Ostrowski and Trapezoid Type Inequalities Related to Pompeiu's Mean Value Theorem With Complex Exponential Weight

Inequalities

Volume 11, Number 4 (2017), 947–964 doi:10.7153/jmi-2017-11-72

OSTROWSKI AND TRAPEZOID TYPE INEQUALITIES RELATED TO POMPEIU’S MEAN VALUE THEOREM

WITH COMPLEX EXPONENTIAL WEIGHT

PIETROCERONE, SEVERS. DRAGOMIR ANDEDERKIKIANTY

(Communicated by A. Agli´c Aljinovi´c)

Abstract. We present some inequalities of Ostrowski and trapezoid type with complex exponen-tial weight, for complex-valued absolutely continuous functions. These inequalities are related to Pompeiu’s mean value theorem. Special cases of these inequalities are applied to obtain (i) some approximation results for thefinite Fourier and Laplace transforms; (ii) refinements of the Os-trowski and trapezoid inequalities; and (iii) new OsOs-trowski and trapezoid type inequalities.

1. Introduction

In 1938, Ostrowski [17] proved the following estimate of the integral mean:

THEOREM1. Let f :[a,b]→R be continuous on [a,b] and differentiable on

(a,b)with|f_{(t)|M<∞for all t∈(a,b).Then, for any x∈[a,b], we have}

f(x)−_b1

−a

_b

a f(t)dt

⎡ ⎣1

4+

x−a+b

2

b−a

2⎤

⎦M(b−a). (1)

The constant 1₄ is best possible, in the sense that it cannot be replaced by a smaller quantity.

Inequality (1) is referred to as Ostrowski’s inequality. For its generalisations and related results we refer the readers to Dragomir and Rassias [15]. Another estimate of the integral mean is given by the trapezoid rule as follows.

THEOREM2. (Cerone and Dragomir [7]) Under the assumptions of Theorem1,

we have

(x−a)f(a)+ (b_b−a −x)f(b)−_b−1_a b

a f(t)dt

⎡ ⎣1

4+

x−a+b

2

b−a

2⎤

⎦M(b−a),

(2) for any x∈[a,b]. The constant 1

4 is best possible.

Mathematics subject classification(2010): Primary: 26D10, 26D15.

Keywords and phrases: Ostrowski inequality, trapezoid inequality, complex exponential weight, Laplace transform, Fourier transform.

c

, Zagreb

Inequality (2) is known as the trapezoid inequality. For its generalisations and related results, we refer the readers to Cerone and Dragomir [7].

It is important to note that the bounds in inequalities (1) and (2) are the same. Cerone [6, Remark 1] stated that there is a strong relationship between the Ostrowski and the trapezoidal functionals which is highlighted by the symmetric transformations amongst their kernels.

In 1946, Pompeiu [19] derived a variant of Lagrange’s mean value theorem, known asPompeiu’s mean value theorem(cf. Sahoo and Riedel [21, p. 83]), as given below:

THEOREM3. For every real-valued function f differentiable on an interval[a,b]

not containing0 and for all pairs x1=x2 in[a,b], there exists a pointξ between x1

and x2such that

x1f(x2)−x2f(x1)

x1−x2 =f(ξ)−ξf

_{(ξ). (3)}

Pompeiu’s mean value theorem is utilised in order to provide another approxima-tion of the integral mean, as given below.

THEOREM4. (Dragomir, 2005 [9]) Let f:[a,b]→Rbe continuous on[a,b]and

differentiable on (a,b) with [a,b] not containing 0. Then for any x∈[a,b], we have the inequality

a+₂b· f(_xx)−_b−1_a b

a f(t)dt

b_|−_x|a

⎡ ⎣1

4+

x−a+b

2

b−a

2⎤

⎦f−f_∞. (4)

where(t) =t, t∈[a,b].The constant 1

4 is best possible.

We refer the readers to Popa [20], Pecari´c and Ungar [ˇ 18], Acu and Sofonea [1], and Acu et al. [2] for the generalisations and extensions of Theorem4. Inequalities of Ostrowski type which are related to the Pompeiu’s mean value theorem are given in the papers by Dragomir [10,11]. Further inequalities of Ostrowski and trapezoid types which are related to the Pompeiu’s mean value theorem can be found in Cerone, Dragomir, and Kikianty [8].

Some exponential Pompeiu type inequalities for complex-valued absolutely con-tinuous functions are given in Dragomir [12], with applications to obtain some new Ostrowski type inequalities. We recall the results on the Ostrowski type inequalities, in the next theorem.

THEOREM5. Let f :[a,b]→C be an absolutely continuous function on the

in-terval[a,b]andα=β+iγ∈Cwithβ >0. Then, for any x∈[a,b]we have

f(x)exp(αb)−exp(αa)

α −exp(αx)

_b

a f(t)dt

(5)

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

|β|B1(a,b,x,α)f−αf∞, f−αf ∈L∞[a,b], q1/q_|β|1/q_(b−a)1/p_{|Bq(a,b,x,α)|}1/q_{f−αfp, f−αf ∈Lp[a,b],}

p>1, 1

where

Bq(a,b,x,α) =2

exqβx−a+₂b+_q1 β

ebqβ_+eaqβ 2 −exqβ

,

for q1 and B∞(a,b,x,α):=exp(xβ)(x−a) +β−1[exp(bβ)−exp(xβ)].If β =0, then for any x∈[a,b] we have

f(x)exp(iγb)−_i exp(iγa)

γ −exp(iγx) _b

a f(t)dt

(6)

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

1

4+

x−a+b 2 b−a

2_{(b−a)2f−i}

γf∞, f−iγf ∈L∞[a,b], _b−x

b−a

q+1 q ₊x−a

b−a

q+1

q q

q+1(b−a)

q+1

q _f−iγ_{fp, f−i}γ_{f ∈Lp[a,b],} p>1, 1

p+1q=1,

(b−a)f_−iγf1.

In this paper, we give refinements of the inequalities in Theorem5. We also present similar results for trapezoid type inequalities with complex exponential weights.

The paper is organised as follows. We present the main theorems concerning in-equalities with complex weights in Section2. We consider special cases of the main theorems, by choosing x= (a+b)/2, in Section 3. The inequalities involving the p-norms (with imaginary weights) wherep=1 are proven to be sharp.

The Fourier transform has been a principal analytical tool in many fields of re-search, such as probability theory, quantum physics, and boundary-value problems [3]. The approximations of thefinite Fourier transform of different classes of functions have been considered by employing integral inequalities of Ostrowski type. We refer to Bar-nett and Dragomir [4] for the approximations of the Fourier transform of absolutely continuous functions; to Barnett, Dragomir, and Hanna [5] for functions of bounded variation; and to Dragomir, Cho, and Kim [13] for Lebesgue integrable mappings. Us-ing a pre-Gr¨uss type inequality, Dragomir, Hanna, and Roumeliotis [14] obtained some approximations of thefinite Fourier transform for complex-valued functions. We apply the inequalities in Section3to obtain some approximation results for thefinite Fourier and Laplace transforms.

Finally, we provide some refinements of the Ostrowski and trapezoid inequalities in Section4by considering special cases of the main theorems in Section2. We also obtain some new Ostrowski and trapezoid type inequalities.

2. Main theorems

The main results concerning the Ostrowski and trapezoid type inequalities with complex exponential weights are given in this section.

2.1. Ostrowski type inequalities

We recall the definition of the Gamma and incomplete Gamma functions:

Γ(t) = _∞

0 x

t−1_e−x_{dx, and Γ(s,x) =} ∞ x t

Throughout the text, for α =β+iγ∈C and 1q<∞, we use the following notation:

Ψ+

q,α(s,t) =e−sβ(qβ)− q+1

q [Γ(q+₁)−Γ(q+₁,_qβ(t−s))]1q_;

and

Ψ−q,α(s,t) =e−tβ(−qβ)− q+1

q _{[Γ(q+1)−Γ(q+1,−q}β_(t−s))]1q_.

THEOREM6. Let f :[a,b]→C be an absolutely continuous function on [a,b],

α=β+iγ∈Cand1<p∞. Let1q<∞be the H¨older conjugate of p. Ifβ=0, then

f_e(x)αx(b−a)− _b

a f(t) eαt dt

Ψ+q,α(a,x)f−αf[a,x],p+Ψ−q,α(x,b)f−αf[x,b],p (7)

Ψ+q,α(a,x) +Ψ−q,α(x,b)f−αf[a,b],p,

for x∈[a,b]. If β=0, then

f_e(ixxγ)(b−a)− _b

a f(t) eitγ dt

(x−a)

q+1 q

(q+1)1q

f−iγf[a,x],p+(b−x) q+1

q

(q+1)q1

f−iγf[x,b],p

(x−a) q+1

q + (b−x)q+q1

(q+1)q1

f−iγf[a,b],p,

(8)

for x∈[a,b]. The inequalities in(8)are sharp.

Proof. We use the Montgomery identity for the absolutely continuous function g:[a,b]→C(cf. Mitrinovi´c, Peˇcari´c, and Fink [16, p. 565]):

g(x)(b−a)− _b

a g(t)dt= _x

a(t−a)g

_(t)dt+ b

x (t−b)g

_{(t)dt, (9)}

wherex∈[a,b]. If g(t) = f(t)/eαt_{, then g(t) = (f(t)−αf(t))/e}αt_{; and with this} choice ofg, (9) becomes:

f(x)

eαx(b−a)− _b

a f(t) eαt dt =

_x

a (t−a)

f_(t)−αf(t)

eαt dt+ _b

x (t−b)

f_(t)−αf(t)

eαt dt.

Take the modulus of (10) and make use of the H¨older’s inequality to obtain the follow-ing inequalities for 1<p∞and its H¨older’s conjugateq:

f_e(x)αx (b−a)− _b

a f(t) eαt dt

x

a

(t−a)|e−αt_|q_dt 1 q

f−αf[a,x],p+

b

x

(b−t)|eαt_|q_dt 1 q

f−αf[x,b],p

=

_x

a(t−a)

q_e−tqβ_dt 1 q

f−αf[a,x],p+

_b

x (b−t)

q_e−tqβ_dt 1 q

f−αf[x,b],p.

We evaluate the integral _x

a(t−a)

q_e−tqβ_dt=e−aqβ_(qβ)−q−1 qβ(x−a)

0 z

q_e−z_dz

=e−aqβ_(qβ)−q−1_{[Γ(q+1)−Γ(q+1,qβ(x−a))],}

by lettingz= (t−a)qβ, and thus x

a (t−a)

q_e−tqβ_dt 1 q

=e−aβ(qβ)−q+q1_{[Γ(q+1)−Γ(q+1,q}β_(x−a))]1q _=Ψ+ q,α(a,x).

Now, we evaluate the integral _b

x (b−t)

q_e−tqβ_{dt =e}−bqβ_(−qβ)−q−1 q(−β)(b−x)

0 z

q_e−z_dz

=e−bqβ(−qβ)−q−1_{[Γ(q+1)−Γ(q+1,−qβ(b−x))],}

by lettingz=−(b−t)q_β, and thus _b

x (b−t)

q_e−tqβ_dt 1 q

=e−bβ(−qβ)−q+1q [Γ(q+₁)−Γ(q+₁,−_qβ(b−x))]1q =Ψ− q,α(x,b), and this proves (7). If_β =0, then

f_e(x)ixγ (b−a)− _b

a f(t) eitγ dt

_x

a(t−a) q_dt

1 q

f−iγf[a,x],p+

_b

x (b−t) q_dt

1 q

f−iγf[x,b],p

=(x−a)

q+1 q

(q+1)q1

f−iγf[a,x],p+(b−x) q+1

q

(q+1)q1

f−iγf[x,b],p

(x−a) q+1

q _{+ (b−x)}q+q1

(q+1)1q

for anyx∈[a,b].This completes the proof. The sharpness of the inequalities in (8) is given by Proposition1.

In particular, when p=∞ (q=1) in Theorem6, inequalities (7) and (8) take simpler forms as follows:

COROLLARY1. Let f:[a,b]→Cbe an absolutely continuous function on [a,b]

and_α=β+i_γ∈C. If _β=0, then we have

f(x)(_eαbx−a)− _b

a f(t) eαt dt

e−aβ−[(x−a)β+1]e−xβ

β2 f−αf[a,x],∞

+e−bβ+ [(b−x)β−1]e−xβ

β2 f−αf[x,b],∞ (11)

1

β2

e−aβ+e−bβ+2

a+b 2 −x

β−1

e−xβ

f−αf[a,b],∞,

for any x∈[a,b]. If β=0, then we have

f(x)(b_eixγ−a)− _b

a f(t) eitγ dt

1₂(x−a)2_f−iγf

[a,x],∞+ (b−x)2f−iγf[x,b],∞ (12)

1

4(b−a)2+

x−a+₂b 2

f−i_γf[a,b],∞,

for any x∈[a,b]. The constants 1

2 and 14 in(12)are sharp.

The case for the 1-norm is as follows:

THEOREM7. Let f :[a,b]→C be an absolutely continuous function on [a,b],

α=β+i_γ∈C. If_β =0, then

f_e(αxx)(b−a)− _b

a f(t) eαt dt

(13)

⎧ ⎪ ⎨ ⎪ ⎩

1

βe−(aβ+1)f−αf[a,x],1+ (b−x)e−xβf−αf[x,b],1, β >0 &a+_β1 x,

(x−a)e−xβ_f−αf

[a,x],1+₋1_βe−(bβ+1)f−αf[x,b],1,β <0 &b+_β1 x,

(x−a)e−xβ_f−αf

[a,x],1+ (b−x)e−xβf−αf[x,b],1, otherwise,

⎧ ⎪ ⎨ ⎪ ⎩

₁

βe−(aβ+1)+ (b−x)e−xβ

f_−αf

[a,b],1, β >0 &a+_β1 x,

(x−a)e−xβ₊ 1

−βe−(bβ+1)

f−αf[a,b],1,β <0 &b+_β1 x,

(b−a)e−xβ_f−αf

for any x∈[a,b]. If β=0, then

f_e(ixxγ)(b−a)− _b

a f(t) eitγ dt

(x−a)f−iγf[a,x],1+ (b−x)f−iγf[x,b],1

(b−a)f−i_γf[a,b],1,

(14)

for any x∈[a,b].

Proof. Take the modulus of (10) and make use of the H¨older’s inequality to obtain:

f_e(x)αx (b−a)− _b

a f(t) eαt dt

sup

t∈[a,x](t−a)e

−tβ_f−αf

[a,x],1+ sup

t∈[x,b](b−t)e

−tβ_f−αf [x,b],1.

Define the functions: A(t) = (t−a)e−tβ _{fort∈[a,x] andB(t) = (b−t)e}−tβ _fort∈ [x,b].

Case 1: β >0.We haveB(t) =−e−βt_{(1+β(b−t)). If β>0, thenB(t)0}

for all t∈[x,b], thus, the supremum is attained at t=x. We have A_{(t) =e}−βt₍₁₋ β(t−a)).The stationary point ofAista= (aβ+1)/β.

Case 1a: tax (a+_β1 x) , we have A(ta) =−βe−(aβ+1)0; thus, the supremum is attained att=ta.

Case 1b: ta>x(a+_β1 >x) , we have 1−β(t−a)>1−β(x−a)>0, which implies thatA(t)>0 for allt∈[a,x]. Thus, the supremum is attained att=x.

Case 2: β<0.We haveA(t)0 for allt∈[a,x], thus, the supremum is attained att=x. The stationary point ofBistb= (bβ+1)/β.

Case 2a: tbx(b+_β1 x) , we have B(ta) =βe−(bβ+1)0; thus, the supre-mum is attained att=tb.

Case 2b: tb<x(b+_β1 <x) , we have 0<1+β(b−x)<1+β(t−x), which implies thatB_{(t)<0 for allt∈[x,b]. Thus, the supremum is attained att=x. This}

completes the proof of (13). If β=0, then

f_e(ixxγ)(b−a)− _b

a f(t) eitγ dt

sup

t∈[a,x](t−a)f _−iγf

[a,x],1+ sup

t∈[x,b](b−t)f _−iγf

[x,b],1

(x−a)f−iγf[a,x],1+ (b−x)f−iγf[x,b],1

(b−a)f−iγf[a,b],1.

This completes the proof.

2.2. Trapezoid type inequalities

The proofs for Theorems8,9, and Corollary2below are similar to Theorems6, 7, and Corollary1. We utilise the trapezoid identity for absolutely continuous function g:[a,b]→C

g(b)(b−x) +g(a)(x−a)− _b

a g(t)dt= _b

a (t−x)g

instead of the Montgomery identity. We omit the proofs.

THEOREM8. Let f :[a,b]→C be an absolutely continuous function on [a,b],

α=β+i_γ∈Cand1<p∞. Let1q<∞be a real number such that 1

p+1q=1. Ifβ=0, then

f_e(b)bα (b−x) + f(a)

eaα (x−a)− _b

a f(t) eαt dt

Ψ−q,α(a,x)f−αf[a,x],p+Ψ+q,α(x,b)f−αf[x,b],p (16)

Ψ−q,α(a,x) +Ψ+q,α(x,b)f−αf[a,b],p, for any x∈[a,b]. If β=0, then

f_e(b)ibγ (b−x) + f(a)

eiaγ (x−a)− _b

a f(t) eitγ dt

(x−a)

q+1 q

(q+1)1q

f−iγf[a,x],p+(b−x) q+1

q

(q+1)q1

f−iγf[x,b],p

(x−a) q+1

q _{+ (b−x)}q+q1

(q+1)q1

f−i_γf[a,b],p,

(17)

for any x∈[a,b]. The inequalities in(17)are sharp.

COROLLARY2. Let f:[a,b]→Cbe an absolutely continuous function on [a,b]

andα=β+iγ∈C. If β=0, then we have

f_e(b)bα (b−x) + f(a)

eaα (x−a)− _b

a f(t) eαt dt

e−xβ+ [(x−a)β−1]e−aβ

β2 f−αf[a,x],∞

+e−xβ−[(b−x)β+1]e−bβ

β2 f−αf[x,b],∞ (18)

1

β2

2e−xβ_{+ [(x−a)β−1]e}−aβ_{−[(b−x)β+1]e}−bβ_f−αf [a,b],∞,

for any x∈[a,b]. If β=0, then we have

f_e(ibbγ)(b−x) + f(a)

eiaγ (x−a)− _b

a f(t) eitγ dt

1₂(x−a)2_f−iγf

[a,x],∞+ (b−x)2f−iγf[x,b],∞ (19)

1

4(b−a)2+

x−a+₂b 2

f−iγf[a,b],∞,

for any x∈[a,b]. The constants 1

THEOREM9. Let f :[a,b]→C be an absolutely continuous function on [a,b], α=β+iγ∈C. Ifβ =0, then

f_e(b)bα (b−x) + f(a)

eaα (x−a)− _b

a f(t) eαt dt

(20)

⎧ ⎪ ⎨ ⎪ ⎩

(x−a)e−aβ_f−αf

[a,x],1+_β1e−(xβ+1)f−αf[x,b],1, β >0 &x+1_β a, 1

−βe−(xβ+1)f−αf[a,x],1+ (b−x)e−bβf−αf[x,b],1,β <0 &x+1_β b,

(x−a)e−aβ_f−αf

[a,x],1+ (b−x)e−bβf−αf[x,b],1,otherwise,

⎧ ⎪ ⎨ ⎪ ⎩

(x−a)e−aβ₊1

βe−(xβ+1)

f−αf[a,b],1, β >0 &x+_β1a,

₁

−βe−(xβ+1)+ (b−x)e−bβ

f_−αf

[a,b],1, β <0 &x+_β1b,

(x−a)e−aβ_{+ (b−x)e}−bβ_f−αf

[a,b],1,otherwise,

for x∈[a,b]. If β=0, then

f_e(ibbγ)(b−x) + f(a)

eiaγ (x−a)− _b

a f(t) eitγ dt

(x−a)f−i_γf[a,x],1+ (b−x)f−iγf[x,b],1

(b−a)f−iγf[a,b],1,

(21)

for x∈[a,b].

REMARK1. Note the similarity of the bounds in Corollaries1and2[also,

Theo-rems6and8, and similarly, Theorems7and9]. Thefirst set of upper bounds in (11) [(7) and (13), respectively] and (12) [(8) and (14), respectively] can be obtained by let-ting a=x, x=b in thefirst term, and x=a, b=x in the second term in (18) [(16) and (20), respectively] and (19) [(17) and (21), respectively].

3. Special cases of the main theorems and approximations of the Laplace and Fourier transforms

One may obtain simpler inequalities from the main theorems, by choosing x= (a+b)/2. These special cases are applied to approximate the Laplace and Fourier transforms. For inequalities involving thep-norms where p=1, the choice ofx= (a+ b)/2 proves that the inequalities are sharp. We summarise the results in this section.

COROLLARY3. Let f:[a,b]→Cbe an absolutely continuous function on[a,b],

α=β+iγ∈Cand1<p<∞. Let q>1 be a real number such that 1

p+1q=1. If β=0, then the following inequalities hold:

fa+b

2

(b−a)

eα(a2+b)

− _b

a f(t) eαt dt

e−aβ_−[b−a

2 β+1]e−

a+b

2 β

β2 f−αf[a,a+b

2 ],∞(22)

+e−bβ+ [b−2aβ−1]e−

a+b

2 β

β2 f−αf[a+b 2 ,b],∞

1

β2

e−aβ+e−bβ−2e−a+2bβ

For the case of the p-norms (1<p<∞), we have:

fa+b

2

eα(a2+b)

(b−a)−

_b

a f(t) eαt dt

(23)

Ψ+

q,α

a,a+₂b

f−αf_[a,a+b

2 ],p+Ψ

−

q,α _a

+b

2 ,b

f−αf_[a+b 2 ,b],p

Ψ+q,α

a,a+₂b

+Ψ−q,α _a

+b

2 ,b

f−αf[a,b],p.

For the case of the1-norm, we have:

fa+b

2

eα(a2+b)

(b−a)− b

a f(t) eαt dt

(24) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1

βe−(aβ+1)f−αf[a,a+b

2 ],1+

b−a

2 e−

a+b

2 βf−αf

[a+b

2 ,b],1, β >0 &a+

1

β x, b−a

2 e−

a+b

2 βf−αf

[a,a+b

2 ],1+

1

−βe−(bβ+1)f−αf[a+b

2 ,b],1,β <0 &b+

1

β x, b−a

2 e−

a+b

2 β

f_−αf

[a,a+2b],1+f

_−αf [a+2b,b],1

, otherwise,

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ₁

βe−(aβ+1)+b−2ae−

a+b

2 βf−αf_[a,b],1, β >0 &a+1 β x, _b−a

2 e−

a+b

2 β+ 1

−βe−(bβ+1)

f−αf[a,b],1,β <0 &b+_β1 x,

(b−a)e−a+b

2 βf−αf_[a,b],1, otherwise,

Ifβ=0, then the following inequalities hold:

fa+b

2

(b−a)

eia+₂bγ − _b

a f(t) eitγ dt

1

8(b−a)2

f−i_γf_[a,a+b

2 ],∞+f

_−iγf [a+b

2 ,b],∞

1₄(b−a)2_f−iγf

[a,b],∞. (25)

For the case of the p-norms (1<p<∞), we have:

fa+₂b

eixγ (b−a)− _b

a f(t) eitγ dt

(b−a)q+1q 2(2q+2)q1

f−iγf_[a,a+b

2 ],p+f

_−iγf [a+b

2 ,b],p

(b−a) q+1

q

(2q+2)1q

f−i_γf[a,b],p. (26)

For the case of the1-norm, we have:

fa+b

2

eia+b

2 γ (b−a)−

_b

a f(t) eitγ dt

b−a 2

f−iγf_[a,a+b

2 ],1+f

_−iγf [a+b

2 ,b],1

PROPOSITION1. Inequalities(25)and(26)are sharp.

Proof. We show that the constants 1₈ and 1₄ in (25) are sharp. We assume that the inequalities hold forA,B>0 instead of 1₈ and 1₄, respectively:

fa+b

2

(b−a)

eia+b

2 γ −

_b

a f(t) eitγ dt

A(b−a)2

f−iγf_[a,a+b

2 ],∞+f

_−iγf [a+b

2 ,b],∞

B(b−a)2f−iγf[a,b],∞.

Let_γ=0 and choose f(x) =|x−a+b

2 |on[a,b]. Thus we have (b−a)

2

4 2A(b−a)2

B(b−a)2_{; which yields A}1

8 and B14. It implies that the constants 12 and 14 in

(12) are sharp.

We show that the inequalities in (26) are sharp. We assume the inequalities hold forC,D>0 instead of 1₂ and 1, respectively:

fa+b

2

eixγ (b−a)− _b

a f(t) eitγ dt

C(

b−a)q+q1 (2q+2)q1

f−i_γf_[a,a+b

2 ],p+f

_−iγf [a+b

2 ,b],p

D(b−a)

q+1 q

(2q+2)1q

f−iγf[a,b],p.

Letγ=0 and take f(x) =|x−a+b

2 | on[a,b], and we now have

(b−a)2

4 2C

(b−a)q+q1

(2q+2)1q

D(b−a) q+1

q

(2q+2)q1 .

Take q→1, then we have (b−₄a)2 2C(b−₄a)2 D(b−₄a)2, which yields C 1

2 and

D1. It implies that the inequalities in (8) are sharp.

Let f:[a,b]→K (K=C,R)be a Lebesgue integrable mapping defined on thefi -nite interval[a,b]. LetL(f)andF(f)be theirfinite Laplace and Fourier transforms, respectively, defined by

L(f)(α):= b a f(s)e

−αs_{ds, α∈C,}

F(f)(t):= b a f(s)e

−2πits_{ds, t∈R.}

REMARK2. (Laplace transform approximations) By rewriting _abf(s)e−αt_{dt =}

L(f_)(α) in (22), (23), and (24), we obtain error bounds in terms of the p-norms (1p∞), for the approximation ofL(f)(α) by f(a+b

2 )(b−a)e−

α(a+b)

REMARK3. (Fourier transform approximations) Let u∈R. Choose γ=2πu in (25), (26), and (27). By rewriting_abf(t)e−2πiut_{dt=F(f)(u), we obtain error bounds} in terms of the p-norms(1p∞), for the approximation of F(f)(t)by

f

a+b 2

(b−a)e−iπ(a+b)t_.

COROLLARY4. Let f:[a,b]→Cbe an absolutely continuous function on[a,b],

α=β+iγ∈Cand1<p<∞. Let q>1 be a real number such that 1_p+1

q=1. If β=0, then the following inequalities hold:

b−₂af_e(aaα)+ f(b) ebα

−

_b

a f(t) eαt dt

e−

a+b

2 β+ [b−a

2 β−1]e−aβ

β2 f−αf[a,a+b 2 ],∞

+e−

a+b

2 β−[b−a

2 β+1]e−bβ

β2 f−αf[a+b

2 ,b],∞ (28)

1

β2

2e−a+2bβ−e−aβ−e−bβ+b−a 2 β

e−aβ_−e−bβ_f−αf [a,b],∞.

For the case of the p-norms (1<p<∞), we have

b−₂af_e(aaα)+ f(b) ebα

− b

a f(t) eαt dt

(29)

Ψ−q,α

a,a+₂b

f−αf_[a,a+b

2 ],p+Ψ

+

q,α _a

+b

2 ,b

f−αf_[a+b 2 ,b],p

Ψ−

q,α

a,a+₂b

+Ψ+

q,α

a+b 2 ,b

f−αf[a,b],p.

For the case of the1-norm, we have

b−₂af_e(a)aα + f(b)

ebα

− _b

a f(t) eαt dt

(30) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

b−a

2 e−aβf−αf[a,a+b

2 ],1+

1

βe−( a+b

2 β+1)f−αf

[a+b

2 ,b],1, β>0 &x+

1

β a,

1 −βe−(

a+b

2 β+1)f−αf

[a,a+b

2 ],1+

b−a

2 e−bβf−αf[a+b

2 ,b],1,β<0 &x+

1

β b, b−a

2

e−aβ_f−αf [a,a+b

2 ],1+e

−bβ_f−αf [a+b

2 ,b],1

, otherwise, ⎧ ⎪ ⎨ ⎪ ⎩

_b−a

2 e−aβ+β1e−( a+b

2 β+1)f−αf_[a,b],1, β >0 &x+1 β a, ₁

−βe−( a+b

2 β+1)+b−₂ae−bβf−αf_[a,b],1,β <0 &x+1 β b, b−a

2

e−aβ_+e−bβ_f−αf

Ifβ=0, then the following inequalities hold:

b−₂af_e(iaaγ)+ f(b) eibγ

− b

a f(t) eαt dt

1₈(b−a)2_f−iγf

[a,a+b

2 ],∞+f

_−iγf [a+b

2 ,b],∞

(31)

1₄(b−a)2_f−iγf [a,b],∞.

For the case of the p-norms (1<p<∞), we have

b−₂af_e(iaaγ)+ f(b) eibγ

− b

a f(t) eαt dt

(b−a) q+1

q

2(2q+2)1q

f−i_γf_[a,a+b

2 ],p+f

_−iγf [a+b

2 ,b],p

(b−a)

q+1 q

(2q+2)1q

f−iγf[a,b],p.

(32)

For the case of the1-norm, we have

b−₂af_e(iaaγ)+ f(b) eibγ

− b

a f(t) eαt dt

b−₂af−iγf_[a,a+b

2 ],1+f

_−iγf [a+b

2 ,b],1

(b−a)f−iγf[a,b],1.

(33)

PROPOSITION2. The inequalities(31)and(32)are sharp.

The proof follows similarly to that of Proposition1; and we omit the proof. It follows from Proposition2that inequalities (19), and (17) are sharp.

REMARK4. (Laplace transform approximations) By rewriting _abf(s)e−αtdt =

L(f)(α) in (28), (29), and (30), we obtain error bounds in terms of the p-norms (1p∞), for the approximation ofL(f_)(α) by b−a

2

_f

(a)

eaα +f_e(bbα)

.

REMARK5. (Fourier transform approximations) Let u∈R. Choose γ=2πu in

(31), (32) and (33). By rewriting_abf(t)e−2πiut_{dt=F(f)(u), we obtain error bounds} in terms of the p-norms(1p∞), for the approximation of F(f)(t)by

b−a 2

f(a) e2aπti+

f(b) e2bπti

4. Some new and refined Ostrowski and trapezoid type inequalities

Using the results in Section2, we obtain inequalities of Ostrowski and trapezoid type; and we present the result in the following subsections.

4.1. Refinements of Ostrowski’s inequalities

Ifα=0 in Corollary1, Theorems6and7, respectively, then we have refinements for the Ostrowski inequality, as follows:

f(x)(b−a)− b

a f(t)dt

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

1 2

(x−a)2_f

[a,x],∞+ (b−x)2f[x,b],∞,

(x−a)q+q1

(q+1)q1 f

[a,x],p+(b−x) q+1

q

(q+1)1q f

[x,b],p, p>1, 1p+1q=1,

(x−a)f

[a,x],1+ (b−x)f[x,b],1,

(34)

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

1

4(b−a)2+

x−a+b

2

2

f [a,b],∞,

(x−a)q+q1_+(b−x)q+q1

(q+1)1q f

[a,b],p, p>1, 1p+1q=1,

(b−a)f

[a,b],1,

(35)

for anyx∈[a,b]. The constants in cases 1 and 2 in (34) and (35) are sharp (cf. Propo-sition1).

4.2. New Ostrowski type inequalities

Leth_α(t) =eαt _{fort∈[a,b]. If f(t) =g(t)h}

α(t) =g(t)eαt in Theorem6, Corol-lary1, and Theorem7, then we have the Ostrowski inequalities: Ifβ=0, we have

g(x)(b−a)− b

a g(t)dt

Ψ+

q,α(a,x)ghα[a,x],p+Ψ−q,α(x,b)ghα[x,b],p

Ψ+

q,α(a,x) +Ψ−q,α(x,b)ghα[a,b],p, (36)

wherep>1 and 1_p+1

q=1,

g(x)(b−a)− b

a g(t)dt

(37)

e−aβ−[(x−a)β+1]e−xβ

β2 ghα[a,x],∞+e

−bβ_{+ [(b−x)β−1]e}−xβ

β2 ghα[x,b],∞

1

β2

e−aβ+e−bβ+2

a+b 2 −x

β−1

e−xβ

and

g(x)(b−a)− b

a g(t)dt

(38) ⎧ ⎪ ⎨ ⎪ ⎩ 1

βe−(aβ+1)ghα[a,x],1+ (b−x)e−xβghα[x,b],1, β >0 &a+_β1 x,

(x−a)e−xβ_ghα

[a,x],1+−1βe−(bβ+1)ghα[x,b],1,β <0 &b+_β1 x,

(x−a)e−xβ_gh

α[a,x],1+ (b−x)e−xβghα[x,b],1, otherwise,

⎧ ⎪ ⎨ ⎪ ⎩ ₁

βe−(aβ+1)+ (b−x)e−xβ

gh_α[a,b],1, β >0 &a+_β1 x,

(x−a)e−xβ₊ 1

−βe−(bβ+1)

g_hα

[a,b],1,β <0 &b+_β1 x,

(b−a)e−xβ_gh

α[a,b],1, otherwise,

forx∈[a,b]. If_β =0, then we have

g(x)(b−a)− b

a g(t)dt ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 2

(x−a)2_gh

α[a,x],∞+ (b−x)2ghα[x,b],∞,

(x−a)q+q1

(q+1)q1 g

_h

α[a,x],p+(b−x) q+1

q

(q+1)1q g

_h

α[x,b],p, p>1, 1p+1q=1, (x−a)gh_α[a,x],1+ (b−x)ghα[x,b],1,

(39) ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 1

4(b−a)2+

x−a+b

2

2

g_hα [a,b],∞,

(x−a)q+q1_+(b−x)q+q1

(q+1)1q g

_h

α[a,b],p, p>1, 1p+1q=1,

(b−a)g_hα

[a,b],1,

(40)

for anyx∈[a,b].

4.3. Refinements of the trapezoid inequalities

Ifα=0 in Corollary2, Theorems8and9, respectively, then we have refinements for the trapezoid inequality as follows:

f(a)(x−a) +f(b)(b−x)− b

a f(t)dt

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 2

(x−a)2_f

[a,x],∞+ (b−x)2f[x,b],∞,

(x−a)q+q1

(q+1)q1 f

[a,x],p+(b−x) q+1

q

(q+1)1q f

[x,b],p, p>1, 1p+1q=1,

(x−a)f

[a,x],1+ (b−x)f[x,b],1,

(41) ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1

4(b−a)2+

x−a+b

2

2

f [a,b],∞,

(x−a)q+q1_+(b−x)q+q1

(+1)1q f

[a,b],p, p>1, 1p+1q=1,

(b−a)f

[a,b],1,

for anyx∈[a,b]. The constants in cases 1 and 2 in (41) and (42) are sharp (cf. Propo-sition2).

4.4. New trapezoid type inequalities

Leth_α(t) =eαt _{fort∈[a,b]. If f(t) =g(t)h}

α(t) =g(t)eαt in Theorem8, Corol-lary2, and Theorem9, then we have the trapezoid inequalities: If β =0, then we have

g(a)(x−a) +g(b)(b−x)−b

a g(t)dt

Ψ−q,α(a,x)ghα[a,x],p+Ψ+q,α(x,b)ghα[x,b],p (43)

Ψ−q,α(a,x) +Ψ+q,α(x,b)

gh_α[a,b],p,

wherep>1 and 1_p+1

q=1,

g(a)(x−a) +g(b)(b−x)−b

a g(t)dt

(44)

e−xβ+ [(x−a)β−1]e−aβ

β2 ghα|[a,x],∞+e

−xβ_{−[(b−x)β+1]e}−bβ

β2 ghα[x,b],∞

1

β2

2e−xβ+ [(x−a)β−1]e−aβ_{−[(b−x)β+1]e}−bβ_gh

α[a,b],∞,

and

g(a)(x−a) +g(b)(b−x)− b

a g(t)dt

(45)

⎧ ⎪ ⎨ ⎪ ⎩

(x−a)e−aβ_gh

α[a,x],1+_β1e−(xβ+1)ghα[x,b],1, β >0 &x+1_β a, 1

−βe−(xβ+1)ghα[a,x],1+ (b−x)e−bβghα[x,b],1,β <0 &x+1_β b,

(x−a)e−aβ_ghα

[a,x],1+ (b−x)e−bβghα[x,b],1, otherwise,

⎧ ⎪ ⎨ ⎪ ⎩

(x−a)e−aβ₊1

βe−(xβ+1)

g_h

α[a,b],1, β >0 &x+_β1a,

₁

−βe−(xβ+1)+ (b−x)e−bβ

g_hα

[a,b],1, β <0 &x+_β1b,

(x−a)e−aβ_{+ (b−x)e}−bβ_gh

α[a,b],1, otherwise,

forx∈[a,b]. Ifβ =0, then

g(a)(x−a) +g(b)(b−x)− b

a g(t)dt

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

1 2

(x−a)2_ghα

[a,x],∞+ (b−x)2ghα[x,b],∞,

(x−a)q+q1

(q+1)q1 g

_hα

[a,x],p+(b−x) q+1

q

(q+1)1q g

_hα|

[x,b],p, p>1, 1p+1q=1, (x−a)g_hα

[a,x],1+ (b−x)ghα[x,b],1,

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

1

4(b−a)2+

x−a+b

2

2

g_hα [a,b],∞,

(x−a)q+q1_+(b−x)q+q1

(q+1)1q g

_hα

[a,b],p, p>1, 1p+1q=1, (b−a)g_hα

[a,b],1,

(47)

forx∈[a,b].

Acknowledgement. The authors would like to thank the anonymous referees for valuable suggestions that have been incorporated in thefinal version of the manuscript.

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(Received January 7, 2015) Pietro Cerone

Department of Mathematics and Statistics La Trobe University Bundoora 3086, Australia e-mail:p.cerone@latrobe.edu.au Sever Dragomir School of Engineering and Science Victoria University PO Box 14428, Melbourne 8001, Victoria, Australia, and DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences School of Computer Science and Applied Mathematics University of the Witwatersrand Private Bag X3, 2050 Wits, South Africa e-mail:sever.dragomir@vu.edu.au Eder Kikianty Department of Mathematics and Applied Mathematics University of Pretoria Private bag X20, Hatfield 0028, South Africa e-mail:eder.kikianty@up.ac.za; eder.kikianty@gmail.com

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