1 [exp (bx) − exp (ax)] .

Moreover, if x and y are commuting and y − x is invertible, then

∫
1

0

exp ((1 − s) x + sy) ds =

∫
1

0

exp (s (y − x)) exp (x) ds

=

(∫
1

0

exp (s (y − x)) ds

)

exp (x)

= (y − x)
−1

[exp (y − x) − I] exp (x)

= (y − x)
−1

[exp (y) − exp (x)] .

Inequalities for functions of operators in Hilbert spaces may be found in the papers [12], [11]

and in the recent monographs [22], [23], [32] and the references therein.

The following inequality of Grüss-Lupaş type in Banach algebras holds:



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Inequalities for Čeby šev Functional in Banach Algebras 57

Theorem 2.1. Let B be a Banach algebra over K (=R,C) , ai, bi ∈ B and αi ∈ K (i = 1, ..., n) .

Then we have the inequality:

∥
∥
∥
∥
∥

n∑

i=1

αi

n∑

i=1

αiaibi −
n∑

i=1

αiai

n∑

i=1

αibi

∥
∥
∥
∥
∥

(2.1)

≤ max
1≤j≤n−1

∥aj+1 − aj∥ max
1≤j≤n−1

∥bj+1 − bj∥

×

⎡

⎣
n∑

i=1

|αi|
n∑

i=1

i2 |αi| −

(
n∑

i=1

i |αi|

)2
⎤

⎦

The inequality (2.1) is sharp in the sense that the multiplicative constant C = 1 in the right

membership can not be replaced by a smaller one.

Let αn be nonzero complex numbers and let

R :=
1

lim sup |αn|
1

n

.

Clearly 0 ≤ R ≤ ∞, but we consider only the case 0 < R ≤ ∞.

Denote by:

D(0, R) =

{
{λ ∈ C : |λ| < R}, if R < ∞

C, if R = ∞,

consider the functions:

λ &→ f(λ) : D(0, R) → C, f(λ) :=
∞∑

n=0

αnλ
n

and

λ &→ fA(λ) : D(0, R) → C, fA(λ) :=
∞∑

n=0

|αn|λ
n.

Let B be a unital Banach algebra and 1 its unity. Denote by

B(0, R) =

{
{x ∈ B : ∥x∥ < R}, if R < ∞

B, if R = ∞.

We associate to f the map:

x &→ f̃(x) : B(0, R) → B, f̃(x) :=
∞∑

n=0

αnx
n.



58 S. S. Dragomir, M. V. Boldea, M. Megan CUBO
19, 1 (2017)

Obviously, f̃ is correctly defined because the series
∑

∞

n=0 αnx
n

is absolutely convergent, since
∑

∞

n=0 ∥αnx
n∥ ≤

∑
∞

n=0 |αn| ∥x∥
n
.

Making use of Theorem 2.1 we have the following inequality for power series:

Theorem 2.2. Let f(λ) =
∑

∞

n=0 αnλ
n be a power series that is convergent on the open disk

D(0, R), with R > 0. If x, y ∈ B with xy = yx and ∥x∥ , ∥y∥ ≤ 1, then we have for λ ∈ C with

|λ| < R the inequality:

∥
∥
∥f̃ (λ · 1) f̃ (λxy) − f̃ (λx) f̃ (λy)

∥
∥
∥ (2.2)

≤ ∥x − 1∥ ∥y − 1∥
{
fA (|λ|)

[
|λ| f′A (|λ|) + |λ|

2
f′′A (|λ|)

]
− [|λ| f′A (|λ|)]

2
}
.

Motivated by the above results we establish in this paper other similar inequalities for the

norm of the Chebyshev difference

f̃ (λ · 1) f̃ (λxy) − f̃ (λx) f̃ (λy)

by the use of some discrete inequalities of Grüss’ type.

First we establish some identities of interest in Banach algebras,

3 Identities in Banach Algebras

Consider the Chebyshev functional defined for p = (p1, ..., pn) ∈ K
n, a = (a1, ..., an) ∈ B

n
and

b = (b1, ..., bn) ∈ B
n, where B is a Banach algebra over the real or complex number field K:

Tn (p; a, b) := Pn

n∑

i=1

piaibi −
n∑

i=1

piai

n∑

i=1

pibi, (3.1)

where Pn :=
∑n

i=1 pi.

The following particular identities for unweighted means hold as well, where Tn (a, x) is defined

as follows:

Tn (a, b) :=
1

n

n∑

i=1

αibi −
1

n

n∑

i=1

ai
1

n

n∑

i=1

bi. (3.2)

If α,β are scalars and x, y are vectors in a Banach algebra B, we denote by

det

(
α β

x y

)

:= αy − βx ∈ B.

The first result is embodied in the following



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Inequalities for Čeby šev Functional in Banach Algebras 59

Theorem 3.1. Let p = (p1, ..., pn) ∈ K
n, and a = (a1, ..., an) , b = (b1, ..., bn) ∈ B

n. If we

define

Pi :=
i∑

k=1

pk, P̄i := Pn − Pi, i ∈ {1, ..., n − 1} ,

Ai (p) :=
i∑

k=1

pkak, Āi (p) := An (p) − Ai (p) , i ∈ {1, ..., n − 1} ,

then we have the identity

Tn (p; a, b) =
n−1∑

i=1

det

(
Pi Pn

Ai (p) An (p)

)

∆bi (3.3)

= Pn

n−1∑

i=1

Pi

(
An (p)

Pn
−

Ai (p)

Pi

)

∆bi

(if Pi ̸= 0, i ∈ {1, ..., n})

=

n−1∑

i=1

PiP̄i

(
Āi (p)

P̄i
−

Ai (p)

Pi

)

∆bi

(
if Pi, P̄i ̸= 0, i ∈ {1, ..., n − 1}

)
;

where ∆bi := bi+1 − bi (i ∈ {1, ..., n − 1}) is the forward difference.

Proof. We use the following well known summation by parts formula

q−1∑

l=p

dl∆vl = dlvl|
q

p
−

q−1∑

l=p

∆dlvl+1, (3.4)

where dl and vl are vectors in a linear space, l = p, ..., q (q > p; p, q are natural numbers) .

If we choose in (3.4), p = 1, q = n, di = PiAn (p) − PnAi (p) and vi = bi (i ∈ {1, ..., n − 1}) ,



60 S. S. Dragomir, M. V. Boldea, M. Megan CUBO
19, 1 (2017)

then we get

n−1∑

i=1

(PiAn (p) − PnAi (p))∆bi

= [PiAn (p) − PnAi (p)] bi|
n

1
−

n−1∑

i=1

∆(PiAn (p) − PnAi (p)) bi+1

= [PnAn (p) − PnAn (p)] bn − [P1An (p) − PnA1 (p)] b1

−

n−1∑

i=1

[Pi+1An (p) − PnAi+1 (p) − PiAn (p) + PnAi (p)] bi+1

= Pnp1a1b1 − p1An (p) b1 −
n−1∑

i=1

(pi+1An (p) − Pnpi+1ai+1) bi+1

= Pnp1a1b1 − p1An (p) b1 − An (p)
n−1∑

i=1

pi+1bi+1 + Pn

n−1∑

i=1

pi+1ai+1bi+1

= Pn

n∑

i=1

piaibi −
n∑

i=1

piai

n∑

i=1

pibi

= Tn (p; a, b) ,

which produce the first identity in (3.3) .

The second and the third identities are obvious and we omit the details.

Before we prove the second result, we need the following lemma providing an identity that is

interesting in itself as well.

Lemma 3.1. Let p = (p1, ..., pn) ∈ K
n and a = (a1, ..., an) ∈ B

n. Then we have the equality

det

(
Pi Pn

Ai (p) An (p)

)

=

n−1∑

j=1

Pmin{i,j}P̄max{i,j}∆aj, (3.5)

for each i ∈ {1, ..., n − 1} .

Proof. Define, for i ∈ {1, ..., n − 1} ,

K (i) :=
n−1∑

j=1

Pmin{i,j}P̄max{i,j}∆aj.



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Inequalities for Čeby šev Functional in Banach Algebras 61

We have

K (i) =
i∑

j=1

Pmin{i,j}P̄max{i,j}∆aj +
n−1∑

j=i+1

Pmin{i,j}P̄max{i,j}∆aj (3.6)

=

i∑

j=1

PjP̄i∆aj +
n−1∑

j=i+1

PiP̄j∆aj

= P̄i

i∑

j=1

Pj∆aj + Pi

n−1∑

j=i+1

P̄j∆aj.

Using the summation by parts formula, we have

i∑

j=1

Pj∆aj = Pjaj|
i+1
1

−

i∑

j=1

(Pj+1 − Pj) aj+1 (3.7)

= Pi+1ai+1 − p1a1 −
i∑

j=1

pj+1aj+1

= Pi+1ai+1 −
i+1∑

j=1

pjaj

and

n−1∑

j=i+1

P̄j∆aj = P̄jaj
∣
∣n
i+1

−

n−1∑

j=i+1

(
P̄j+1 − P̄j

)
aj+1 (3.8)

= P̄nan − P̄i+1ai+1 −
n−1∑

j=i+1

(Pn − Pj+1 − Pn + Pj) aj+1

= −P̄i+1ai+1 +
n−1∑

j=i+1

pj+1aj+1.

Using (3.7) and (3.8) we have

K (i) = P̄i

⎛

⎝Pi+1ai+1 −
i+1∑

j=1

pjaj

⎞

⎠ + Pi

⎛

⎝
n−1∑

j=i+1

pj+1aj+1 − P̄i+1ai+1

⎞

⎠

= P̄iPi+1ai+1 − PiP̄i+1ai+1 − P̄i

i+1∑

j=1

pjaj + Pi

n−1∑

j=i+1

pj+1aj+1

= [(Pn − Pi) Pi+1 − Pi (Pn − Pi+1)] ai+1

+ Pi

n−1∑

j=i+1

pj+1aj+1 − P̄i

i+1∑

j=1

pjaj



62 S. S. Dragomir, M. V. Boldea, M. Megan CUBO
19, 1 (2017)

= Pnpi+1ai+1 + Pi

n−1∑

j=i+1

pj+1aj+1 − P̄i

i+1∑

j=1

pjaj

=
(
Pi + P̄i

)
pi+1ai+1 + Pi

n−1∑

j=i+1

pj+1aj+1 − P̄i

i+1∑

j=1

pjaj

= Pi

n−1∑

j=i+1

pjaj − P̄i

i∑

j=1

pjaj = PiĀi (p) − P̄iAi (p)

= det

(
Pi Pn

Ai (p) An (p)

)

;

and the identity is proved.

We are able now to state and prove the second identity for the Čebyšev functional

Theorem 3.2. With the assumptions of Theorem 3.1, we have the equality

Tn (p; a, b) =
n−1∑

i=1

n−1∑

j=1

Pmin{i,j}P̄max{i,j}∆aj∆bi. (3.9)

The proof is obvious by Theorem 3.1 and Lemma 3.1.

4 Some New Inequalities

The following result holds

Theorem 4.1. Let p = (p1, ..., pn) ∈ K
n, and a = (a1, ..., an) , b = (b1, ..., bn) ∈ B

n. Then we

have the inequalities

∥Tn (p; a, b)∥ (4.1)

≤

⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

max1≤i≤n−1

∥
∥
∥
∥
∥
det

(
Pi Pn

Ai (p) An (p)

)∥
∥
∥
∥
∥

∑
n−1

i=1 ∥∆bi∥ ;

(
∑

n−1

i=1

∥
∥
∥
∥
∥
det

(
Pi Pn

Ai (p) An (p)

)∥
∥
∥
∥
∥

q)1/q (∑
n−1

i=1 ∥∆bi∥
p
)1/p

for p > 1, 1
p
+ 1

q
= 1;

∑n−1
i=1

∥
∥
∥
∥
∥
det

(
Pi Pn

Ai (p) An (p)

)∥
∥
∥
∥
∥
max1≤i≤n−1 ∥∆bi∥ .



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Inequalities for Čeby šev Functional in Banach Algebras 63

All the inequalities in (4.1) are sharp in the sense that the constants 1 cannot be replaced by smaller

constants.

Proof. Using the first identity in (3.3), we have

∥Tn (p; a, b)∥ ≤
n−1∑

i=1

∥
∥
∥
∥
∥
det

(
Pi Pn

Ai (p) An (p)

)∥
∥
∥
∥
∥
∥∆bi∥ .

Using Hölder’s inequality, we deduce the desired result (4.1) .

Let prove, for instance, that the constant 1 in the second inequality is best possible.

Assume, for C > 0, we have that

∥Tn (p; a, b)∥ ≤ C

(
n−1∑

i=1

∥
∥
∥
∥
∥
det

(
Pi Pn

Ai (p) An (p)

)∥
∥
∥
∥
∥

q)1/q (
n−1∑

i=1

∥∆bi∥
p

)1/p

(4.2)

for p > 1, 1
p
+ 1

q
= 1, n ≥ 2.

If we choose n = 2, then we get

T2 (p; a, b) = p1p2 (a2 − a1) (b2 − b1) .

Also, for n = 2,

(
n−1∑

i=1

∥
∥
∥
∥
∥
det

(
Pi Pn

Ai (p) An (p)

)∥
∥
∥
∥
∥

q)1/q

= |p1p2| ∥a2 − a1∥

and

⎛

⎝
n−1∑

j=1

∥∆bj∥
p

⎞

⎠

1/p

= ∥b2 − b1∥ .

Then by (4.2), holding for n = 2, p1, p2 ̸= 0, a1 ̸= a2, b2 ̸= b1, we deduce C ≥ 1, proving that 1 is

the best possible constant in that inequality.

The following corollary for the uniform distribution of the probability p holds.



64 S. S. Dragomir, M. V. Boldea, M. Megan CUBO
19, 1 (2017)

Corolary 1. With the assumptions of Theorem 4.1 for a and b, we have the inequalities

∥Tn (a, b)∥ (4.3)

≤
1

n2
×

⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

max1≤i≤n−1

∥
∥
∥
∥
∥
det

(
i n

∑
i

k=1 ak
∑

n

k=1 ak

)∥
∥
∥
∥
∥

∑
n−1

i=1 ∥∆bi∥ ;

(
∑

n−1

i=1

∥
∥
∥
∥
∥
det

(
i n

∑i
k=1 ak

∑n
k=1 ak

)∥
∥
∥
∥
∥

q)1/q

×
(∑

n−1

i=1 ∥∆bi∥
p
)1/p

for p > 1, 1
p
+ 1

q
= 1;

∑n−1
i=1

∥
∥
∥
∥
∥
det

(
i n

∑i
k=1 ak

∑n
k=1 ak

)∥
∥
∥
∥
∥
max1≤i≤n−1 ∥∆bi∥ .

The following result may be stated as well.

Theorem 4.2. With the assumptions of Theorem 4.1 and if Pi ̸= 0 (i = 1, ..., n) , then we have

the inequalities

∥Tn (p; a, b)∥ (4.4)

≤ |Pn| ×

⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

max1≤i≤n−1

∥
∥
∥
An(p)
Pn

−
Ai(p)
Pi

∥
∥
∥
∑

n−1

i=1 |Pi| ∥∆bi∥ ;

(∑
n−1

i=1 |Pi|
∥
∥
∥
An(p)
Pn

−
Ai(p)
Pi

∥
∥
∥
q)1/q (∑

n−1

i=1 |Pi| ∥∆bi∥
p
)1/p

for p > 1, 1
p
+ 1

q
= 1;

∑
n−1

i=1 |Pi|
∥
∥
∥
An(p)
Pn

−
Ai(p)
Pi

∥
∥
∥max1≤i≤n−1 ∥∆bi∥ .

All the inequalities in (4.4) are sharp in the sense that the constant 1 cannot be replaced by a

smaller constant.

Proof. Follows by the second identity in (3.3) and taking into account that

∥Tn (p; a, b)∥ ≤ |Pn|
n−1∑

i=1

∥
∥
∥
∥
An (p)

Pn
−

Ai (p)

Pi

∥
∥
∥
∥ |Pi| ∥∆bi∥ .

Using Hölder’s weighted inequality, we easily deduce (4.4) .

The sharpness of the constant may be shown in a similar manner. We omit the details.



CUBO
19, 1 (2017)

Inequalities for Čeby šev Functional in Banach Algebras 65

The following corollary containing the unweighted inequalities holds.

Corolary 2. With the above assumptions for a and b one, has

∥Tn (a, b)∥ (4.5)

≤
1

n
×

⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

max1≤i≤n−1

∥
∥
∥ 1
n

∑
n

k=1 ak −
1

i

∑
i

k=1 ak

∥
∥
∥
∑

n−1

i=1 i ∥∆bi∥ ;

(∑
n−1

i=1 i
∥
∥
∥ 1
n

∑
n

k=1 ak −
1

i

∑
i

k=1 ak

∥
∥
∥
q)1/q (∑

n−1

i=1 i ∥∆bi∥
p
)1/p

for p > 1, 1
p
+ 1

q
= 1;

∑n−1
i=1 i

∥
∥
∥ 1
n

∑n
k=1 ak −

1

i

∑i
k=1 ak

∥
∥
∥max1≤i≤n−1 ∥∆bi∥ .

The inequalities in (4.5) are sharp in the sense mentioned above.

Another type of inequalities may be stated if one uses the third identity in (3.3).

Theorem 4.3. With the assumptions in Theorem 4.1 and if Pi, Pi ̸= 0, i ∈ {1, ..., n − 1} , then we

have the inequalities

∥Tn (p; a, b)∥ (4.6)

≤

⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

max1≤i≤n−1

∥
∥
∥
Ai(p)

Pi
−

Ai(p)
Pi

∥
∥
∥
∑

n−1

i=1 |Pi|
∣
∣Pi
∣
∣∥∆bi∥ ;

(∑n−1
i=1 |Pi|

∣
∣Pi
∣
∣
∥
∥
∥
Ai(p)

Pi
−

Ai(p)
Pi

∥
∥
∥
q)1/q (∑n−1

i=1 |Pi|
∣
∣Pi
∣
∣∥∆bi∥

p
)1/p

for p > 1, 1
p
+ 1

q
= 1;

∑
n−1

i=1 |Pi|
∣
∣Pi
∣
∣
∥
∥
∥
Ai(p)

Pi
−

Ai(p)
Pi

∥
∥
∥max1≤i≤n−1 ∥∆bi∥ .



66 S. S. Dragomir, M. V. Boldea, M. Megan CUBO
19, 1 (2017)

In particular, if pi =
1

n
, i ∈ {1, ..., n} , then we have

∥Tn (a, b)∥ (4.7)

≤
1

n2

⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

max1≤i≤n−1

∥
∥
∥ 1
n−i

∑
n

k=i+1 ak −
1

i

∑
i

k=1 ak

∥
∥
∥

×
∑n−1

i=1 i (n − i) ∥∆bi∥ ;

(∑n−1
i=1 i (n − i)

∥
∥
∥ 1
n−i

∑n
k=i+1 ak −

1

i

∑i
k=1 ak

∥
∥
∥
q)1/q

×
(∑n−1

i=1 i (n − i) ∥∆bi∥
p
)1/p

for p > 1, 1
p
+ 1

q
= 1;

∑n−1
i=1 i (n − i)

∥
∥
∥ 1
n−i

∑n
k=i+1 ak −

1

i

∑i
k=1 ak

∥
∥
∥

× max1≤i≤n−1 ∥∆bi∥ .

The inequalities in (4.6)and (4.7) are sharp in the above mentioned sense.

A different approach may be considered if one uses the representation in terms of double sums

for the Chebyshev functional provided by the Theorem 3.2.

The following result holds.

Theorem 4.4. With the assumptions in Theorem 4.1, we have the inequalities

∥Tn (p; a, b)∥ (4.8)

≤

⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

max1≤i,j≤n−1
{∣
∣Pmin{i,j}

∣
∣
∣
∣P̄max{i,j}

∣
∣
} ∑n−1

i=1 ∥∆ai∥
∑n−1

i=1 ∥∆bi∥ ;

(∑n−1
i=1

∑n−1
j=1

∣
∣Pmin{i,j}

∣
∣q
∣
∣P̄max{i,j}

∣
∣q
)1/q

×
(∑n−1

i=1 ∥∆ai∥
p
)1/p (∑n−1

i=1 ∥∆bi∥
p
)1/p

for p > 1, 1
p
+ 1

q
= 1;

∑n−1
i=1

∑n−1
j=1

∣
∣Pmin{i,j}

∣
∣
∣
∣P̄max{i,j}

∣
∣

× max1≤i≤n−1 ∥∆ai∥ max1≤i≤n−1 ∥∆bi∥ .

The inequalities are sharp in the sense mentioned above.

The proof follows by the identity (3.9) on using Hölder’s inequality for double sums and we

omit the details.

The above theorem has some important consequences as follows:



CUBO
19, 1 (2017)

Inequalities for Čeby šev Functional in Banach Algebras 67

Corolary 3. Let p = (p1, ..., pn) ∈ K
n, and a = (a1, ..., an) , b = (b1, ..., bn) ∈ B

n. Then we

have the inequality

∥Tn (p; a, b)∥ ≤
1

4

(
n∑

k=1

|pk|

)2 n−1∑

i=1

∥∆ai∥
n−1∑

i=1

∥∆bi∥ . (4.9)

The constant 1
4
is best possible in (4.9).

Proof. We observe that

∣
∣Pmin{i,j}

∣
∣ =

∣
∣
∣
∣
∣
∣

min{i,j}∑

k=1

pk

∣
∣
∣
∣
∣
∣
≤

min{i,j}∑

k=1

|pk|

and

∣
∣P̄max{i,j}

∣
∣ =

∣
∣
∣
∣
∣
∣

n∑

k=max{i,j}

pk

∣
∣
∣
∣
∣
∣
≤

n∑

k=max{i,j}

|pk|

for any 1 ≤ i, j ≤ n − 1.

This implies that

∣
∣Pmin{i,j}

∣
∣
∣
∣P̄max{i,j}

∣
∣ ≤

min{i,j}∑

k=1

|pk|
n∑

k=max{i,j}

|pk|

≤
1

4

⎛

⎝
min{i,j}∑

k=1

|pk| +
n∑

k=max{i,j}

|pk|

⎞

⎠

2

≤
1

4

(
n∑

k=1

|pk|

)2

for any 1 ≤ i, j ≤ n − 1.

Therefore

max
1≤i,j≤n−1

{∣
∣Pmin{i,j}

∣
∣
∣
∣P̄max{i,j}

∣
∣
}
≤

1

4

(
n∑

k=1

|pk|

)2

and by the first inequality in (4.8) we get (4.9).

To prove the sharpness of the constant
1

4
, assume that (4.9) holds with a constant C > 0, i.e.

∥Tn (p; a, b)∥ ≤ C

(
n∑

k=1

|pk|

)2
n−1∑

i=1

∥∆ai∥
n−1∑

i=1

∥∆bi∥ . (4.10)

If we take in (4.10) n = 2, then we have

∥p1p2 (a2 − a1) (b2 − b1)∥ ≤ C (|p1| + |p2|)
2
∥a2 − a1∥ ∥b2 − b1∥ .

If we take p1 = p2 =
1

2
, a2 − a1 = α · 1, b2 − b1 = β · 1 with α,β ̸= 0 then we get C ≥

1

4
, and the

proof is completed.



68 S. S. Dragomir, M. V. Boldea, M. Megan CUBO
19, 1 (2017)

We have:

Corolary 4. Let p = (p1, ..., pn) ∈ K
n, and a = (a1, ..., an) , b = (b1, ..., bn) ∈ B

n. Then we

have the inequality

∥Tn (p; a, b)∥ ≤
1

4p
(n + 1)

2p

⎡

⎣
n∑

i=1

|pi|
q

n∑

i=1

i2 |pi|
q
−

(
n∑

i=1

i |pi|
q

)2
⎤

⎦

1/q

(4.11)

×

(
n−1∑

i=1

∥∆ai∥
p

)1/p (
n−1∑

i=1

∥∆bi∥
p

)1/p

,

for p > 1, 1
p
+ 1

q
= 1.

Proof. We observe that by Hölder’s inequality we have

∣
∣Pmin{i,j}

∣
∣q =

∣
∣
∣
∣
∣
∣

min{i,j}∑

k=1

pk

∣
∣
∣
∣
∣
∣

q

≤ (min {i, j})
q−1

min{i,j}∑

k=1

|pk|
q

and

∣
∣P̄max{i,j}

∣
∣q =

∣
∣
∣
∣
∣
∣

n∑

k=max{i,j}

pk

∣
∣
∣
∣
∣
∣

q

≤ (n − max {i, j} + 1)
q−1

n∑

k=max{i,j}

|pk|
q

for any 1 ≤ i, j ≤ n − 1.

Then

∣
∣Pmin{i,j}

∣
∣q
∣
∣P̄max{i,j}

∣
∣q (4.12)

≤ (n − max {i, j} + 1)
q−1

(min {i, j})
q−1

min{i,j}∑

k=1

|pk|
q

n∑

k=max{i,j}

|pk|
q

= [(n − max {i, j} + 1) (min {i, j})]
q−1

min{i,j}∑

k=1

|pk|
q

n∑

k=max{i,j}

|pk|
q
.

Observe that

(n − max {i, j} + 1) (min {i, j}) ≤
1

4
(n − max {i, j} + 1 + min {i, j})

2
.

Since for any 1 ≤ i, j ≤ n − 1 we have

max {i, j} − min {i, j} = |i − j|

then

(n − max {i, j} + 1) (min {i, j}) ≤
1

4
(n + 1 − |i − j|)

2
≤

1

4
(n + 1)

2



CUBO
19, 1 (2017)

Inequalities for Čeby šev Functional in Banach Algebras 69

and by (4.12) we get

∣
∣Pmin{i,j}

∣
∣q
∣
∣P̄max{i,j}

∣
∣q ≤

[
1

4
(n + 1)

2

]q−1 min{i,j}∑

k=1

|pk|
q

n∑

k=max{i,j}

|pk|
q

or any 1 ≤ i, j ≤ n − 1.

Making use of the second inequality in (4.8) we have

∥Tn (p; a, b)∥ (4.13)

≤

⎛

⎝
n−1∑

i=1

n−1∑

j=1

∣
∣Pmin{i,j}

∣
∣q
∣
∣P̄max{i,j}

∣
∣q

⎞

⎠

1/q

×

(
n−1∑

i=1

∥∆ai∥
p

)1/p (
n−1∑

i=1

∥∆bi∥
p

)1/p

≤

⎛

⎝
[
1

4
(n + 1)

2

]q−1 n−1∑

i=1

n−1∑

j=1

⎛

⎝
min{i,j}∑

k=1

|pk|
q

n∑

k=max{i,j}

|pk|
q

⎞

⎠

⎞

⎠

1/q

×

(
n−1∑

i=1

∥∆ai∥
p

)1/p (
n−1∑

i=1

∥∆bi∥
p

)1/p

=

[
1

4
(n + 1)

2

]q−1
q

⎛

⎝
n−1∑

i=1

n−1∑

j=1

⎛

⎝
min{i,j}∑

k=1

|pk|
q

n∑

k=max{i,j}

|pk|
q

⎞

⎠

⎞

⎠

1/q

×

(
n−1∑

i=1

∥∆ai∥
p

)1/p (
n−1∑

i=1

∥∆bi∥
p

)1/p

=
1

4p
(n + 1)

2p

⎛

⎝
n−1∑

i=1

n−1∑

j=1

⎛

⎝
min{i,j}∑

k=1

|pk|
q

n∑

k=max{i,j}

|pk|
q

⎞

⎠

⎞

⎠

1/q

×

(
n−1∑

i=1

∥∆ai∥
p

)1/p (
n−1∑

i=1

∥∆bi∥
p

)1/p

.

If we use the identity for real numbers

Tn (q; a, b) =
n−1∑

i=1

n−1∑

j=1

Qmin{i,j}Q̄max{i,j}∆aj∆bi.

and the choices aj = bj = j and qk = |pk|
q
, then we get

n−1∑

i=1

n−1∑

j=1

⎛

⎝
min{i,j}∑

k=1

|pk|
q

n∑

k=max{i,j}

|pk|
q

⎞

⎠ =
n∑

i=1

|pi|
q

n∑

i=1

i2 |pi|
q
−

(
n∑

i=1

i |pi|
q

)2

.



70 S. S. Dragomir, M. V. Boldea, M. Megan CUBO
19, 1 (2017)

Replacing this in (4.13) produces the desired result (4.11).

The following corollary also holds. It was obtained earlier in the paper with a different proof.

Corolary 5. Let p = (p1, ..., pn) ∈ K
n, and a = (a1, ..., an) , b = (b1, ..., bn) ∈ B

n. Then we

have the inequality

∥Tn (p; a, b)∥ ≤

⎡

⎣
n∑

i=1

|pi|
n∑

i=1

i2 |pi| −

(
n∑

i=1

i |pi|

)2
⎤

⎦ (4.14)

× max
1≤i≤n−1

∥∆ai∥ max
1≤i≤n−1

∥∆bi∥ .

The inequality is sharp.

Proof. From the third inequality in (4.8) we have

∥Tn (p; a, b)∥

≤ max
1≤i≤n−1

∥∆ai∥ max
1≤i≤n−1

∥∆bi∥
n−1∑

i=1

n−1∑

j=1

∣
∣Pmin{i,j}

∣
∣
∣
∣P̄max{i,j}

∣
∣

= max
1≤i≤n−1

∥∆ai∥ max
1≤i≤n−1

∥∆bi∥
n−1∑

i=1

n−1∑

j=1

∣
∣
∣
∣
∣
∣

min{i,j}∑

k=1

pk

∣
∣
∣
∣
∣
∣

∣
∣
∣
∣
∣
∣

n∑

k=max{i,j}

pk

∣
∣
∣
∣
∣
∣

≤ max
1≤i≤n−1

∥∆ai∥ max
1≤i≤n−1

∥∆bi∥
n−1∑

i=1

n−1∑

j=1

⎛

⎝
min{i,j}∑

k=1

|pk|
n∑

k=max{i,j}

|pk|

⎞

⎠

=

⎡

⎣
n∑

i=1

|pi|
n∑

i=1

i2 |pi| −

(
n∑

i=1

i |pi|

)2
⎤

⎦ max
1≤i≤n−1

∥∆ai∥ max
1≤i≤n−1

∥∆bi∥

and the desired inequality (4.14) is proved.

The sharpness of the inequality follows as above and the details are omitted.

5 Inequalities for Power Series

We have:

Theorem 5.1. Let f(λ) =
∑

∞

n=0 αnλ
n be a power series that is convergent on the open disk

D(0, R), with R > 0. If x, y ∈ B with xy = yx and ∥x∥ , ∥y∥ < 1, then we have for λ ∈ C with

|λ| < R the inequality:

∥
∥
∥f̃ (λ · 1) f̃ (λxy) − f̃ (λx) f̃ (λy)

∥
∥
∥ ≤

1

4
·

∥x − 1∥ ∥y − 1∥

(1 − ∥x∥) (1 − ∥y∥)
f2A (|λ|) . (5.1)



CUBO
19, 1 (2017)

Inequalities for Čeby šev Functional in Banach Algebras 71

Proof. Utilising the inequality (4.9) we have for all n ≥ 1

∥
∥
∥
∥
∥

n∑

i=0

αiλ
i

n∑

i=0

αiλ
ixiyi −

n∑

i=0

αiλ
ixi

n∑

i=0

αiλ
iyi

∥
∥
∥
∥
∥

(5.2)

≤
1

4

(
n∑

k=0

|αk| |λ|
k

)2
n−1∑

i=0

∥
∥xi+1 − xi

∥
∥

n−1∑

i=0

∥
∥yi+1 − yi

∥
∥ .

Observe that

n−1∑

i=0

∥
∥xi+1 − xi

∥
∥ =

n−1∑

i=0

∥
∥xi (x − 1)

∥
∥ ≤ ∥x − 1∥

n−1∑

i=0

∥
∥xi
∥
∥

≤ ∥x − 1∥
n−1∑

i=0

∥x∥
i
= ∥x − 1∥

1 − ∥x∥
n

1 − ∥x∥

and, similarly,

n−1∑

i=0

∥
∥yi+1 − yi

∥
∥ ≤ ∥y − 1∥

1 − ∥y∥
n

1 − ∥y∥
.

Utilizing (5.2) and the fact that xy = yx, we have

∥
∥
∥
∥
∥

n∑

i=0

αiλ
i

n∑

i=0

αiλ
i (xy)

i
−

n∑

i=0

αiλ
ixi

n∑

i=0

αiλ
iyi

∥
∥
∥
∥
∥

(5.3)

≤
1

4
∥x − 1∥ ∥y − 1∥

(
n∑

k=0

|αk| |λ|
k

)2
1 − ∥x∥

n

1 − ∥x∥

1 − ∥y∥
n

1 − ∥y∥

for any n ≥ 1.

Since all the series whose partial sums are involved in (5.3) are convergent, then by letting

n → ∞ in (5.3) we deduce the desired inequality (5.1).

Corolary 6. Let f(λ) =
∑

∞

n=0 αnλ
n be a power series that is convergent on the open disk D(0, R),

with R > 0. If x ∈ B and ∥x∥ < 1, then we have for λ ∈ C with |λ| < R the inequality:

∥
∥
∥
∥f̃ (λ · 1) f̃

(
λx2
)
−
[
f̃ (λx)

]2
∥
∥
∥
∥ ≤

1

4
·

∥x − 1∥
2

(1 − ∥x∥)
2
f2
A
(|λ|) . (5.4)



72 S. S. Dragomir, M. V. Boldea, M. Megan CUBO
19, 1 (2017)

As some natural examples that are useful for applications, we can point out that, if

f (λ) =
∞∑

n=1

(−1)
n

n
λn = ln

1

1 + λ
, λ ∈ D (0, 1) ; (5.5)

g (λ) =
∞∑

n=0

(−1)
n

(2n) !
λ2n = cosλ, λ ∈ C;

h (λ) =
∞∑

n=0

(−1)
n

(2n + 1) !
λ2n+1 = sinλ, λ ∈ C;

l (λ) =
∞∑

n=0

(−1)
n
λn =

1

1 + λ
, λ ∈ D (0, 1) ;

then the corresponding functions constructed by the use of the absolute values of the coefficients

are

fA (λ) =
∞∑

n=1

1

n
λn = ln

1

1 − λ
, λ ∈ D (0, 1) ; (5.6)

gA (λ) =
∞∑

n=0

1

(2n) !
λ2n = coshλ, λ ∈ C;

hA (λ) =
∞∑

n=0

1

(2n + 1) !
λ2n+1 = sinhλ, λ ∈ C;

lA (λ) =
∞∑

n=0

λn =
1

1 − λ
, λ ∈ D (0, 1) .

Other important examples of functions as power series representations with nonnegative coefficients

are:

exp (λ) =
∞∑

n=0

1

n!
λn λ ∈ C, (5.7)

1

2
ln

(
1 + λ

1 − λ

)

=

∞∑

n=1

1

2n − 1
λ2n−1, λ ∈ D (0, 1) ;

sin
−1 (λ) =

∞∑

n=0

Γ
(
n + 1

2

)

√
π(2n + 1) n!

λ2n+1, λ ∈ D (0, 1) ;

tanh
−1

(λ) =
∞∑

n=1

1

2n − 1
λ2n−1, λ ∈ D (0, 1)

2F1 (α,β,γ,λ) =
∞∑

n=0

Γ (n + α)Γ (n + β)Γ (γ)

n!Γ (α)Γ (β)Γ (n + γ)
λn,α,β,γ > 0,

λ ∈ D (0, 1) ;

where Γ is Gamma function.



CUBO
19, 1 (2017)

Inequalities for Čeby šev Functional in Banach Algebras 73

Example 1. a) If x, y ∈ B with xy = yx and ∥x∥ , ∥y∥ < 1, then we have for λ ∈ C the inequality:

∥exp [λ(1 + xy)] − exp [λ(x + y)]∥ ≤
1

4
·

∥x − 1∥ ∥y − 1∥

(1 − ∥x∥) (1 − ∥y∥)
exp (2 |λ|) . (5.8)

In particular, we have

∥
∥exp

[
λ
(
1 + x2

)]
− exp [2λx]

∥
∥ ≤

1

4
·

∥x − 1∥
2

(1 − ∥x∥)
2
exp (2 |λ|) (5.9)

and
∥
∥exp

[
λ
(
1 − x2

)]
− 1
∥
∥ ≤

1

4
·
∥x − 1∥ ∥x + 1∥

(1 − ∥x∥)
2

exp (2 |λ|) (5.10)

for any x ∈ B with ∥x∥ < 1 and λ ∈ C.

b) We have the inequality

∥
∥
∥(1 − λ)

−1
(1 − λxy)

−1
− (1 − λx)

−1
(1 − λy)

−1
∥
∥
∥ (5.11)

≤
1

4
·

∥x − 1∥ ∥y − 1∥

(1 − ∥x∥) (1 − ∥y∥) (1 − |λ|)
2

for any x, y ∈ B with xy = yx, ∥x∥ , ∥y∥ < 1 and λ ∈ C with |λ| < 1.

In particular, we have

∥
∥
∥(1 − λ)

−1
(
1 − λx2

)
−1

− (1 − λx)
−2
∥
∥
∥ ≤

1

4
·

∥x − 1∥
2

(1 − ∥x∥)
2
(1 − |λ|)

2
(5.12)

and
∥
∥
∥(1 − λ)

−1
(
1 + λx2

)
−1

− (1 − λx)
−1

(1 + λx)
−1
∥
∥
∥ (5.13)

≤
1

4
·

∥x − 1∥ ∥x + 1∥

(1 − ∥x∥)
2
(1 − |λ|)

2

for any x ∈ B with ∥x∥ < 1 and λ ∈ C with |λ| < 1.

c) We have the inequality

∥
∥
∥ln (1 − λ)

−1
ln (1 − λxy)

−1
− ln (1 − λx)

−1
ln (1 − λy)

−1
∥
∥
∥ (5.14)

≤
1

4
·

∥x − 1∥ ∥y − 1∥

(1 − ∥x∥) (1 − ∥y∥)
[ln (1 − |λ|)]

2

for any x, y ∈ B with xy = yx, ∥x∥ , ∥y∥ < 1 and λ ∈ C with |λ| < 1.

In particular, we have

∥
∥
∥(1 − λ)

−1
(
1 − λx2

)
−1

− (1 − λx)
−2
∥
∥
∥ ≤

1

4
·

∥x − 1∥
2

(1 − ∥x∥)
2
[ln (1 − |λ|)]

2
(5.15)



74 S. S. Dragomir, M. V. Boldea, M. Megan CUBO
19, 1 (2017)

and
∥
∥
∥(1 − λ)

−1
(
1 + λx2

)
−1

− (1 − λx)
−1

(1 + λx)
−1
∥
∥
∥ (5.16)

≤
1

4
·
∥x − 1∥ ∥x + 1∥

(1 − ∥x∥)
2

[ln (1 − |λ|)]
2

for any x ∈ B with ∥x∥ < 1 and λ ∈ C with |λ| < 1.

Acknowledgement. The authors would like to thank the anonymous referee for valuable

suggestions that have been implemented in the final version of the paper.

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