Some Bounds for the Complex Čebyšev Functional of Functions of Bounded Variation

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of Functions of Bounded Variation

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Dragomir, Sever S (2021) Some Bounds for the Complex Čebyšev Functional of Functions of Bounded Variation. Symmetry, 13 (6). ISSN 2073-8994

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Article

Some Bounds for the Complex ˇ Cebyšev Functional of Functions of Bounded Variation

Silvestru Sever Dragomir^1,2

Citation: Dragomir, S.S. Some Bounds for the Complex ˇCebyšev Functional of Functions of Bounded Variation.Symmetry2021,13, 990.

https://doi.org/10.3390/sym13060990

Academic Editors: Alina Alb Lupas and Roman Ger

Received: 27 April 2021 Accepted: 24 May 2021 Published: 2 June 2021

Publisher’s Note:MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations.

Copyright: c 2021 by the authors.

Licensee MDPI, Basel, Switzerland.

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://

creativecommons.org/licenses/by/

4.0/).

1 Mathematics, College of Engineering & Science Victoria University, P.O. Box 14428, Melbourne City 8001, Australia; [email protected]

2 DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences, School of Computer Science &

Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa

Abstract: In this paper, we provide several bounds for the modulus of thecomplex ˇCebyšev functional. Applications to the trapezoid and mid-point inequalities, that are symmetric inequalities, are also provided.

Keywords: measurable functions; integrable functions; functions of bounded variation; Cauchy- Bunyakovsky-Schwarz inequality; Grüss’ type inequalities; ˇCebyšev functional

1. Introduction

For Lebesgue integrable functions f, g : [a,b] → C we consider the complex Cebyšev functionalˇ

C(f,g):= ¹ b−a

Z b

a f(t)g(t)dt− ¹ b−a

Z b

a f(t)dt 1 b−a

Z b a g(t)dt.

For two integrable real-valued functions f,g : [a,b] → R, in order to compare the integral mean of the product with the product of the integral means, in 1934, G. Grüss [1]

showed that

|C(f,g)| ≤ ¹

4(M−m)(N−n), (1)

providedm,M,n,Nare real numbers with the property that

−_∞<m≤ f ≤M<_∞, −_∞<n≤g≤N<_∞ a.e. on [a,b]. (2) The constant ¹₄ is most possible in (1) in the sense that it cannot be replaced by a smaller one. For other results, see [2–7].

To extend this symmetric inequality for complex-valued functions we need the following preparations.

Forφ,Φ∈C^and[a,b]an interval of real numbers, define the sets of complex-valued functions (see [4,8,9])

U¯_[a,b](φ,Φ):=ⁿg:[a,b]→C| Reh

(Φ−g(t))g(t)−φ

i≥0 for a.e.t∈[a,b]^o and

∆¯[a,b](φ,Φ):=

g:[a,b]→C|

g(t)− ^φ+Φ 2

≤ ¹

2|_Φ−φ|for a.e.t∈[a,b]

. For anyφ,Φ∈ C,φ 6= Φ, we have that ¯U_[a,b](φ,Φ)and ¯∆_[a,b](φ,Φ)are nonempty, convex and closed sets and

U¯_[a,b](φ,Φ) =∆^¯_[a,b](φ,Φ). (3)

Symmetry2021,13, 990. https://doi.org/10.3390/sym13060990 https://www.mdpi.com/journal/symmetry

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We observe that for anyz∈Cwe have the equivalence

z− ^φ+Φ 2

≤ ¹ 2|Φ−φ| if and only if

Re

(Φ−z) z¯−φ

≥0.

This follows from the equality 1

4|Φ−φ|²−

z−^φ+_Φ 2

2

=Re

(Φ−z) z¯−φ that holds for anyz∈C.

The equality (3) is thus a simple consequence of this fact.

For anyφ,Φ∈C^,^φ6=Φ,we also have that

U¯_[a,b](φ,Φ) ={g:[a,b]→C|(ReΦ−Reg(t))(Reg(t)−Reφ) (4) +(ImΦ−Img(t))(Img(t)−Imφ)≥0 for a.e.t∈[a,b]}.

Now, if we assume that Re(_Φ)≥Re(φ)and Im(_Φ)≥Im(φ), then we can define the following set of functions as well:

S¯_[a,b](φ,Φ):={g:[a,b]→C| Re(Φ)≥Reg(t)≥Re(φ) (5) and Im(Φ)≥Img(t)≥Im(φ)for a.e.t∈[a,b]}.

One can easily observe that ¯S_[a,b](φ,Φ)is closed, convex and

∅6=S^¯_[a,b](φ,Φ)⊆U^¯_[a,b](φ,Φ). (6)

This fact provides also numerous example of complex functions belonging to the class

∆¯_[a,b](φ,Φ).

In [4] we obtained the following complex version of Grüss’ inequality:

|C(f,g)| ≤ ¹

4|Φ−φ||Ψ−ψ| (7)

provided f ∈ _∆^¯_[a,b](φ,Φ)andg ∈ _∆^¯_[a,b](ψ,Ψ), where g denotes the complex conjugate function ofg.

We denotethe varianceof the complex-valued function f : [a,b] → C^by^D(f)and defined as

D(f) =C f, ¯f1/2

=

"

1 b−a

Z _b

a |f(t)|²dt−

1 b−a

Z _b a f(t)dt

2#1/2

, where ¯f denotes the complex conjugate function of f.

If we apply the inequality (7) forg= f, then we obtain D(f)≤ ¹

2|Φ−φ|. (8)

We observe that, ifg ∈ _∆^¯_[a,b](ψ,Ψ), then

g(t)−^ψ+₂^Ψ ≤ ¹₂|Ψ−ψ|for a.e. t ∈ [a,b] that is equivalent to

g(t)−^ψ+₂^Ψ≤ ¹₂Ψ−ψ

meaning thatg ∈ ∆^¯_[a,b] ψ,Ψ

and by (7), forginstead ofgwe also have

|C(f,g)| ≤ ¹

4|Φ−φ||Ψ−ψ| (9)

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provided f ∈∆^¯_[a,b](φ,Φ)andg∈∆^¯_[a,b](ψ,Ψ).

We can also consider the following quantity associated with a complex-valued function f :[a,b]→C,

E(f):=|C(f,f)|^1/2=

1 b−a

Z _b

a f²(t)dt− 1

b−a Z _b

a f(t)dt 2

1/2

. Using (9) we also have

E(f)≤ ¹

2|Φ−φ|. (10)

For an integrable functionf :[a,b]→C, consider themean deviationof f defined by R(f):= ¹

b−a Z _b

a

f(t)− ¹ b−a

Z _b a f(s)ds

dt.

The following result holds (see [10] or the more extensive preprint version [11]).

Theorem 1. Let f :[a,b]→Cbe of bounded variation on[a,b]and g:[a,b] →C^{a Lebesgue} integrable function on[a,b].Then

|C(f,g)| ≤ ¹ 2

b _

a

(f)R(g)≤ ¹ 2

b _

a

(f)D(g), (11)

where

b _

a

(f)denotes the total variation of f on the interval[a,b].The constant ¹₂is best possible in (11).

Corollary 1. If f,g:[a,b]→Care of bounded variation on[a,b],then

|C(f,g)| ≤ ¹ 2

b _

a

(f)R(g)≤ ¹ 2

b _

a

(f)D(g)≤ ¹ 4

b _

a

(f)

b _

a

(g). (12)

The constant¹₄is best possible in (12).

We also have

D(f)≤ ¹ 2

b _

a

(f), (13)

and the constant¹₂ is best possible in (13).

Using the above results we can state, for a function of bounded variationf :[a,b]→C, that E²(f)≤ ¹

2

b _

a

(f)R(f)≤ ¹ 2

b _

a

(f)D(f)≤ ¹ 4

"_b _

a

(f)

#2

. (14)

In the recent paper [12] we obtained the following result that extends to complex functions the inequalities obtained in [13]:

Theorem 2. Let f,g:[a,b]→Cbe measurable on[a,b].Then

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|C(f,g)| ≤











γ∈infC

kg−γk_∞R(f)if g∈L_∞[a,b]and f ∈L[a,b],

1 (b−a)^1/q inf

γ∈C

kg−γk_qRp(f), g∈ Lq[a,b], f ∈ Lp[a,b], and p, q>1with ¹_p+¹_q =1,

b−a1 inf

γ∈C

kg−γk₁R_∞(f)if g∈L[a,b]and f ∈ L_∞[a,b].

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An important corollary of this result is:

Corollary 2. Assume that g :[a,b] →Cis measurable on[a,b]and g∈∆^¯_[a,b](ψ,Ψ)for some distinct complex numbersψ,Ψ.Then

|C(f,g)| ≤ ¹

2|Ψ−ψ|R(f) (16)

if f ∈L[a,b].

In particular, we have

D²(g)≤ ¹

2|Ψ−ψ|R(g). (17)

This generalizes the following result obtained by Cheng and Sun [14] by a more complicated technique

|C(f,g)| ≤ ¹

2(M−m)R(f), (18)

providedm≤g≤Mfor a.e. x∈[a,b]. The constant¹₂is best in (18) as shown by Cerone and Dragomir in [15] where a general version for Lebesgue integral and measurable spaces was also given.

Motivated by the above results, in this paper we establish other bounds for the absolute value of the ˇCebyšev functional when the complex-valued functions are of bounded variation. Applications to the trapezoid and mid-point inequalities are also provided.

2. Main Results

We have the following inequality for the complex ˇCebyšev functional that extends naturally the real case:

Theorem 3. If f,g:[a,b]→Care Lebesgue integrable on[a,b],then

max{|C(f,g)|,|C(|f|,g)|,|C(f,|g|)|,|C(|f|,|g|)|} ≤D(f)D(g) (19) and

maxn

|C(f,g)|,|C(|f|,g)|, C

f,|g|^o≤D(f)D(g). (20) Proof. As in the real case, we haveKorkine’s identity

C(f,g):= ¹ 2(b−a)²

Z _b a

Z _b

a (f(t)− f(s))(g(t)−g(s))dtds, that can be proved directly by doing the calculations in the right hand side.

By the properties of modulus, we have

|(f(t)−f(s))(g(t)−g(s))|=|f(t)−f(s)||g(t)−g(s)|

≥







|(|f(t)| − |f(s)|)(g(t)−g(s))|,

|(f(t)− f(s))(|g(t)| − |g(s)|)|,

|(|f(t)| − |f(s)|)(|g(t)| − |g(s)|)|

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for anyt,s∈[a,b].

Using the properties of the integral versus the modulus, we also have 1

2(b−a)² Z _b

a Z _b

a |f(t)−f(s)||g(t)−g(s)|dtds (21)

≥ ¹

2(b−a)²









 Rb

a

Rb

a|(f(t)− f(s))(g(t)−g(s))|dtds, Rb

a

Rb

a|(|f(t)| − |f(s)|)(g(t)−g(s))|dtds, Rb

a

Rb

a|(f(t)− f(s))(|g(t)| − |g(s)|)|dtds, Rb

a

Rb

a|(|f(t)| − |f(s)|)(|g(t)| − |g(s)|)|dtds,

≥ ¹

2(b−a)²











Rb a

Rb

a(f(t)−f(s))(g(t)−g(s))dtds ,

Rb a

Rb

a(|f(t)| − |f(s)|)(g(t)−g(s))dtds ,

Rb a

Rb

a(f(t)−f(s))(|g(t)| − |g(s)|)dtds ,

Rb a

Rb

a(|f(t)| − |f(s)|)(|g(t)| − |g(s)|)dtds ,

=











|C(f,g)|,

|C(|f|,g)|,

|C(f,|g|)|,

|C(|f|,|g|)|.

Using the Cauchy-Bunyakovsky-Schwarz integral inequality, we have 1

2(b−a)² Z _b

a Z _b

a |f(t)− f(s)||g(t)−g(s)|dtds (22)

≤

"

1 2(b−a)²

Z _b a

Z _b

a |f(t)− f(s)|²dtds

#1/2

×

"

1 2(b−a)²

Z _b a

Z _b

a |g(t)−g(s)|²dtds

#1/2

and since

1 2(b−a)²

Z b a

Z b

a |f(t)− f(s)|²dtds

= ¹

2(b−a)² Z b

a Z b

a (f(t)−f(s))(f(t)−f(s))dtds

= ¹

2(b−a)² Z _b

a Z _b

a (f(t)−f(s))f(t)−f(s)dtds

=C f,f

=D²(f),

and a similar equality forg, hence we obtain from (21) and (22) the desired result (19).

The inequality (20) follows from (19).

For a function ofbounded variation f :[a,b] →Cwe define theCumulative Variation Function(CVF) of f byV_f :[a,b]→[_0,_∞)_whereV_f(t):=^W^t_a(f), the total variation of f on the interval[a,t]witht∈[a,b].

It is know that the CVF is monotonic nondecreasing on[a,b]and is continuous in a pointc∈[a,b]if and only if the generating function f is continuous in that point. If f is Lipschitzianwith the constantL>0, i.e.,

|f(t)− f(s)| ≤L|t−s|for anyt, s∈[a,b],

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thenVf is also Lipschitzian with the same constant.

Theorem 4. Let f,g:[a,b]→Cbe Lebesgue measurable on[a,b].

(i) If f is of bounded variation and g is Lebesgue integrable on[a,b],then max{|C(f,g)|,|C(|f|,g)|,|C(f,|g|)|,|C(|f|,|g|)|} ≤D

V_f

D(g); (23) (ii) If f and g are of bounded variation on[a,b],then

max{|C(f,g)|,|C(|f|,g)|,|C(f,|g|)|,|C(|f|,|g|)|} ≤C Vf,Vg

; (24)

(iii) If f is Lipschitzian with the constant L>0and g is of bounded variation on[a,b],then max{|C(f,g)|,|C(|f|,g)|,|C(f,|g|)|,|C(|f|,|g|)|} ≤LC `,Vg

, (25)

where`is the identity mapping of the interval[a,b],namely`(t) =t,t∈[a,b]. Proof. (i) If f is of bounded variation then fora≤s<t≤bwe have

|f(t)− f(s)| ≤

t _

s

(f) =Vf(t)−Vf(s). Ifa≤t<s≤bthen also

|f(t)− f(s)| ≤

s _

t

(f) =V_f(s)−V_f(t). Therefore

|f(t)− f(s)| ≤V_f(t)−V_f(s) for anyt,s∈[a,b].

Using the Cauchy-Bunyakovsky-Schwarz integral inequality, we have 1

2(b−a)² Z _b

a Z _b

a |f(t)−f(s)||g(t)−g(s)|dtds (26)

≤

"

1 2(b−a)²

Z _b a

Z _b

a |f(t)−f(s)|²dtds

#1/2

×

"

1 2(b−a)²

Z _b a

Z _b

#1/2

≤

"

1 2(b−a)²

Z _b a

V_f(t)−V_f(s)²dtds

#1/2

×

"

1 2(b−a)²

Z _b a

Z _b

#1/2

=D V_f

D(g).

Using the inequality (21) and (26) we obtain the desired result (23).

(ii) If f andgare of bounded variation on[a,b], then for anyt,s∈[a,b]we have

|f(t)−f(s)||g(t)−g(s)| ≤Vf(t)−Vf(s)Vg(t)−Vg(s)

=V_f(t)−V_f(s) Vg(t)−Vg(s)

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sinceVf(·)andVg(·)are monotonic nondecreasing on[a,b]. Then

1 2(b−a)²

Z _b a

Z _b

a |f(t)−f(s)||g(t)−g(s)|dtds (27)

≤ ¹

2(b−a)² Z _b

a Z _b

a

Vf(t)−Vf(s) Vg(t)−Vg(s)dtds

=C V_f,Vg

.

(iii) If f is Lipschitzian with the constant L > 0 and gis of bounded variation on [a,b], then

|f(t)− f(s)||g(t)−g(s)| ≤L|t−s|Vg(t)−Vg(s)

=L(t−s) Vg(t)−Vg(s) sinceVf(·)is monotonic nondecreasing on[a,b].

Then

1 2(b−a)²

Z _b a

Z _b

a |f(t)− f(s)||g(t)−g(s)|dtds (28)

≤ ¹

2(b−a)²^L Z _b

a Z _b

a (t−s) Vg(t)−Vg(s)dtds

=LC V_f,Vg

.

In 1970, A. M. Ostrowski [16] proved among other things the following result, which is somehow a mixture of the ˇCebyšev and Grüss results

|C(f,g)| ≤ ¹

8(b−a)(M−m)g⁰

_∞, (29) provided f is Lebesgue integrable on [a,b] and satisfying (2) while g : [a,b] → R ^is absolutely continuous andg⁰∈ L_∞[a,b]. Here the constant¹₈ is also sharp.

The following lemma for real-valued functions holds [17].

Lemma 1. Let h:[a,b]→Rbe an integrable function on[a,b]such that

−_∞<γ≤h(x)≤_Γ<_∞for a.e. x on[a,b]. (30) Then we have the inequality

1 b−a

Z _b a

Z _x

a h(t)dt− ^x−a b−a

Z _b a h(u)du

dx (31)

≤ ¹ 2

1 b−a

Z _b

a h(u)du−γ

Γ− ¹ b−a

Z _b a h(u)du

b−a Γ−γ

≤ ¹

8(_Γ−γ)(b−a).

The constants¹₂and ¹₈are sharp in the sense that they cannot be replaced by smaller constants.

When one function is complex-valued, we can state the following refinement and extension of Ostrowski’s inequality (29). This extends the corresponding result from [17] in which both functions are real-valued.

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Theorem 5. Let f : [a,b] → R be measurable and such that there exist the constants m, M∈Rwith

−∞<m≤ f(x)≤M<_∞for a.e. x on[a,b]. (32) If g : [a,b] → C is absolutely continuous on [a,b] with g⁰ ∈ L_∞[a,b] then we have the inequality

|C(f,g)| ≤ ¹ 2 g⁰

_∞ 1

b−a

Rb

a f(x)dx−m

M−_b−a¹ Rb

a f(x)dx

M−m (b−a) (33)

≤ ¹

8(b−a)(M−m)g⁰ _∞. The constants¹₂and ¹₈are sharp in the above sense.

Proof. Integrating by parts, we have 1

b−a Z _b

a

Z _x

a f(t)dt− ^x−a b−a

Z _b a f(u)du

g⁰(x)dx

= ¹

b−a

"

Z _x

Z _b a f(u)du

g(x)

b

a

− Z _b

a g(x)

f(x)− ¹ b−a

Z _b a f(u)du

dx

=− ¹

b−a Z _b

a g(x)f(x)dx+ ¹ b−a

Z _b

a g(x)dx· ¹ b−a

Z _b a f(x)dx

=−C(f,g).

Taking the modulus, we have

|C(f,g)| ≤ ¹ b−a

Z _b a

Z _x

Z _b a f(u)du

g⁰(x)dx

≤g⁰ _∞

1 b−a

Z _b a

Z _x

a f(t)dt−^x−a b−a

Z _b a f(u)du

dx (by Lemma1)

≤ ¹ 2 g⁰

_∞ 1

b−a

Rb

a f(x)dx−m

M−_b−a¹ Rb

a f(x)dx

M−m (b−a).

The sharpness of the constants follows from the real-valued case outlined in [17].

3. Some Examples

Assume that the function f : [a,b] →Cis of bounded variation on[a,b]. Since the functionV_f is monotonic nondecreasing on[a,b], then 0≤V_f(t)≤^W^b_a(f)for anyt∈[a,b], and by (8) we haveD

Vf

≤ ¹₂^W^b_a(f). Using (23) we have

max{|C(f,g)|,|C(|f|,g)|,|C(f,|g|)|,|C(|f|,|g|)|} ≤ ¹ 2

b _

a

(f)D(g) (34) for anygLebesgue integrable function on[a,b].

Ifg∈_∆^¯_[a,b](ψ,Ψ)for some distinct complex numbersψ,Ψ, then by (34) we have max{|C(f,g)|,|C(|f|,g)|,|C(f,|g|)|,|C(|f|,|g|)|} ≤ ¹

4|_Ψ−ψ|

b _

a

(f). (35)

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If f andgare of bounded variation on[a,b], thenVf andVgare monotonic nondecreasing on[a,b]and by (12) we have

C

V_f,Vg

≤ ¹

2

b _

a

(f)R Vg

≤ ¹ 2

b _

a

(f)D Vg

≤ ¹ 4

b _

a

(f)

b _

a

(g). (36) Using the inequality (24) we then have

max{|C(f,g)|,|C(|f|,g)|,|C(f,|g|)|,|C(|f|,|g|)|} (37)

≤ ¹ 2

b _

a

(f)R Vg

≤ ¹ 2

b _

a

(f)D Vg

≤ ¹ 4

b _

a

(f)

b _

a

(g). If f andgare of bounded variation on[a,b], then by (15) we have

C

Vf,Vg

≤











γ∈infC

Vg−γ _∞R

V_f ,

1 (b−a)^1/q inf

γ∈C

Vg−γ _qRp

V_f , wherep, q>1 with¹_p+¹_q =1,

b−a1 inf

γ∈C

Vg−γ ₁R_∞

V_f .

(38)

Using the inequality (24) we then have

≤











γ∈infC

Vg−γ _∞R

V_f ,

1 (b−a)^1/q inf

γ∈C

Vg−γ _qRp

Vf

, wherep, q>1 with¹_p+¹_q =1,

1 b−a inf

γ∈C

Vg−γ ₁R_∞

Vf

. From (39) we obtain in particular

≤

Vg−¹ 2

b _

a

(g) _∞

R V_f

≤ ¹ 2

b _

a

(g)R V_f

.

Now, we observe that forf =`, where`is the identity mapping of the interval[a,b], namely`(t) =t,t∈[a,b], we have

R(`) = ¹ b−a

Z _b a

t− ^a+_b 2

dt= ¹

4(b−a). Then we have by (18) that

C `_,Vg ≤ ¹

8(b−a)

b _

a

(g). (41)

Therefore, iff is Lipschitzian with the constantL>0 andgis of bounded variation on[a,b], then by (25) and (41) we have

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max{|C(f,g)|,|C(|f|,g)|,|C(f,|g|)|,|C(|f|,|g|)|} ≤ ¹

8(b−a)L

b _

a

(g). (42)

From (33) forf =V_handg=`we havem=0 andM=

b _

a

(h)and

|C(`_,V_h)| ≤ ¹ 2







1− ¹

(b−a)

b _

a

(h) Z _b

a V_h(t)dt





 Z _b

a V_h(t)dt (43)

≤ ¹ 8(b−a)

b _

a

(h),

provided thath:[a,b]→Cis of bounded variation on[a,b].

Therefore, iff is Lipschitzian with the constantL>0 andgis of bounded variation on[a,b], then by (25) and (43) we have

≤ ¹ 2L







1− ¹

(b−a)

b _

a

(g) Z _b

a Vg(t)dt





 Z _b

a Vg(t)dt≤ ¹

8(b−a)L

b _

a

(g).

This is an improvement of the inequality (42) above.

It is known that, if h : [a,b] → C is absolutely continuous on [a,b], then [18]

(Theorem 16), V_h is absolutely continuous and V_h(x) = Rx

a|h⁰(t)|dt for any x ∈ [a,b]. Moreover,V_h⁰(x) =|h⁰(x)|for a.e.x∈[a,b].

Assume thatg:[a,b]→Cis absolutely continuous withg⁰ ∈L_∞[a,b]and f :[a,b]→ Cis of bounded variation. From (33) forV_f,Vgwe have

C

V_f,Vg

≤ ¹

2 g⁰

_∞







1− ¹

(b−a)

b _

a

(g) Z _b

a Vg(t)dt





 Z _b

a Vg(t)dt (45)

≤ ¹ 8(b−a)

b _

a

(f)g⁰ _∞. Using (24) we then have

≤ ¹ 2 g⁰

_∞







1− ¹

(b−a)

b _

a

(g) Z _b

a Vg(t)dt





 Z _b

a Vg(t)dt

≤ ¹ 8(b−a)

b _

a

(f)g⁰ _∞,

provided thatg:[a,b]→Cis absolutely continuous withg⁰∈ L_∞[a,b]andf :[a,b]→C is of bounded variation on[a,b].

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4. Applications to Trapezoid Inequality

Leth :[a,b] →Cbe an absolutely continuous function on[a,b]. Then we have the trapezoid equality

h(a) +h(b)

2 − ¹

b−a Z _b

a h(t)dt= ¹ b−a

Z _b a

t−^a+b 2

h⁰(t)−δ

dt (47)

for anyδ∈C. This is obvious integrating by parts in the right hand side of the equality.

Using the inequality (12) for f =h⁰−δandg=`−^a+b₂ we obtain 1

b−a

Z _b a

t−^a+b 2

h⁰(t)−δ dt

≤ ¹ 2

b _

a

h⁰−δ R

`− ^a+b

2

. (48)

Since R

`−^a+b

2

= ¹

b−a Z _b

a

t−^a+b

2 − ¹

b−a Z _b

a

s−^a+b 2

ds

= ¹

b−a Z _b

a

t−^a+b 2

dt= ¹

4(b−a), then by (47) and (48) we obtain

h(a) +h(b)

2 − ¹

b−a Z _b

a h(t)dt

≤ ¹ 8(b−a)

b _

a

h⁰−δ

(49) provided thath⁰is of bounded variation andδ∈C^.

If b

_

a

h⁰−^Ψ+ψ 2

≤ ¹

2(b−a)|Ψ−ψ| (50)

for some complex numbersψ,Ψ∈C, then by (49) we obtain

h(a) +h(b)

2 − ¹

b−a Z _b

a h(t)dt

≤ ¹

16(b−a)²|Ψ−ψ|, (51) provided h satisfies the condition (50). We observe that, a sufficient condition for the condition (50) to hold is thathis twice differentiable on(a,b)andh⁰⁰∈_∆^¯_[a,b](ψ,Ψ).

Using the inequality (12) for f =`−^a+b₂ andg=h⁰−δwe obtain 1

b−a

Z b a

t− ^a+b 2

h⁰(t)−δ dt

≤ ¹ 2

b _

a

`− ^a+b

2

R h⁰−δ

(52) Since

b _

a

`−^a+b

2

=b−a and

R h⁰−δ

= ¹

b−a Z _b

a

h⁰(t)−^h(b)−h(a) b−a

dt, then by (52) and (47) we obtain

h(a) +h(b)

2 − ¹

b−a Z _b

a h(t)dt

≤ ¹ 2

Z _b a

dt (53)

providedh:[a,b]→Cis an absolutely continuous function on[a,b].

If we use the inequality (16) for f =h⁰withh⁰∈_∆^¯_[a,b](φ,Φ)for some distinct complex

(13)

numbersφ,Φandg=`−^a+b₂ , then 1

b−a

Z _b a

t−^a+b 2

h⁰(t)dt

≤ ¹

2|Φ−φ|R

`−^a+b

2

(54)

= ¹

8(b−a)|Φ−φ|. Using (47) we then obtain

h(a) +h(b)

2 − ¹

b−a Z _b

a h(t)dt

≤ ¹

8(b−a)|_Φ−φ|

providedh :[a,b] →Cis an absolutely continuous function on[a,b]and there exist the distinct complex numbersφ,Φsuch thath⁰∈∆^¯_[a,b](φ,Φ).

Forg=`−^a+b₂ we have Vg(t) =

t _

a

`− ^a+b

2

=t−a, t∈[a,b]. If we use the inequality (44) for f =h⁰andg=`−^a+b₂ we obtain

1 b−a

Z _b a

t−^a+b 2

h⁰(t)dt

(55)

≤ ¹ 2K







1− ¹

(b−a)

b _

a

(g) Z _b

a (t−a)dt





 Z _b

a (t−a)dt

namely

1 b−a

Z _b a

t−^a+b 2

h⁰(t)dt

≤ ¹

8(b−a)²K, (56)

providedh⁰is Lipschitzian with the constantK>0.

If we use (47) and (56), we obtain

h(a) +h(b)

2 − ¹

b−a Z _b

a h(t)dt

≤ ¹

8(b−a)²K providedh⁰is Lipschitzian with the constantK>0.

5. Applications to Mid-Point Inequality

Leth :[a,b] →Cbe an absolutely continuous function on[a,b]. Then we have the mid-point equality

h a+b

2

− ¹ b−a

Z _b

a h(t)dt= ¹ b−a

Z _b

a p(t) h⁰(t)−δ

dt (57)

for anyδ∈C, where the kernelp:[a,b]→Ris given by

p(t):=











t−aift∈^ha,^a+b₂ i , t−bift∈^a+b₂ ,bi .

(58)

This is obvious integrating by parts in the right hand side of the equality.

Using the inequality (12) for f =h⁰−δandg=pwe obtain

(14)

1 b−a

Z b

a p(t) h⁰(t)−δ dt

≤ ¹ 2

b _

a

h⁰−δ

R(p). (59)

Since

R(p) = ¹ b−a

Z _b

a |p(t)|dt= ¹

4(b−a), then by (57) and (59) we obtain

h

a+b 2

− ¹ b−a

Z _b a h(t)dt

≤ ¹ 8(b−a)

b _

a

h⁰−δ

(60) provided thath⁰is of bounded variation andδ∈C.

Ifh satisfies the condition (50) for some complex numbersψ,Ψ ∈ C, then by (60) we obtain

h

a+b 2

− ¹ b−a

Z _b a h(t)dt

≤ ¹

16(b−a)²|Ψ−ψ| (61) provided h satisfies the condition (50). We observe that, a sufficient condition for the condition (50) to hold is thathis twice differentiable on(a,b)andh⁰⁰∈∆^¯_[a,b](ψ,Ψ).

Using the inequality (12) for f = pandg=h⁰−δwe obtain 1

b−a

Z _b

a p(t) h⁰(t)−δ dt

≤ ¹ 2

b _

a

(p)R h⁰−δ

. (62)

Since

b _

a

(p) =b−aand

R h⁰−δ

= ¹

b−a Z _b

a

dt, then by (57) and (62) we obtain

h

a+b 2

− ¹ b−a

Z _b a h(t)dt

≤ ¹ 2

Z _b a

dt (63)

providedh:[a,b]→Cis an absolutely continuous function on[a,b].

If we use the inequality (16) for f =h⁰withh⁰∈_∆^¯_[a,b](φ,Φ)for some distinct complex numbersφ,Φandg= p, then

1 b−a

Z _b

a p(t)h⁰(t)dt

≤ ¹

2|Φ−φ|R(p) = ¹

8(b−a)|Φ−φ|. (64) By (57) and (64) we obtain

h

a+b 2

− ¹ b−a

Z b a h(t)dt

≤ ¹

8(b−a)|_Φ−φ|,

provided that h : [a,b] → C is an absolutely continuous function on [a,b] and h⁰ ∈

∆¯[a,b](φ,Φ)for some distinct complex numbersφ,Φ.

The kernelpdefined by (58) is of bounded variation and we have fort∈^ha,^a+b₂ i that Vp(t) =^W^t_a(p) =t−a and fort∈^a+b₂ ,bi

Vp(t) =

t _

a

(p) =

a+b 2

_

a

(p) +

t _

a+b 2

(p) =t−a,

(15)

therefore, for anyt∈[a,b]we haveVp(t) =t−a. We observe that, this is an example of a function of bounded variation for the CVF is differentiable for everyt∈[a,b].

Now, if we takeg= pand f =h⁰−δand use the inequality (24), then we obtain

h

a+b 2

− ¹ b−a

Z _b a h(t)dt

(65)

≤ ¹ b−a

Z b

a V_h⁰−δ(t)(t−a)dt−¹ 2

Z b

a V_h⁰−δ(t)dt.

Using the inequality (33) we also have

|C(Vh⁰−δ,`−a)| (66)

≤ ¹ 2

1 b−a

Rb

a V_h⁰_−δ(x)dx W_b

a(h⁰−δ)− _b−a¹ Rb

a V_h⁰_−δ(x)dx W_b

a(h⁰−δ) (b−a)

≤ ¹ 8(b−a)

b _

a

h⁰−δ . Then by (57) and (66) we obtain

h

a+b 2

− ¹ b−a

Z _b a h(t)dt

(67)

≤ ¹

2 1− ¹

(b−a)^W^b_a(h⁰−δ) Z _b

a V_h⁰−δ(x)dx

! Z _b

a V_h⁰−δ(x)dx

≤ ¹ 8(b−a)

b _

a

h⁰−δ

for anyδ∈C, whereh:[a,b]→Cis an absolutely continuous function on[a,b]and the derivative is of bounded variation. This is a refinement of (60).

6. Conclusions

In this paper, we provide several bounds for the modulus of thecomplex ˇCebyšev functional C(f,g):= ¹

b−a Z _b

a f(t)g(t)dt− ¹ b−a

Z _b a f(t)dt

Z _b a g(t)dt

under various assumptions for the integrable functions f,g:[a,b]→C. We show among other things that, if f andgare of bounded variation on[a,b], then

max{|C(f,g)|,|C(|f|,g)|,|C(f,|g|)|,|C(|f|,|g|)|} ≤C V_f,Vg

,

where the cumulative variation functionVf :[a,b]→[0,∞)is defined byVf(t):=^W^t_a(f). Applications to the trapezoid and mid-point inequalities are also provided.

Funding:This research received no external funding.

Institutional Review Board Statement:Not applicable.

Informed Consent Statement:Not applicable.

Data Availability Statement:Not applicable.

Conflicts of Interest:The author declares no conflict of interest.

(16)

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