GENERALIZED POWER POMPEIU TYPE INEQUALITIES FOR LOCAL FRACTIONAL INTEGRALS WITH APPLICATIONS TO OSTROWSKI'S INEQUALITY

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GENERALIZED POWER POMPEIU TYPE INEQUALITIES FOR LOCAL FRACTIONAL INTEGRALS WITH APPLICATIONS TO

OSTROWSKI’S INEQUALITY

S. ERDEN1, M. Z. SARIKAYA2, S. S. DRAGOMIR3 §

Abstract. We establish some generalizations of power Pompeiu’s inequality for local fractional integral. Afterwards, these results gave some new generalized Ostrowski type inequalities. Finally, some applications of these inequalities for generalized special means are obtained.

Keywords: Ostrowski’s inequality, Pompeiu’s mean value theorem, Local fractional inte-gral, Fractal space, Special Means.

AMS Subject Classification: 26D10, 26D15, 26A33.

1. Introduction

In 1938, it was obtained the following result by Ostrowski in [8].

Theorem 1.1. Let f : [a, b]→ R be a differentiable mapping on (a, b) whose derivative f0 : (a, b)→ R is bounded on (a, b), i.e., kf0k_∞ = sup

t∈(a,b)

|f0(t)| < ∞. Then, the following inequality holds: f (x) − 1 b − a b Z a f (t)dt ≤ " 1 4+ (x − a+b₂ )2 (b − a)2 # (b − a) f0 _∞ (1) for all x ∈ [a, b]. The constant 1₄ is the best possible.

Inequality (1) has wide applications in numerical analysis and in the theory of spe-cial means; estimating error bounds for mid-point, trapezoid and Simpson rules and other quadrature rules, etc. It has attracted considerable attention and interest from mathemati-cians and other researchers as shown by hundreds of papers published in the last decade.

1 _{Department of Mathematics, Faculty of Science, Bartın University, BARTIN-TURKEY.}

e-mail: erdensmt@gmail.com;

ORDIC: https://orcid.org/0000-0001-8430-7533;

2

Department of Mathematics, Faculty of Science and Arts, D¨uzce University, Konuralp Campus, D¨uzce-TURKEY.

e-mail: sarikayamz@gmail.com;

ORDIC: http://orcid.org/0000-0002-6165-9242;

3 _{Mathematics,School of Engineering & Science, Victoria University, PO Box 14428, Melbourne City,}

MC 8001, Australia.

e-mail: sever.dragomir@vu.edu.au;

ORDIC: https://orcid.org/0000-0003-2902-6805;

§ Manuscript received: Month( December) Day (27), 2011.

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As a result of these studies, one can find by making a simple search in the MathSciNet database of the American Mathematical Society.

In 1946, Pompeiu [9] derived a variant of Lagrange’s mean value theorem, now known as Pompeiu’s mean value theorem. It can be stated as follows:

Theorem 1.2. For every real valued function f differentiable on an interval [a, b] not containing 0 and for all pairs x1 6= x2 in [a, b], there exist a point ξ between x1 and x2

such that

x1f (x2) − x2f (x1)

x1− x2

= f (ξ) − ξf0(ξ).

It has been obtained the following Pompeiu type inequality by Dragomir in [4].

Theorem 1.3. Let f : [a, b] → R be 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 + b 2 f (x) x + 1 b − a b Z a f (t)dt ≤ b − a |x| " 1 4+ (x −a+b₂ )2 (b − a)2 # f − lf0 ∞.

where l(t) = t for all t ∈ [a, b] . The constant 1₄ is sharp in the sense that it cannot be replaced by a smaller constant.

Many researcher studied on inequailities by using Pompeiu mean value theorem. For example, it is established OStrowski type inequalities via Pompeiu mean value theorem in [1], [2], [4], [5], [10], [12]. Furthermore, Sarikaya obtained an inequality of Gr¨uss type via variant Pompeiu mean value theorem in [11]. Also, a large number of Pompeiu type inequalities have been studied by mathematicians.

2. Preliminaries

Recall the set Rα of real line numbers and use the Gao-Yang-Kang’s idea to describe the definition of the local fractional derivative and local fractional integral, see [14, 16] and so on.

Recently, the theory of Yang’s fractional sets [14] was introduced as follows. For 0 < α ≤ 1, we have the following α-type set of element sets:

Zα: The α-type set of integer is defined as the set {0α, ±1α, ±2α, ..., ±nα, ...} . Qα : The α-type set of the rational numbers is defined as the set {mα =

p q α : p, q ∈ Z, q 6= 0}.

Jα : The α-type set of the irrational numbers is defined as the set {mα 6= p_qα : p, q ∈ Z, q 6= 0}.

Rα: The α-type set of the real line numbers is defined as the set Rα = Qα∪ Jα_.

If aα_{, b}α _{and c}α _{belongs the set R}α _{of real line numbers, then}

(1) aα+ bα and aαbα belongs the set Rα; (2) aα+ bα = bα+ aα= (a + b)α= (b + a)α; (3) aα+ (bα+ cα) = (a + b)α+ cα;

(4) aαbα = bαaα= (ab)α= (ba)α; (5) aα(bαcα) = (aαbα) cα;

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(7) aα+ 0α= 0α+ aα = aα and aα1α = 1αaα= aα.

The definition of the local fractional derivative and local fractional integral can be given as follows.

Definition 2.1. [14] A non-differentiable function f : R → Rα_{, x → f (x) is called to be}

local fractional continuous at x0, if for any ε > 0, there exists δ > 0, such that

|f (x) − f (x₀)| < εα

holds for |x − x0| < δ, where ε, δ ∈ R. If f (x) is local continuous on the interval (a, b) , we

denote f (x) ∈ Cα(a, b).

Definition 2.2. [14] The local fractional derivative of f (x) of order α at x = x0 is defined

by f(α)(x0) = dαf (x) dxα x=x0 = lim x→x0 ∆α(f (x) − f (x0)) (x − x0)α , where ∆α(f (x) − f (x0))=Γ(α + 1) (f (x) − f (xe 0)) . If there exists f(k+1)α(x) = k+1 times z }| {

Dα_x...D_xαf (x) for any x ∈ I ⊆ R, then we denote f ∈ D_(k+1)α(I), where k = 0, 1, 2, ...

Lemma 2.1. [15] Suppose that f (x) ∈ Cα[a, b] and f (x) ∈ Dα(a, b), then for 0 < α ≤ 1

we have a α−differential form

dαf (x) = f(α)(x)dxα.

Lemma 2.2. [15] Let I be an interval, f, g : I ⊂ R → Rα (I◦ is the interior of I) such that f, g ∈ Dα(I◦). Then, the following differentiation rules are valid.

(1) dα[f (x)±g(x)]_dxα = f(α)(x) ± g(α)(x); (2) dαf (x)g(x)_dxα = f(α)(x)g(x) + f (x)g(α)(x); (3) d αf (x) g(x) dxα = f(α)(x)g(x)−f (x)g(α)(x) [g(x)]2 where g(x) 6= 0; (4) dα[cf (x)]_dxα = cf(α)(x) where c is a constant; (5) If y(x) = (f ◦ g) (x), then dαy(x) dxα = f (α)_(g(x))_g(1)_(x)α_.

Theorem 2.1 (Generalized mean value theorem). [18]Suppose that f (x) ∈ Cα[a, b] ,

f(α)(x) ∈ C (a, b) , then we have

f (x) − f (x0) (x − x0)α = f (α)_(ξ) Γ(α + 1) where a < x0< ξ < x < b.

Definition 2.3. [14] Let f (x) ∈ Cα[a, b] . Then the local fractional integral is defined by,

aIbαf (x) = 1 Γ(α + 1) b Z a f (t)(dt)α = 1 Γ(α + 1)∆t→0lim N −1 X j=0 f (tj)(∆tj)α,

with ∆tj = tj+1− tj and ∆t = max {∆t1, ∆t2, ..., ∆tN −1} , where [tj, tj+1] , j = 0, ..., N − 1

and a = t0< t1< ... < tN −1< tN = b is partition of interval [a, b] .

Here, it follows that aI_bαf (x) = 0 if a = b and aI_bαf (x) = −bIaαf (x) if a < b. If for any

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Lemma 2.3. [14]

(1) (Local fractional integration is anti-differentiation) Suppose that f (x) = g(α)(x) ∈ Cα[a, b] , then we have

aIbαf (x) = g(b) − g(a).

(2) (Local fractional integration by parts) Suppose that f (x), g(x) ∈ Dα[a, b] and f(α)(x),

g(α)(x) ∈ Cα[a, b] , then we have

aIbαf (x)g(α)(x) = f (x)g(x)|ab −aIbαf(α)(x)g(x). Lemma 2.4. [14] We have i) d α_xkα dxα = Γ(1 + kα) Γ(1 + (k − 1) α)x (k−1)α_; ii) 1 Γ(α + 1) b R a xkα(dx)α = Γ(1 + kα) Γ(1 + (k + 1) α) b (k+1)α_{− a}(k+1)α_{, k ∈ R.}

Theorem 2.2 (Generalized Ostrowski inequality). [13]Let I ⊆ R be an interval, f : I0 ⊆ R → Rα (I0 is the interior of I) such that f ∈ Dα(I0) and f(α) ∈ Cα[a, b] for a, b ∈ I0

with a < b Then. for all x ∈ [a, b] , we have the inequality f (x) −Γ (1 + α) (b − a)α aI α bf (t) ≤ 2α Γ (1 + α) Γ (1 + 2α)   1 4α + x − a+b₂ b − a !2α (b − a)α f (α) _∞.

In [6], Erden and Sarikaya proved the following identity and also they established the following inequality by using this identity.

Theorem 2.3 (Generalized Pompeiu’s mean value theorem). Let f : [a, b] ⊆ R → Rα _be

a mapping such that f ∈ Dα(a, b), with [a, b] not containing 0 and for all pairs x1 6= x2 in

[a, b], there exist a point ξ in (x1, x2) such that the following equality holds:

xα₁f (x2) − xα2f (x1) (x1− x2)α = f (ξ) − ξ α Γ (1 + α)f (α)_(ξ).

Theorem 2.4. Let f : [a, b] ⊆ R → Rα be a mapping such that f ∈ Cα[a, b] and f ∈

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

Γ(1 + α) Γ(1 + 2α) f (x) xα (a + b) α₋ 1 (b − a)α aI α bf (t) ≤ 2 α_{Γ(1 + α)(b − a)}α Γ(1 + 2α) |x|α " 1 4α + x −a+b₂ 2α (b − a)2α # f − lf (α) _∞ where l(t) = _Γ(1+α)tα , t ∈ [a, b] , and f − lf(α)

_∞= sup

ξ∈(a,b)

f (ξ) − lf(α)_(ξ) < ∞.

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The interested reader is invited to look over the references [3], [7], [14]-[19] for local fractional theory. Also, many researcher studied on generalized Ostrowski type inequalities for local fractions integrals (see, [13]). In addition, Erden and Sarikaya give generalized Pompeiu mean value theorem and some generalized Pompeiu type inequalities for local fractional calculus in [6].

In this study, some generalization of power Pompeiu’s type inequalities involving local fractional integrals are obtained and also some new generalized Ostrowski type inequalities are obtained. Finally, applications of these inequalities for special means are also given.

3. Generalized Power Pompeiu’s Type Inequalities

Generalized Ostrowski type inequalities can be derived using the following inequality. Corollary 3.1 (Generalized Pompeiu’s Inequality). With the assumptions of Theorem 2.3 and if f − lf(α) _∞= sup t∈(a,b) f (t) − lf(α)(t)< ∞ where l(t) = t α Γ(1+α), t ∈ [a, b] , then |tαf (x) − xαf (t)| ≤ f − lf (α) _∞|x − t| α

for any t, x ∈ [a, b] .

We can generalize the above inequality for the power function as follows.

Theorem 3.1. Let f : [a, b] → Rα be f ∈ Dα(a, b) and f ∈ Cα[a, b] , b > a > 0. If r ∈ R,

r 6= 0, then for any x ∈ [a, b] , we have the inequality

|trαf (x) − xrαf (t)| ≤ 1 |r|α|x rα_{− t}rα_| lf (α)_{− r}α_f _∞ (2)

where l(t) = _Γ(1+α)tα , t ∈ [a, b] and lf(α)_{− r}α_f

_∞= sup

s∈[a,b]

f(α)_{(s)l(s) − r}α_{f (s)} .

Proof. Because of f ∈ Dα(a, b) and f ∈ Cα[a, b] , H ∈ Dα(a, b) and H ∈ Cα[a, b] defined

as H(s) = f (s)_srα. Then, for any t, x ∈ [a, b] with x 6= t, we have

1 Γ(α + 1) x Z t H(α)(s)(ds)α = f (x) xrα − f (t) trα . (3)

On the other side, using the second and fifth items of Theorem 2.2, we obtain

H(α)(s) = f

(α)_(s)sα_{− r}α_{Γ(1 + α)f (s)}

s(r+1)α . (4)

From (3) and (4), we get

trαf (x) − xrαf (t) = xrαtrαΓ(α + 1) Γ(α + 1) x Z t f(α)(s)_Γ(1+α)sα − rα_{f (s)} s(r+1)α (ds) α_. ₍₅₎

Taking the modulus in (5), we have

|trαf (x) − xrαf (t)| (6) ≤ xrαtrαΓ(α + 1) Γ(α + 1) x Z t f(α)(s)l(s) − rαf (s) s(r+1)α (ds) α

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and therefore we get the inequality |trαf (x) − xrαf (t)| ≤ xrαtrαΓ(α + 1) Γ(α + 1) x Z t 1 s(r+1)α(ds) α sup s∈[x,t]([t,x]) f (α)_{(s)l(s) − r}α_{f (s)} .

Applying Lemma 2.4(ii), we can write

|trαf (x) − xrαf (t)| ≤ 1 |r|αx rα_trα lf (α)_{− r}α_f ∞ 1 trα − 1 xrα

which competes the proof.

Theorem 3.2. Let f : [a, b] → Rα _{be f ∈ D}

α(a, b) and f ∈ Cα[a, b] , b > a > 0. If r ∈ R,

r 6= 0, then for any x ∈ [a, b] , we have

|trαf (x) − xrαf (t)| ≤ x rα_trα_{Γ(α + 1)} minx(r+1)α_{, t}(r+1)α lf (α)_{− r}α_f ₁ where l(t) = _Γ(1+α)tα , t ∈ [a, b] , and lf(α)_{− r}α_f

1 is defined by lf (α)_{− r}α_f 1 = 1 Γ(α + 1) x Z t f (α)_{(s)l(s) − r}α_{f (s)} (ds) α_.

Proof. If we utilize the inequality (6), then we obtain the inequality

|trαf (x) − xrαf (t)| ≤ xrαtrαΓ(α + 1) Γ(α + 1) x Z t f(α)(s)l(s) − rαf (s) s(r+1)α (ds) α ≤ xrαtrαΓ(α + 1) 1 Γ(α + 1) x Z t f (α)_{(s)l(s) − r}α_{f (s)} (ds) α × sup s∈[x,t]([t,x]) 1 s(r+1)α = x rα_trα_{Γ(α + 1)} minx(r+1)α_{, t}(r+1)α lf (α)_{− r}α_f 1.

The proof is thus completed.

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r 6= 0 and r 6= −1_p, 1_p+ 1_q = 1 with p > 1, then for any x ∈ [a, b] , we have the inequality |trα_{f (x) − x}rα_{f (t)|} ≤ x rα_trα_{Γ(α + 1)}1p |1 − q (r + 1)|αq x (1−q(r+1))α_{− t}(1−q(r+1))α 1 q lf (α)_{− r}α_f p,

where l(t) = _Γ(1+α)tα , t ∈ [a, b] , and lf(α)− rα_f p is defined by lf (α)_{− r}α_f p=   1 Γ(α + 1) x Z t f (α)_{(s)l(s) − r}α_{f (s)} p (ds)α   1 p .

Proof. Utilizing the inequality (6) and H¨older’s integral inequality, we deduce

|trα_{f (x) − x}rα_{f (t)| ≤ x}rα_trαΓ(α + 1) Γ(α + 1) x Z t f(α)(s)l(s) − rαf (s) s(r+1)α (ds) α ≤ xrαtrαΓ(α + 1) 1 Γ(α + 1) x Z t 1 sq(r+1)α(ds) α 1 q × 1 Γ(α + 1) x Z t f (α)_{(s)l(s) − r}α_{f (s)} p (ds)α 1 p .

Afterwards, should we apply Lemma 2.4(ii), then we get the inequality |trα_{f (x) − x}rα_{f (t)|} ≤ x rα_trα_{Γ(α + 1)}1p |1 − q (r + 1)|αq lf (α)_{− r}α_f p x (1−q(r+1))α_{− t}(1−q(r+1))α 1 q ,

which completes the proof.

4. Generalized Ostrowski Type Results

We give several Ostrowski type inequalities involving local fractional integral.

r 6= 0 and r 6= −1, then for any x ∈ [a, b] , we have Γ(1 + rα) br+1− ar+1α Γ(1 + (r + 1) α) f (x) − x rα aIbαf (t) (7) ≤ lf(α)− rα_f _∞ |r|α × Mr(x), if r > 0 −Mr(x), if r ∈ (−∞, 0) \ {−1}

where l(t) = _Γ(1+α)tα , t ∈ [a, b] and lf(α)− rα_f

_∞ = sup t∈[a,b] f(α)(t)l(t) − rαf (t), and Mr(x) is defined by Mr(x) = Γ(1 + rα)2αx(r+1)α− ar+1_{+ b}r+1α Γ(1 + (r + 1) α) + 2αx(r+1)α− (a + b)α Γ(α + 1) .

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Proof. Integrating both sides of (2) with respect to t from a to b for local fractional integrals, we obtain Γ(1 + rα) br+1− ar+1α Γ(1 + (r + 1) α) f (x) − x rα aIbαf (t) ≤ lf(α)− rα_f _∞ |r|αΓ(α + 1) b Z a |xrα− trα| (dt)α.

Should we take r > 0, then we have

1 Γ(α + 1) b Z a |xrα− trα| (dt)α = 1 Γ(α + 1) x Z a (xrα− trα_{) (dt)}α₊ 1 Γ(α + 1) b Z x (trα− xrα_{) (dt)}α = Γ(1 + rα) Γ(1 + (r + 1) α) h ar+1+ br+1α − 2αx(r+1)αi+2 α_x(r+1)α_{− (a + b)}α Γ(α + 1) .

On the other side, if we take r ∈ (−∞, 0) \ {−1} , then we have the equality

1 Γ(α + 1) b Z a |xrα− trα| (dt)α = 1 Γ(α + 1) x Z a (trα− xrα) (dt)α+ 1 Γ(α + 1) b Z x (xrα− trα) (dt)α = Γ(1 + rα) Γ(1 + (r + 1) α) h 2αx(r+1)α− ar+1+ br+1α i +(a + b) α_{− 2}α_x(r+1)α Γ(α + 1) .

r 6= 0 and r 6= 1, then for any x ∈ [a, b] , we have f (x) xrα (b − a) α₋ aIbα f (t) trα ≤ 1 |r|α lf(α)− rαf _∞ Γ(α + 1) × Sr(x), if r ∈ (0, ∞) \ {1} −Sr(x), if r < 0

where l(t) = _Γ(1+α)tα , t ∈ [a, b] and lf(α)− rα_f

_∞= sup t∈[a,b] f(α)(t)l(t) − rαf (t) and Sr(x) is defined by Sr(x) = 2αx(1−r)α− a1−r_{+ b}1−rα Γ(α + 1) (1 − r)α + (a + b)α− 2α_xα Γ(α + 1)xrα .

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Proof. Dividing both sides of (2) with trαxrα and integrating over t ∈ [a, b] for local fractional integrals, we obtain

f (x) xrα (b − a) α₋ aIbα f (t) trα ≤ 1 |r|α lf(α)− rα_f _∞ Γ(α + 1) b Z a 1 trα − 1 xrα (dt)α. For r ∈ (0, ∞) \ {1} , we observe that

1 Γ(α + 1) b Z a 1 trα − 1 xrα (dt)α = 1 Γ(α + 1) x Z a 1 trα − 1 xrα (dt)α+ 1 Γ(α + 1) b Z x 1 xrα − 1 trα (dt)α. Also, using the Lemma 2.4(ii), we can write

1 Γ(α + 1) b Z a 1 trα − 1 xrα (dt)α = 2 α_x(1−r)α_{− a}1−r_{+ b}1−rα Γ(α + 1) (1 − r)α + (a + b)α− 2α_xα Γ(α + 1)xrα

for any x ∈ [a, b] .

On the other side, for r < 0, we also have

1 Γ(α + 1) b Z a 1 trα − 1 xrα (dt)α = a 1−r_{+ b}1−rα − 2α_x(1−r)α Γ(α + 1) (1 − r)α + 2αxα− (a + b)α Γ(α + 1)xrα

for any x ∈ [a, b] .

5. Applications For Some Special Means Let us recall some generalized means:

Aα(a, b) = aα+ bα 2α ; Ln(a, b) = " Γ (1 + nα) Γ (1 + (n + 1)α) " b(n+1)α_{− a}(n+1)α (b − a)α ##_n1 , n ∈ Z\ {−1, 0} , a, b ∈ R, a 6= b.

Now, let us reconsider the inequality (7): Γ(1 + rα) br+1− ar+1α Γ(1 + (r + 1) α) f (x) − x rα aIbαf (t) ≤ lf(α)− rαf _∞ |r|α × Mr(x), if r > 0 −Mr(x), if r ∈ (−∞, 0) \ {−1}

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where Mr(x) is defined by Mr(x) = Γ(1 + rα)2αx(r+1)α− ar+1_{+ b}r+1α Γ(1 + (r + 1) α) + 2αx(r+1)α− (a + b)α Γ(α + 1) .

Consider the mapping f : (0, ∞) → Rα_{, f (t) = t}nα_{, n ∈ Z\ {−1, 0} . Then, 0 < a < b,}

we have f a + b 2 = [Aα(a, b)]n and 1 (b − a)α aI α bf (t) = [Ln(a, b)]n.

Now, should we use the Lemma 2.4, we obtain

lf (α)_{− r}α_f _∞=        Γ(1+nα) Γ(1+α)Γ(1+(n−1)α) − r α b nα_, _{n > 1} Γ(1+nα) Γ(1+α)Γ(1+(n−1)α) − r α a nα_, _{n ∈ (−∞, 1]\ {−1, 0}}

and then we can write the inequality Γ(1 + rα) br+1− ar+1α Γ(1 + (r + 1) α) [Aα(a, b)] n_{− (b − a)}α_[A α(a, b)]r[Ln(a, b)]n ≤ 2 α_δ n(a, b) |r|α × Mr(x), if r > 0 −Mr(x), if r ∈ (−∞, 0) \ {−1}

where δn(a, b) is defined by

δn(a, b) =        Γ(1+nα) Γ(1+α)Γ(1+(n−1)α)− r α b nα_, _{n > 1} Γ(1+nα) Γ(1+α)Γ(1+(n−1)α)− r α a nα_, _{n ∈ (−∞, 1]\ {−1, 0}} and Mr(x) is defined as Mr(x) = Γ(1 + rα)h[Aα(a, b)](r+1)− Aα(ar+1, br+1) i Γ(1 + (r + 1) α) +[Aα(a, b)] (r+1)_{− A} α(a, b) Γ(α + 1) . References

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[2] Ahmad, F., Mir, N. A. and Sarikaya, M. Z., (2014) An inequality of Ostrowski type via variant of Pompeiu’s mean value theorem, J. Basic. Appl. Sci. Res, 4(4), pp. 204-211.

[3] Chen, G-S., (2013), Generalizations of H¨older’s and some related integral inequalities on fractal space. Journal of Function Spaces and Applications, Article ID 198405, 9 pages.

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Samet ERDEN graduated from Sakarya University, Sakarya, Turkey in 2010. He received his M.Sc. from Sakarya University in 2013. Also, he recieved PhD degree from DuzceUniversity in 2017. He worked a research assistant at Bartn University from 2012 to 2017. His research interests fo-cus on integral inequalities, fractional integrals and Momenths of random variable.

Mehmet Zeki SARIKAYA received his BSc (Maths), MSc (Maths) and PhD (Maths) degrees from Afyon Kocatepe University, Afyonkarahisar, Turkey in 2000, 2002 and 2007 respectively. At present, he is working as a professor in the Department of Mathematics at Duzce University (Turkey) and is the head of the department. Moreover, he is the founder and Editor-in-Chief of Konuralp Journal of Mathematics (KJM). He is the author or coau-thor of more than 200 papers in the field of theory of inequalities, potential theory, integral equations and transforms, special functions, time-scales.

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Silvestru Sever DRAGOMIR is the Chair of the international Research Group in Mathematical Inequalities and Applications (RGMIA) and the Ed-itor in Chief of the Australian Journal of Mathematical Analysis and Ap-plications (AJMAA). He is a member of the editorial boards of more than 30 international journals. As confirmed by the American Mathematical So-ciety database he is the author and editor of 20 books and more than 800 publications.

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