In this work, firstly, we established Hermite-Hadamard’s inequalities for harmonically convex functions via Katugampola fractional integrals

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Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 20 (2019), No. 1, pp. 409–424 DOI: 10.18514/MMN.2019.2722

HERMITE-HADAMARD TYPE INEQUALITIES FOR

HARMONICALLY CONVEX FUNCTIONS VIA KATUGAMPOLA FRACTIONAL INTEGRALS

˙ILKER MUMCU, ERHAN SET, AND AHMET OCAK AKDEMIR Received 25 October, 2018

Abstract. In this work, firstly, we established Hermite-Hadamard’s inequalities for harmonically convex functions via Katugampola fractional integrals. Then we give some Hermite-Hadamard type inequalities of these classes functions.

2010Mathematics Subject Classification: 26A33; 26A51; 26D10

Keywords: Hermite-Hadamard inequality, Riemann-Liouville fractional integrals, Katugampola fractional integrals

1. INTRODUCTION AND PRELIMINARIES

We will start with a definition of mathematical analysis that has a high degree precedence for the inequality theory.

A function f WI R!Ris said to be convex if the inequality f .uC.1 / v/f .u/C.1 / f .v/

holds for allu; v2I and2Œ0; 1.

This definition has been used in the celebrated Hermite-Hadamard inequality;

Let f WIR!Rbe a convex function anda; b2I witha < b, then f

aCb 2

1

b a Z b

a

f .x/ dx f .a/Cf .b/

2 : (1.1)

In addition to giving upper and lower bounds for the mean value of a convex function, this double inequality has many applications.

Convexity plays an important role in different fields of pure and applied sciences.

In recent years we have noticed that theory of convexity developed rapidly. Con- sequently several new generalizations of convex functions have been proposed in the literature. Recently Is¸can [4] introduced the notion of harmonic convex function.

c 2019 Miskolc University Press

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Definition 1. LetI R=f0gbe a real interval. A functionf WI!Ris said to be harmonically convex, if

f

xy txC.1 t /y

tf .y/C.1 t /f .x/ (1.2) for allx; y2I andt2Œ0; 1.

The following theorem involve a different variant of Hadamard’s inequality for harmonically convex functions.

Theorem 1 ([4]). Let I R=f0g !R be a harmonically convex function and a; b2I witha < b. Iff 2LŒa; bthen the following inequalities hold.

f 2ab

aCb

ab b a

Z b a

f .x/

x² dx f .a/Cf .b/

2 :

To prove our results, we will use the following concepts and definitions.

The Beta function [11, p.18]:

B .a; b/D .a/ .b/

.aCb/ D Z 1

0

t^{a 1}.1 t /^{b 1}dt; a; b > 0;

where .˛/DR₁

0 e ^tu^{˛ 1}duis Gamma function.

The hypergeometric function [7]:

2F1.a; bIcI´/D 1 ˇ.b; c b/

Z 1 0

t^{b 1}.1 t /^{c b 1}.1 ´t / ^adt; c > b > 0; ´ < 1:

Lemma 1([10]). For0 < ˛1and0a < b, we have ja^˛ b^˛j .b a/^˛:

Definition 2. Letf 2L1Œa; b. The Riemann-Liouville integralsJ_a_Cf andJ_b f of order˛ > 0are defined by

J_a_Cf .x/D 1 ./

Z x a

.x t /¹f .t /dt; x > a and

J_b f .x/D 1 ./

Z b x

.t x/¹f .t /dt; x < b respectively where ./DR₁

0 e ^tu¹du:HereJ_a⁰_Cf .x/DJ_b⁰ f .x/Df .x/

In the case ofD1, the fractional integral reduces to classical integral.

The great impact of fractional calculus in pure and applied sciences can not be denied.

Resultantly many researchers used the techniques of fractional calculus intensively to get the new refinements of the previously known results. For example, we refer the reader to [1–3] and references cited therein. In [12], Sarıkaya et. al. proved a new

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version of Hermite-Hadamard’s inequalities in Riemann-Liouville fractional integral form as follows:

Theorem 2. Letf WŒa; b!Rbe a positive function with 0a < b andf 2 L1Œa; b:Iff is a convex function onŒa; b, then the following inequalities for fractional integrals holds:

f

aCb 2

.˛C1/

2.b a/^˛ŒJ_a^˛_Cf .b/CJ_b^˛ f .a/f .a/Cf .b/

2 (1.3)

with˛ > 0.

For further results related to Hermite-Hadamard type inequalities involving fractional integrals on can see [8,9,12–19].

In [5], Iscan et al. gave a generalization of (1.3) for harmonically convex functions as follows:

Theorem 3. Letf WI .0;1/!Rbe a function such thatf 2LŒa; b, where a; b2Iwitha < b. Iff is a harmonically convex function onŒa; b, then the following inequalities for fractional integrals hold:

f 2ab

aCb

.˛C1/

2

ab b a

˛

n

J_1=a^˛ .f ıg/.1=b/CJ_1=b^˛ _C.f ıg/.1=a/

o

f .a/Cf .b/

2 (1.4)

whereg.x/D1=x:

Katugampola gave a new fractional integral that generalizes the Riemann-Liouville and the Hadamard fractional integrals into a single form.

Definition 3([6]). LetŒa; bRbe a finite interval. Then, the left- and right-side Katugampola fractional integrals of order.˛ > 0/off 2X_c^p.a; b/are defined:

I_a^˛_Cf .x/D ^{1 ˛} .˛/

Z x a

t¹

.x t/^{1 ˛}f .t /dt and

I_b^˛ f .x/D ^{1 ˛} .˛/

Z b x

t¹

.t x/^{1 ˛}f .t /dt witha < x < band > 0, if the integral exist.

Theorem 4([6]). Let˛ > 0and > 0. Then forx > a, 1.lim!1I_a^˛_Cf .x/DJ_a^˛_Cf .x/,

2.lim_!₀CI^˛_a_Cf .x/DH_a^˛_Cf .x/.

Similar results also hold for right-sided operators.

The main purpose of this paper is to establish Hermite-Hadamard’s inequalities for harmonically convex functions via Katugampola fractional integral. We also obtain Hermite-Hadamard type inequalities of these classes functions.

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2. HERMITE-HADAMARD INEQUALITIES FOR HARMONICALLY CONVEXITY VIA

KATUGAMPOLA FRACTIONAL INTEGRALS

Consider the spaceX_c^p.a; b/ .c2R,1p 1/consist of those complex-valued Lebesque measurable functions' on.a; b/for whichk'kX_c^p <1, with

k'kXc^p D Z b

a jx^c'.x/j^pdx x

!1=p

.1p <1/ and

k'kXc^p Desssup_x₂_.a;b/Œx^cj'.x/j:

Hermite-Hadamard’s inequalities for harmonically convex functions can be repres- ented in Katugampola fractional integral forms as follows:

Theorem 5. Let˛ > 0and > 0. Letf WI .0;1/!Rbe a function such that f 2X_c^p.a; b/, wherea; b2I witha < b. Iff is a harmonically convex function onŒa; b, then the following inequalities hold:

f

2ab aCb

^˛ .˛C1/

2

ab b a

˛

nI_1=a^˛ .f ıg/.1=b/CI^˛_1=b_C.f ıg/.1=a/o f .a/Cf .b/

2 :

(2.1)

whereg.x/D1=x:

Proof. Lett2Œ0; 1. Considerx; y2Œa; b; a0, choosingxD_t_b_C^a_{.1 t}^b_/a, yD _t_a_C^a_{.1 t}^b _/b. Since f is harmonically convex function onŒa; b, and from definition, we can write

f

2xy xCy

f .x/Cf .y/ 2 Then we have

f

2ab aCb

f ._tbC^a.1 t^b/a/Cf ._taC^a.1 t^b /b/

2 (2.2)

Multiplying both sides of (2.2) byt^{˛ 1}, then integrating the resulting inequality with respect totoverŒ0; 1, we obtain

f

2ab aCb

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˛ 2

Z 1 0

t^{˛ 1}f

ab tbC.1 t/a

dtC

Z 1 0

t^{˛ 1}f

aa tbC.1 t/b

dt

D˛ 2

ab b a

˛ Z 1=a 1=b

x¹ x _b¹

^{1 ˛}f 1

x

dx C

Z 1=b 1=a

x¹

1

a x^{1 ˛}f 1

x

dx

D^˛ .˛C1/

2

ab b a

˛

nI_1=a^˛ .f ıg/.1=b/CI_1=b^˛ _C.f ıg/.1=a/o whereg.x/D1=x. So the first inequality is proved.

For the proof of the second inequality in (2.1), we first note that that for a harmonically convex functionf, we have

f

ab tbC.1 t/a

tf .a/C.1 t/b and

f

ab taC.1 t/b

tf .b/C.1 t/a: By adding these inequalities, we have

f

ab tbC.1 t/a

Cf

ab taC.1 t/b

f .a/Cf .b/: (2.3) Then multiplying both sides of (2.3) byt^{˛ 1}, and integrating the resulting inequality with respect totoverŒ0; 1, we get

Z 1 0

f

ab tbC.1 t/a

t^{˛ 1}dtC Z 1

0

f

ab taC.1 t/b

t^{˛ 1}dt Œf .a/Cf .b/

Z 1 0

t^{˛ 1}dt i.e.

^˛ .˛C1/

2

ab b a

˛

n

I^˛_1=a .f ıg/.1=b/CI_1=b^˛ _C.f ıg/.1=a/

o

f .a/Cf .b/

2 :

The proof is completed.

Remark1. In Theorem5, taking limit!1we obtain inequality of (1.4).

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3. HERMITE-HADAMARD TYPE INEQUALITIES FORKATUGAMPOLA FRACTIONAL INTEGRALS

Let f W I .0;1/!R be a differentiable function on I^ı, the interior of I, throughout this section we will take

If.gI˛; a; b/Df .a/Cb 2

^˛ .˛C1/

2

ab b a

˛

nI_1=a^˛ .f ıg/.1=b/CI_1=b^˛ _C.f ıg/.1=a/o

; wherea; b2I witha < b.g.x/D1=xand is Euler Gamma function.

Lemma 2. Let ˛ > 0 and > 0. Let f W I .0;1/!R be a differentiable function such thatf 2Xc^p.a; b/, wherea; b2I witha < b. Then the following equality holds:

If.gI˛; a; b/Dab.b a/ 2

Z 1 0

Œt^˛ .1 t/^˛t¹ ŒtaC.1 t/b²f⁰

ab taC.1 t/b

dt:

(3.1) Proof. LetAtDtaC.1 t/bandBtDtbC.1 t/a. It suffices to note that

I_f.gI˛; a; b/Dab.b a/ 2

Z 1 0

Œt^˛ .1 t/^˛t¹ ŒtaC.1 t/b²f⁰

ab taC.1 t/b

dt Dab.b a/

2

Z 1 0

t^˛t¹ A²_t f⁰

ab At

dt ab.b a/

2

Z 1 0

.1 t/^˛t¹ A²_t f⁰

ab At

dt

DI1CI2: (3.2)

By integrating by part, we get I1D1

2

"

t^˛f

ab At

ˇ ˇ ˇ ˇ

1 0

˛ Z 1

0

t^{˛ 1}f

ab At

dt

#

D1 2

"

f .b/ ˛

ab b a

˛Z 1=a 1=b

x¹ x _b¹

^{1 ˛}f 1

x

dx

#

D1 2

f .b/ ^˛ .˛C1/

ab b a

˛

I^˛_1=a .f ıg/.1=b/

(3.3)

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and similarly we get I2D 1

2

"

.1 t/^˛f

ab At

ˇ ˇ ˇ ˇ

1 0

C˛ Z 1

0

.1 t/^{˛ 1}t¹f

ab At

dt

#

D1 2

f .a/ ˛ Z 1

0

u^{˛ 1}f

ab Bt

du

D1 2

"

f .a/ ˛

ab b a

˛Z 1=b 1=a

x¹

1

a x^{1 ˛}f 1

x

dx

#

D1 2

f .a/ ^˛ .˛C1/

ab b a

˛

I^˛_1=b_C.f ıg/.1=a/

(3.4) Using (3.3) and (3.4) in (3.2), we get equality (3.1).

Remark2. In Lemma2, taking limit!1we obtain inequality Lemma 3 in [5].

Theorem 6. Let˛ > 0and > 0. Letf WI.0;1/!Rbe a differentiable function such thatf 2Xc^p.a; b/, wherea; b2Iwitha < b. Ifjfj^qis a harmonically convex function onŒa; bfor some fixedq1, then the following inequalities holds:

jI_f.gI˛; a; b/j ab.b a/

2 ^{1 1=q}₁ .˛Ia; b/

2.˛Ia; b/jf⁰.b/j^qC3.˛Ia; b/jf⁰.a/j^q1=q (3.5) where

1.˛Ia; b/D b ² .˛C1/

2F1

2; ˛C1I˛C2I1 a b

C²F1

2; 1I˛C2I1 a b

; 2.˛Ia; b/D b ²

.˛C2/

2F1

2; ˛C2I˛C3I1 a b

C 1

˛C1²F1

2; 2I˛C3I1 a b

; 3.˛Ia; b/D b ²

.˛C2/

1

˛C1²F1

2; ˛C1I˛C3I1 a b

C²F1

2; 1I˛C3I1 a b

:

Proof. Let At DtaC.1 t/b. From Lemma2, using the property of the modulus, the power mean inequality and the harmonically convexity ofjfj^q, we get

jI_f.gI˛; a; b/j

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ab.b a/ 2

Z 1 0

jt^˛ .1 t/^˛jjt¹j A²_t

ˇ ˇ ˇ ˇ

f⁰ ab

At

ˇ ˇ ˇ ˇ

dt ab.b a/

2

Z 1 0

jt^˛ .1 t/^˛jjt¹j

A²_t dt

^{1 1=q}

Z 1

0

ˇ ˇ ˇ ˇ

f⁰ ab

At

ˇ ˇ ˇ ˇ

dt ^1=q

ab.b a/ 2

Z 1 0

jt^˛C.1 t/^˛jjt¹j

A²_t dt

^{1 1=q}

Z 1

0

jt^˛C.1 t/^˛jjt¹j A²_t

tjf⁰.b/j^qC.1 t/jf⁰.a/j^q dt

^1=q

ab.b a/

2 ^{1 1=q}₁ .˛Ia; b/

2.˛Ia; b/jf⁰.b/j^qC3.˛Ia; b/jf⁰.a/j^q1=q

: (3.6) Calculating1.˛Ia; b/,2.˛Ia; b/and3.˛Ia; b/, we have

1.˛Ia; b/

D Z 1

0

Œt^˛C.1 t/^˛t¹

A²_t dt (3.7)

Db ² Z 1

0

.u^˛C.1 u/^˛/

1

1 a b

u

2

dt D b ²

.˛C1/

2F1

2; ˛C1I˛C2I1 a b

C²F1

2; 1I˛C2I1 a b

: Similarly, we get

2.˛Ia; b/

D Z 1

0

Œt^˛C.1 t/^˛t¹

A²_t tdt (3.8)

D b ² .˛C2/

2F1

2; ˛C2I˛C3I1 a b

C 1

˛C1²F1

2; 2I˛C3I1 a b

and

3.˛Ia; b/

D Z 1

0

Œt^˛C.1 t/^˛t¹

A²_t tdt (3.9)

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D b ² .˛C2/

1

˛C1²F1

2; ˛C1I˛C3I1 a b

C²F1

2; 1I˛C3I1 a b

So, if we use (3.7)-(3.9) in (3.6), we obtain the inequality of (3.5). This completes the proof.

Remark3. In Theorem6, taking limit!1we obtain Theorem 5 in [5].

Theorem 7. Let˛ > 0 and > 0. Let f WI .0;1/!Rbe a differentiable function such thatf 2Xc^p.a; b/, wherea; b2I witha < b. Ifjfj^l is a harmonically convex function onŒa; bfor some fixedl 1, then the following inequalities holds:

2 ^{1 1=q}₄ .˛Ia; b/

5.˛Ia; b/jf⁰.b/j^qC6.˛Ia; b/jf⁰.a/j^q1=q (3.10) where

4D b ² .˛C1/

2F1

2; ˛C1I˛C2I1 a b

2F1

2; 1I˛C2I1 a b

C²F1

2; 1I˛C2I1 2

1 a

b

5D b ² .˛C2/

2F1

2; ˛C2I˛C3I1 a b

1

˛C1²F1

2; 2I˛C3I1 a b

C 1

2.˛C1/²F1

2; 2I˛C3I1 2

1 a

b

6D b ² .˛C2/

1

˛C1²F1

2; ˛C1I˛C3I1 a b

2F1

2; 1I˛C3I1 a b

C²F1

2; 1I˛C3I1 2

1 a

b

:

Proof. Let At DtaC.1 t/b. From Lemma2, using the property of the modulus, the power mean inequality and the harmonically convexity of jfj^q, we have

jIf.gI˛; a; b/j ab.b a/

2

Z 1 0

ˇ ˇ ˇ ˇ

f⁰ ab

At

ˇ ˇ ˇ ˇ

dt ab.b a/

2

Z 1 0

A²_t dt

^{1 1=q}

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Z 1

0

ˇ ˇ ˇ ˇ

f⁰ ab

At

ˇ ˇ ˇ ˇ

dt ^1=q

ab.b a/ 2

Z 1 0

A²_t dt

^{1 1=q}

Z 1

0

tjf⁰.b/j^qC.1 t/jf⁰.a/j^q dt

^1=q

ab.b a/

2 K₁^{1 1=q}.˛Ia; b/

K2.˛Ia; b/jf⁰.b/j^qCK3.˛Ia; b/jf⁰.a/j^q1=q

: (3.11) CalculatingK1,K2andK3, by Lemma1, we get

K1D Z 1

0

A²_t dt

D Z 1=2

0

..1 t/^˛ t^˛/t¹ A²_t dtC

Z 1 1=2

.t^˛ .1 t/^˛/t¹ A²_t dt D

Z 1 0

.t^˛ .1 t/^˛/t¹ A²_t dtC2

Z 1=2 0

..1 t/^˛ t^˛/t¹ A²_t dt

Z 1 0

u^˛A_u²du Z 1

0

.1 u/^˛A_u²duC2 Z 1=2

0

.1 2u/^˛A_u²du D b ²

.˛C1/

2F1

2; ˛C1I˛C2I1 a b

2F1

2; 1I˛C2I1 a b

C²F1

2; 1I˛C2I1 2

1 a

b

: (3.12)

and similarly we obtain K2D

Z 1 0

jt^˛ .1 t/^˛jjt¹j A²_t tdt

Z 1 0

u^˛^C¹A_u²du Z 1

0

.1 u/^˛uA_u²duC2 Z 1=2

0

.1 2u/^˛uA_u²du D b ²

.˛C2/

2F1

2; ˛C2I˛C3I1 a b

1

˛C1²F1

2; 2I˛C3I1 a b

C 1

2.˛C1/²F1

2; 2I˛C3I1 2

1 a

b

; (3.13)

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and K3D

Z 1 0

A²_t .1 t /dt

Z 1 0

t^˛.1 u/A_u²du Z 1

0

.1 u/^˛^C¹A_u²duC2 Z 1=2

0

.1 2u/^˛.1 u/A_u²du D b ²

.˛C2/

1

˛C1²F1

2; ˛C1I˛C3I1 a b

2F1

2; 1I˛C3I1 a b

C²F1

2; 1I˛C3I1 2

1 a

b

: (3.14)

So, if we use (3.12)-(3.14) in (3.11), we get the inequality (3.10).

Theorem 8. Let˛ > 0 and > 0. Let f WI .0;1/!Rbe a differentiable function such thatf 2X_c^p.a; b/, wherea; b2I witha < b. Ifjfj^l is a harmonically convex function onŒa; bfor some fixedl > 1,1=kC1= lD1, then the following inequalities holds:

jI_f.gI˛; a; b/j Da.b a/ 2b

^1=k₇ C^1=k₈ jf⁰.a/j^lC jf⁰.b/j^l C1

!1= l

(3.15) where

7DB

k kC1

p ; ˛kC1

2F1

2k;k kC1

I˛kCkC1C1 k

I1 a b

8D 1

˛kCkC^{1 k} 2F1

2k; ˛kCkC1 k

I˛kCkC1C1 k

I1 a b

Proof. LetAt DtaC.1 t/b. From Lemma 2, H¨older inequality and the harmonically convexity ofjfj^q, we have

2

Z 1 0

t^˛t¹ A²_t

ˇ ˇ ˇ ˇ

f⁰ ab

At

ˇ ˇ ˇ ˇ

dtC Z 1

0

.1 t/^˛t¹ A²_t

ˇ ˇ ˇ ˇ

f⁰ ab

At

ˇ ˇ ˇ ˇ

dt

ab.b a/ 2

Z 1 0

t^˛kt^{k. 1/}

A^2k_t dt

!1=k

Z 1 0

ˇ ˇ ˇ ˇ

f⁰ ab A^2k_t

!ˇ ˇ ˇ ˇ

l!1= l

C Z 1

0

.1 t/^˛kt^{k. 1/}

A^2k_t dt

!1=k

Z 1 0

ˇ ˇ ˇ ˇ

f⁰ ab A^2k_t

!ˇ ˇ ˇ ˇ

l!1= l

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ab.b a/ 2

K₄^1=kCK₅^1=kZ 1 0

h

tjf⁰.b/j^lC.1 t/jf⁰.a/j^li dt

^{1= l}

ab.b a/ 2

K₄^1=kCK₅^1=k jf⁰.a/j^lC jf⁰.b/j^l C1

!1= l

: (3.16)

CalculatingK4andK5, we obtain K4D

Z 1 0

.1 t/^˛kt^{k. 1/}

A^2k_t dt D b ^2k

B.^{k k}_p^C¹; ˛kC1/²F1

2k;k kC1

I˛kCkC1C1 k

I1 a b

(3.17)

K5D Z 1

0

t^˛kt^{k. 1/}

A^2k_t dt D

˛kCkC1 k

b ^2k2F1

2k; ˛kCkC1 k

I˛kCkC1C1 k

I1 a b

(3.18) So, if we use (3.17) and (3.18) in (3.16), we get the inequality of (3.15). This com-

pletes the proof.

Theorem 9. Let˛ > 0 and > 0. Let f WI .0;1/!Rbe a differentiable function such that f 2X_c^p.a; b/, wherea; b 2I with a < b. Ifjfj^l is a harmonically convex function onŒa; b for some fixedl > 1, 1=kC1= l D1, then the following inequalities holds:

jI_f.gI˛; a; b/j ab.b a/ 2 ^1=k₉

10jf⁰.b/j^lC11jf⁰.a/j^l1= l

(3.19) where

9Db ^2k2F1

2k;1

IC1

I1 a b

(3.20) 10D 1

2^C1 B

C1

; ˛lC1

C˛lC1 2 ²F1

1

; 1I˛lC2I1 2

(3.21) 11D 1

2¹ B

1

; ˛lC1

2F1

1;1

I˛lC1 C1I1

2

C.˛lC1/.˛lC2/

2²

2F1

1

; 2I˛lC3I1 2

: (3.22)

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Proof. LetAt DtaC.1 t/b. From Lemma1, Lemma2, H¨older inequality and the harmonically convexity ofjfj^q, we have

jI_f.gI˛; a; b/j ab.b a/ 2

Z 1 0

ˇ ˇ ˇ ˇ

f⁰ ab

At

ˇ ˇ ˇ ˇ

dt

ab.b a/ 2

Z 1 0

1 A^2k_t dt

!1=k

Z 1

0 jt^˛ .1 t/^˛j^ljt¹j^l ˇ ˇ ˇ ˇ

f⁰ ab

At

ˇ ˇ ˇ ˇ

l

dt

!1= l

ab.b a/ 2

Z 1 0

1 A^2k_t dt

!1=k

Z 1

0 j1 2tj^˛lh

^{1= l}

ab.b a/ 2 K₆^1=k

K7jf⁰.b/j^lCK8jf⁰.a/j^l1= l

: (3.23) where

9D Z 1

0

1

A^2k_t dtDb ^2k2F1

2k;1

IC1

I1 a b

(3.24) 10D

Z 1

0 j1 2tj^˛ltdt D

Z 1=2¹⁼

0

.1 2t/^˛ltdtC Z 1

1=2¹⁼

.2t 1/^˛ltdt D 1

2^C1 B

C1

; ˛lC1

C˛lC1 2 ²F1

1

; 1I˛lC2I1 2

(3.25)

11D Z 1

0 j1 2tj^˛l.1 t/dt D

Z 1=2¹⁼

0

.1 2t/^˛l.1 t/dtC Z 1

1=2¹⁼

.2t 1/^˛l.1 t/dt D 1

2¹ B

1

; ˛lC1

2F1

1;1

I˛lC1 C1I1

2

C.˛lC1/.˛lC2/

2²

2F1

1

; 2I˛lC3I1 2

: (3.26)

So, if we use (3.24)-(3.26) in (3.23), we get desired result.

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Theorem 10. Let˛ > 0and > 0. Letf WI .0;1/!Rbe a differentiable function such thatf 2Xc^p.a; b/, wherea; b2I with a < b. If jfj^q is a harmonically convex function on Œa; bfor some fixed q > 1, 1=kC1= l D1, then the following inequalities holds:

jI_f.gI˛; a; b/j Dab.b a/ 2 ^1=k₁₂

₁₃jf⁰.b/j^lC₁₄jf⁰.a/j^l1= l

(3.27) where

12D 1

2^{.k k}^C^1/=B

k kC1

; ˛kC1

C 1 2²F1

kC k 1

; 1I˛kC2I1 2

13D 1

.C1/b^{2l 2}F1

2l;C1

I2C1

I1 a b

14D

.C1/b^{2l 2}F1

2l;1

I2C1

I1 a b

:

Proof. LetAt DtaC.1 t/b. From Lemma1, Lemma2, H¨older inequality and the harmonically convexity ofjfj^q, we have

jIf.gI˛; a; b/j ab.b a/ 2

Z 1 0

ˇ ˇ ˇ ˇ

f⁰ ab

At

ˇ ˇ ˇ ˇ

dt ab.b a/

2

Z 1

0 jt^˛ .1 t/^˛j^kjt¹j^kdt ^1=k

Z 1

0

1 A^2l_t

ˇ ˇ ˇ ˇ

f⁰ ab

At

ˇ ˇ ˇ ˇ

l

dt

!1= l

ab.b a/ 2

Z 1

0 j.2t 1/j^˛kt^{k. 1/}dt ^1=k

Z 1

0

1 A^2l_t

h

!1= l

ab.b a/ 2 K₆^1=k

K7jf⁰.b/j^lCK8jf⁰.a/j^l1= l

: (3.28) where

12D Z 1

0 j.2t 1/j^˛kt^{k. 1/}dt

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D

Z 1=2¹⁼

0

.1 2t/^˛kt^{k. 1/}dtC Z 1

1=2¹⁼

.2t 1/^˛kt^{k. 1/}dt

D 1

2^{.k k}^C^1/=B

k kC1

; ˛kC1

(3.29) C 1

2²F1

kC k 1

; 1I˛kC2I1 2

; (3.30)

13D Z 1

0

tA_t^2ldt

D 1

.C1/b^{2l 2}F₁

2l;C1

I2C1

I1 a b

(3.31) ₁₄D

Z 1 0

.1 t/A_t^2ldt

D

.C1/b^{2l 2}F1

2l;1

I2C1

I1 a b

(3.32) So, if we use (3.30)-(3.32) in (3.28), we obtain desired result.

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Authors’ addresses

˙Ilker Mumcu

Ordu University, Department of Mathematics, Faculty of Science and Arts, Ordu, Turkey E-mail address:mumcuilker@msn.com

Erhan Set

Ordu University, Department of Mathematics, Faculty of Science and Arts, Ordu, Turkey E-mail address:erhanset@yahoo.com

Ahmet Ocak Akdemir

Aˇgrı ˙Ibrahim C¸ ec¸en, Department of Mathematics, Faculty of Science and Arts, Aˇgrı, Turkey E-mail address:ahmetakdemir@agri.edu.tr