O R I G I N A L R E S E A R C H
On weighted Iyengar-type inequalities for conformable fractional
integrals
Mehmet Zeki Sarikaya1•Hatice Yaldiz1•Hu¨seyin Budak1
Received: 23 September 2016 / Accepted: 30 August 2017 / Published online: 11 September 2017 Ó The Author(s) 2017. This article is an open access publication
Abstract In this study, we establish some new weighted Iyengar-type integral inequalities using fractional Stef-fensen’s inequality for conformable fractional integral. The results presented here would provide generalizations of those given in earlier works.
Keywords Trapezoid inequality Iyengar inequality Weighted quadrature rule Conformable fractional integral Mathematics Subject Classification 26D15 26A33 26A42
Introduction
The presence of a differentiable mappings was introduced in 1938 for the first time and the results were announced by Iyengar. In recent years, Iyengar-type integral inequalities have been studied by several authors such as [5,6,9,10,12]. In this paper, we concentrate on Iyengar-type integral inequalities and present some generalizations. Section 2 consists of conformable integrals, derivatives, and some properties of them. Steffensen’s inequality for
conformable fractional integral is also given. Finally, in Sect. 3, my means of Steffensen’s inequality, we obtain some main interesting results. We first establish two new identities for conformable integral involving some moments. Generalizing this, using these equalities and considering fractional Steffensen’s inequality for con-formable fractional integrals, we get a new version of weighted Iyengar-type integral inequalities for con-formable integrals.
The results presented here will provide generalizations of those given in earlier works. Iyengar proved a useful inequality which gives an upper bound for the approxi-mation of the integral average by the mean of the values of mapping at end points of the interval that is given below (see [7] or [11, p.471]).
Theorem 1.1 Let f be a differentiable function on a; bð Þ and assume that there is a constant M[ 0, such that
f0ð Þx M, 8x 2 a; bð Þ. Then, we have Zb a f xð Þdx b að Þf að Þ þ f bð Þ 2 M bð aÞ2 4 1 4Mðf bð Þ f að ÞÞ 2 : ð1:1Þ
In [3], Agarwal and Dragomir proved the following generalization of Theorem1.1.
Theorem 1.2 Let f : I R ! R be a differentiable mapping on I (the interior of I), a; b½ I with a\b: Assume that M¼ supx2 a;b½ f
0 x ð Þ\1 and m¼ infx2 a;b½ f 0 x ð Þ [ 1: If M [ m and f0 is integrable on a; b
½ ; then the following inequality holds: & Hatice Yaldiz
yaldizhatice@gmail.com
Mehmet Zeki Sarikaya sarikayamz@gmail.com Hu¨seyin Budak hsyn.budak@gmail.com
1 Department of Mathematics, Faculty of Science and Arts,
Du¨zce University, Du¨zce, Turkey DOI 10.1007/s40096-017-0235-z
Zb a f xð Þdx b að Þf að Þ þ f bð Þ 2 ½M bð aÞ f bð Þ þ f að Þ f b½ ð Þ f að Þ m b að Þ 2 Mð mÞ :
Using a classical Steffensen’s inequality, Liu proved a generalization of the above result involving weighted integrals in terms of bounds involving the first derivative of the function.
The following Lemmas and Theorem was given by him in [9] (or [5]).
Lemma 1.1 Let f : I R ! R be a differentiable map-ping on I (the interior of I) and ½a; b I with M¼ supx2 a;b½ f 0 x ð Þ\1, m ¼ infx2 a;b½ f 0 x ð Þ [ 1, and M[ m. Assume that w xð Þ 0 for all x 2 a; b½ l ¼ Rb
aw xð Þdx\1 and t ¼ Rb
a xw xð Þdx\1 be the zeroth and first moments of wð Þ on a; b: ½ . If f0 is integrable on a; b½ , then the following inequality holds:
M m ½ Q½ b b kð ÞPb Zb a w xð Þf xð Þdx l f að ð Þ maÞ mt M m½ Q½ a k þ að ÞPaþ kl: ð1:2Þ where Pa¼R aþk a w xð Þdx, Qb¼ Rb bkw xð Þdx and k¼ ba Mm f bð Þf að Þ ba m .
Lemma 1.2 Let the conditions be as in Lemma1.1, then the following inequality holds:
M m ½ Q½ b k bð ÞPb kl Zb a w xð Þf xð Þdx l f bð ð Þ mbÞ mt M m½ Q½ a k þ að ÞPa: ð1:3Þ
Let the condition of Lemma1.1and1.2be maintained. Then, the following inequality holds:
M m ½ Qb b kð ÞPb k 2l Zb a w xð Þf xð Þdx l 2ðf að Þ þ f bð Þ m a þ bð ÞÞ mt M m½ Qa k þ að ÞPaþ k 2l : ð1:4Þ
Now, we will introduce the conformable integral and derivative:
Definition and properties of conformable
fractional derivative and integral
The following definitions and theorems with respect to conformable fractional derivative and integral were refer-red in (see, [1,2,4,8]).
Definition 2.1 (Conformable fractional derivative) Given a function f :½0; 1Þ ! R. Then, the ‘‘conformable frac-tional derivative’’ of f of order a is defined by
Dað Þ tf ð Þ ¼ lim e!0
f tð þ et1aÞ f tð Þ
e ð2:1Þ
for all t [ 0; a2 0; 1ð Þ:If f is a-differentiable in some 0; a
ð Þ; a [ 0; limt!0þfð Það Þ exist, then definet
fð Það Þ ¼ lim0 t!0þf
a
ð Þð Þ:t ð2:2Þ
We can write fð Það Þ for Dt að Þ tf ð Þ to denote the con-formable fractional derivatives of f of order a. In addition, if the conformable fractional derivative of f of order a exists, then we simply say f is a-differentiable.
Theorem 2.1 Let a2 0; 1ð and f and g be a-differen-tiable at a point t[ 0. Then
(i) Daðafþ bgÞ ¼ aDað Þ þ bDf að Þ; for all a; b 2 R;g (ii) Dað Þ ¼ 0;for all constant functions f tk ð Þ ¼ k; (iii) Dað Þ ¼ fDfg að Þ þ gDg að Þ;f (iv) Da f g ¼fDað Þ gDg að Þf g2 : If f is differentiable, then Dað Þ tf ð Þ ¼ t1a df dtð Þ:t ð2:3Þ
Definition 2.2 (Conformable fractional integral) Let a2 ð0; 1 and 0 a\b: A function f : ½a; b ! R is a-frac-tional integrable on [a, b] if the integral
Z b a f xð Þdax:¼ Z b a f xð Þxa1dx ð2:4Þ
exists and is finite. All a-fractional integrable on [a, b] is indicated by L1 að½a; bÞ: Remark 2.1 Iaað Þ tf ð Þ ¼ Ia 1 t a1f ¼ Z t a f xð Þ x1adx;
where the integral is the usual Riemann improper integral, and a2 ð0; 1.
Theorem 2.2 Let f :ða; bÞ ! R be differentiable and 0\a 1. Then, for all t [ a, we have
IaaDaaf tð Þ ¼ f tð Þ f að Þ: ð2:5Þ Theorem 2.3 (Integration by parts) Let f ; g :½a; b ! R be two functions, such that fg is differentiable. Then Z b a f xð ÞDa að Þ xg ð Þdax¼ fgjba Z b a g xð ÞDa að Þ xf ð Þdax: ð2:6Þ
The following theorem gave by Anderson [4].
Theorem 2.4 (Fractional Steffensen’s inequality) Let a2 ð0; 1 and a and b be real numbers, such that 0 a\b. Let h : ½a; b ! ½0; 1Þ and g : ½a; b ! 0; 1½ be a-fractional integrable functions on ½a; b with h is decreasing. Then Zb b‘ h xð Þdax Zb a h xð Þg xð Þdax Zaþ‘ a h xð Þdax; ð2:7Þ where‘ :¼a babðaaaÞ Rb ag xð Þdax.
Weighted trapezoidal inequality for conformable
fractional integrals
Some definitions are required to simplify the subsequent work.
Definition 3.1 Let w xð Þ be a positive conformable inte-grable function on a; b½ and a 2 ð0; 1. Let l and t be its zeroth and first moments about zero, so that
la¼1 a Zb a w xð Þdax\1; ð3:1Þ and ta¼ 1 a Zb a xaw xð Þdax\1: ð3:2Þ
Definition 3.2 P and Q will be used to denote the zeroth and first moments of w xð Þ over a subinterval a; b½ . In particular, for ‘ [ 0, the subscript a and b will be used to indicate the intervals a; a½ þ ‘ and b ‘; b½ , respectively. Thus, for example
Paða; ‘Þ :¼ 1 a Zaþl a w xð Þdax; ð3:3Þ Qaða; ‘Þ :¼ 1 a Zaþ‘ a xaw xð Þdax; ð3:4Þ Pað‘; bÞ :¼ 1 a Zb b‘ w xð Þdax; ð3:5Þ and Qað‘; bÞ :¼ 1 a Zb b‘ xaw xð Þdax: ð3:6Þ
Fractional Steffensen’s inequality (2.7) will now be used to obtain inequalities for conformable fractional integrals to give weighted trapezoidal-type quadrature rules. First, the following lemmas will need to be proved by the use of the Fractional Steffensen’s inequality.
Lemma 3.1 Let a2 0; 1ð and a; b 2 R with 0 a\b, f : a; b½ ! R be a conformable differentiable with M¼ supx2 a;b½ Daf xð Þ\1, m¼ infx2 a;b½ Daf xð Þ [ 1 and M[ m. Assume that w xð Þ 0 for all x 2 a; b½ and t¼1
a Rb
a x aw xð Þd
ax\1 be the first moment of w :ð Þ on a; b
½ . If Daf is a-fractional integrable on a; b½ ; then the following inequalities hold:
M m ½ Q½ að‘; bÞ b ‘ð ÞaPað‘; bÞ Zb a w xð Þf xð Þdax laðaf að Þ maaÞ mta M m½ Q½ aða; ‘Þ a þ ‘ð ÞaPaða; ‘Þ þ a þ ‘½ð Þaaala; ð3:7Þ where Pa and Qa are as described in Definition3.2. Proof Let hbð Þ ¼x
Rb
x w uð Þdau and g xð Þ ¼ Daf xð Þm
Mm : Then, from inequality (2.7), we get
Lb Ib Ub; ð3:8Þ where Lb ¼ M mð Þ Zb b‘ hbð Þdx ax; ð3:9Þ Ib¼ Zb a hbð Þ Dxð af xð Þ mÞdax; ð3:10Þ and Ub¼ M mð Þ Zaþ‘ a hbð Þdx ax: ð3:11Þ
Now, an integration by parts and change of order formula gives
Ib¼ Zb a Zb x w uð Þdau 0 @ 1 ADaf xð Þdax m Zb a Zb x w uð Þdau 0 @ 1 Adax ¼ Zb x w uð Þdau 0 @ 1 Af xð Þ b a þ Zb a w xð Þf xð Þdax m Zb a w uð Þ Zu a dau 0 @ 1 A ¼ laðaf að Þ maaÞ mtaþ Zb a w xð Þf xð Þdax: ð3:12Þ In addition ‘¼ a bð aÞ ba aa ð Þ M mð Þ Zb a Daf xð Þ m ð Þdax: ð3:13Þ
It should be noted that 0\‘ b a: h For the lower bound Lb, a change of order of integration gives Lb M m¼ Zb b‘ Zb x w uð Þdau 0 @ 1 Adax ¼ Qað‘; bÞ b ‘ð ÞaPað‘; bÞ; ð3:14Þ where Pað‘; bÞ and Qað‘; bÞ are described in Definition3.2. Similarly, the upper bound Ubmay be obtained through a change of order of integration to give
Ub M m¼ Zaþ‘ a Zb x w uð Þdau 0 @ 1 Adax ¼ Zb a Zb x w uð Þdaudax Zb aþ‘ Zb x w uð Þdaudax ¼ Zb a w uð Þ Zu a dax 0 @ 1 Adau Zb aþ‘ w uð Þ Zu aþ‘ dax 0 @ 1 Adau ¼ Qaða; ‘Þ a þ ‘ð ÞaPaða; ‘Þ þ a þ ‘½ð Þaaa la 2 ; ð3:15Þ where Paða; ‘Þ and Qaða; ‘Þ are described in Definition3.2, and ta is the zeroth moment of w xð Þ on a; b½ .
Using (3.8)–(3.15), the lemma is thus proved. h
Lemma 3.2 Let the conditions be as in Lemma3.1, then the following inequalities hold:
M m ½ Q½ að‘; bÞ b ‘ð ÞaPað‘; bÞ þ b ‘½ð Þabala Zb a w xð Þf xð Þdax laðaf bð Þ mbaÞ mta M m½ Q½ aða; ‘Þ a þ ‘ð ÞaPaða; ‘Þ: ð3:16Þ where Pa and Qa are as described in Definition3.2. Proof The proof follows along similar lines to that of Lemma 3.1. Let hað Þ ¼ x R
x
aw uð Þdau and g xð Þ ¼ Daf xð Þm
Mm : Then, from inequality (2.7)
La Ia Ua; ð3:17Þ where Ia¼ Zb a hað Þ Dxð af xð Þ mÞdax; La ¼ M mð Þ Zb b‘ hað Þdx ax and Ua¼ M mð Þ Zaþ‘ a hað Þdx ax:
Now, a straight forward integration by parts yields
Ia¼ Zb a hað Þ Dxð af xð Þ mÞdax ¼ Zb a w xð Þf xð Þdax laðaf bð Þ mbaÞ mta: ð3:18Þ
Furthermore, an interchange of the order of integration and simplification of results yields
La¼ M mð Þ Zb b‘ hað Þdx ax ¼ M mð Þ Q½ að‘; bÞ b ‘ð ÞaPað‘; bÞ þ b ‘½ð Þabala; ð3:19Þ and Ua ¼ M mð Þ Zaþ‘ a hað Þdx ax¼ Qaða; ‘Þ a þ ‘ð ÞaPaða; ‘Þ: ð3:20Þ
Hence, using (3.17)–(3.20), the lemma is thus proved. h Theorem 3.1 Let the condition of Lemmas3.1and3.2be maintained. Then, the following inequality holds:
M m ½ Qað‘; bÞ b ‘ð ÞaPað‘; bÞ þ b ‘½ð Þaba la 2 h i Zb a w xð Þf xð Þdax la a f að Þ þ f bð Þ 2 m aaþ ba 2 mta M m½ Qaða; ‘Þ a þ ‘ð ÞaPaða; ‘Þ þ a þ ‘½ð Þaaa la 2 h i ; ð3:21Þ where the Pas and Qas are as defined in Definition3.2. Proof Adding (3.7) and (3.16), and resulting inequalities divided by 2. This completes the proof. h Corollary 3.1 Let the conditions be as in the previous lemmas and theorem of this section. Then
Zb a w xð Þf xð Þdax la a f að Þ þ f bð Þ 2 m aaþ ba 2 mta b a aa 2 la:
Proof The corollary follows readily from (3.21) on noting that: Qað‘; bÞ ¼ 1 a Zb b‘ xaw xð Þdax b ‘ ð Þa a Zb b‘ w xð Þdax; Qaða; ‘Þ ¼ 1 a Zaþ‘ a xaw xð Þdax aþ ‘ ð Þa a Zaþ‘ a w xð Þdax; and b ‘ ð Þaba aa ba; a\b ‘\b; aþ ‘ ð Þaaa ba aa; a\aþ ‘\b:
Hence, this completes the proof. h
Remark 3.1 If we take a¼ 1 in Lemma 3.1, then inequality (3.7) reduces inequality (1.2).
Remark 3.2 If we take a¼ 1 in Lemma 3.2, then inequality (3.16) reduces inequality (1.3).
Remark 3.3 If we take a¼ 1 in Theorem 3.1, then inequality (3.21) reduces inequality (1.4).
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