inequality; Riemann-Liouville fractional integral; fractional integral operator; ... by introducing a number of integral inequalities involving various fractional ...
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466
Applications and Applied Mathematics: An International Journal (AAM)
Vol. 12, Issue 2 (December 2017), pp. 1017 - 1035
Ostrowski type fractional integral operators for generalized (๐; ๐, ๐, ๐) โpreinvex functions ๐
๐. ๐๐๐ฌ๐ก๐ฎ๐ซ๐ข and ๐๐. ๐๐ข๐ค๐จ
Faculty of Technical Science Ismail Qemali University Albania 1 artionkashuri@gmail.com; 2rozanaliko86@gmail.com Received: August 4, 2017; Accepted: October 12, 2017
Abstract In the present paper, the notion of generalized (๐; ๐ , ๐, ๐) โpreinvex function is applied to establish some new generalizations of Ostrowski type inequalities via fractional integral operators. These results not only extend the results appeared in the literature but also provide new estimates on these type.
Keywords: Ostrowski type inequality; Hรถlder's inequality; Minkowski's inequality; power mean inequality; Riemann-Liouville fractional integral; fractional integral operator; s โconvex function in the second sense; m โinvex
MSC 2010 No.: 26A33, 26A51, 26D07, 26D10, 26D15
1. Introduction The following notations are used throughout this paper. We use ๐ผ to denote an interval on the real line โ = (โโ, +โ) and ๐ผ โ to denote the interior of ๐ผ. For any subset ๐พ โ โ๐ , ๐พ โ is used to denote the interior of ๐พ. โ๐ is used to denote a ๐ โdimensional vector space. The set of integrable functions on the interval [๐, ๐] is denoted by ๐ฟ1 [๐, ๐]. The following result is known in the literature as the Ostrowski inequality (Liu et al., 2015), which ๐ 1 gives an upper bound for the approximation of the integral average ๐โ๐ โซ๐ ๐(๐ก)๐๐ก by the value ๐(๐ฅ) at point ๐ฅ โ [๐, ๐].
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Theorem 1. Let ๐: ๐ผ โถ โ be a mapping differentiable on ๐ผ โ and let ๐, ๐ โ ๐ผ โ with ๐ < ๐. If |๐ โฒ (๐ฅ)| โค ๐ for all ๐ฅ โ [๐, ๐], then ๐+๐ 2 1 1 (๐ฅ โ 2 ) |๐(๐ฅ) โ โซ ๐(๐ก)๐๐ก| โค ๐(๐ โ ๐) [ + ] , โ ๐ฅ โ [๐, ๐]. (๐ โ ๐)2 ๐โ๐ 4 ๐
(1)
๐
For other recent results concerning Ostrowski type inequalities, see ((Agarval et al., 2016)-(Alomari et al., 2010); (Dragomir et al., 1997)-(Dragomir, 2001); Kashuri et al., 2016; Kashuri et al., 2017; Liu, 2007; Liu, 2009; (รzdemir et al., 2010)-(Pachpatte, 2001); Rafiq et al., 2007; Sarikaya, 2010; Tunรง, 2014; Ujeviฤ, 2004; Yildiz et al., 2016; Zhongxue, 2008). Ostrowski inequality is playing a very important role in all the fields of mathematics, especially in the theory of approximations. Thus such inequalities were studied extensively by many researches and numerous generalizations, extensions and variants of them for various kind of functions like bounded variation, synchronous, Lipschitzian, monotonic, absolutely, continuous and ๐ โtimes differentiable mappings etc. appeared in a number of papers. In recent years, one more dimension has been added to this studies, by introducing a number of integral inequalities involving various fractional operators like Riemann-Liouville, Erdelyi-Kober, Katugampola, conformable fractional integral operators etc. by many authors see (Abdeljawad, 2015; Katugampola, 2014; Khalil et al., 2014; Purohit et al., 2014; Set et al., 2017). Riemann-Liouville fractional integral operators are the most central between these fractional operators. Fractional calculus see ((Chu et al., 2017)-(Dahmani et al., 2010); Kashuri et al., 2017; Raina, 2005), was introduced at the end of the nineteenth century by Liouville and Riemann, the subject of which has become a rapidly growing area and has found applications in diverse fields ranging from physical sciences and engineering to biological sciences and economics. Definition 1. ๐ผ ๐ผ Let ๐ โ ๐ฟ1 [๐, ๐]. The Riemann-Liouville integrals ๐ฝ๐+ ๐ and ๐ฝ๐โ ๐ of order ๐ผ > 0 with ๐ โฅ 0 are defined by ๐ฅ
1 ๐ผ ๐ฝ๐+ ๐(๐ฅ) = โซ(๐ฅ โ ๐ก)๐ผโ1 ๐(๐ก)๐๐ก, ๐ฅ > ๐ ๐ค(๐ผ) ๐
and ๐
๐ผ ๐ฝ๐โ ๐(๐ฅ)
1 = โซ(๐ก โ ๐ฅ)๐ผโ1 ๐(๐ก)๐๐ก, ๐ > ๐ฅ, ๐ค(๐ผ) ๐ฅ
+โ
0 0 where ๐ค(๐ผ) = โซ0 ๐ โ๐ข ๐ข๐ผโ1 ๐๐ข. Here ๐ฝ๐+ ๐(๐ฅ) = ๐ฝ๐โ ๐(๐ฅ) = ๐(๐ฅ). In the case of ๐ผ = 1, the fractional integral reduces to the classical integral.
Due to the wide application of fractional integrals, some authors extended to study fractional Ostrowski type inequalities for functions of different classes see (Liu et al., 2016). In see (Raina, 2005), Raina introduced a class of functions defined formally by
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+โ ๐ (๐ฅ) = โฑ๐,๐
๐(0),๐(1),โฆ (๐ฅ) โฑ๐,๐
=โ ๐=0
๐(๐) ๐ฅ๐ ๐ค(๐๐ + ๐)
(๐, ๐ > 0; |๐ฅ| < โ),
(2)
where the coefficients (๐(๐), ๐ โ โ โช {0}) is a bounded sequence of positive real numbers. With the help of (2), Raina see (Raina, 2005) and see (Agarwal et al., 2016) defined the following leftsided and right-sided fractional integral operators respectively, as follows: ๐ฅ ๐ (๐ฅ๐,๐,๐+;๐ ๐)(๐ฅ)
๐ [๐(๐ฅ โ ๐ก)๐ ]๐(๐ก)๐๐ก, (๐ฅ > ๐ > 0), = โซ(๐ฅ โ ๐ก)๐โ1 โฑ๐,๐
(3)
๐ ๐ ๐ (๐ฅ๐,๐,๐โ;๐ ๐)(๐ฅ)
๐ [๐(๐ก โ ๐ฅ)๐ ]๐(๐ก)๐๐ก, (0 < ๐ฅ < ๐), = โซ(๐ก โ ๐ฅ)๐โ1 โฑ๐,๐
(4)
๐ฅ
where ๐, ๐ > 0, ๐ โ โ and ๐(๐ก) is such that the integral on the right side exits. It is easy to verify ๐ ๐ that (๐ฅ๐,๐,๐+;๐ ๐)(๐ฅ) and (๐ฅ๐,๐,๐โ;๐ ๐)(๐ฅ) are bounded integral operators on ๐ฟ1 [๐, ๐], if ๐ [๐(๐ โ ๐)๐ ] < +โ. โ โ โฑ๐,๐+1
In fact, for ๐ โ ๐ฟ1 (๐, ๐), we have ๐ โ๐ฅ๐,๐,๐+;๐ ๐(๐ฅ)โ โค โ(๐ โ ๐)๐ โ๐โ1 1
and ๐ โ๐ฅ๐,๐,๐โ;๐ ๐(๐ฅ)โ โค โ(๐ โ ๐)๐ โ๐โ1 , 1
where ๐
1 ๐
โ๐โ๐ โ (โซ|๐(๐ก)|๐ ๐๐ก) . ๐
The importance of these operators stems indeed from their generality. Many useful fractional integral operators can be obtained by specializing the coefficient ๐(๐). For instance the classical ๐ผ ๐ผ Riemann-Liouville fractional integrals ๐ฝ๐+ and ๐ฝ๐โ of order ฮฑ follow easily by setting ๐ = ๐ผ, ๐(0) = 1 and ๐ = 0 in (3) and (4). Now, let us evoke some definitions. Definition 2. (Hudzik et al., 1994) A function ๐: [0, +โ[ โถ โ is said to be ๐ โconvex in the second sense, if ๐(๐๐ฅ + (1 โ ๐)๐ฆ) โค ๐๐ ๐(๐ฅ) + (1 โ ๐)๐ ๐(๐ฆ),
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for all ๐ฅ, ๐ฆ โฅ 0, ๐ โ [0,1] and ๐ โ ]0,1]. It is clear that a 1โconvex function must be convex on [0, +โ[ as usual. The ๐ โconvex functions in the second sense have been investigated in see (Hudzik et al., 1994). Definition 3. (Antczak, 2005) A set ๐พ โ โ๐ is said to be invex with respect to the mapping ๐: ๐พ ร ๐พ โถ โ๐ , if ๐ฅ + ๐ก๐(๐ฆ, ๐ฅ) โ ๐พ for every ๐ฅ, ๐ฆ โ ๐พ and ๐ก โ [0,1]. Notice that every convex set is invex with respect to the mapping ๐(๐ฆ, ๐ฅ) = ๐ฆ โ ๐ฅ, but the converse is not necessarily true see (Antczak, 2005; Yang et al., 2003). Definition 4. (Pini, 1991) A function ๐ defined on the invex set ๐พ โ โ๐ is said to be preinvex with respect ๐, if for every ๐ฅ, ๐ฆ โ ๐พ and ๐ก โ [0,1], we have that ๐(๐ฅ + ๐ก๐(๐ฆ, ๐ฅ)) โค (1 โ ๐ก)๐(๐ฅ) + ๐ก๐(๐ฆ). The concept of preinvexity is more general than convexity since every convex function is preinvex with respect to the mapping ๐(๐ฆ, ๐ฅ) = ๐ฆ โ ๐ฅ, but the converse is not true. The aim of this paper is to establish some generalizations of Ostrowski type inequalities using new identity given in Section 2 for generalized (๐; ๐ , ๐, ๐) โpreinvex functions via generalized fractional integral operators. In Section 3, some conclusions and future research are given. These results not only extend the results appeared in the literature see (Yildiz et al., 2016) but also provide new estimates on these type.
2. Main Results Definition 5. (Du et al., 2016) A set ๐พ โ โ๐ is said to be ๐ โinvex with respect to the mapping ๐: ๐พ ร ๐พ ร ]0,1] โถ โ๐ , for some fixed ๐ โ ]0,1], if ๐๐ฅ + ๐ก๐(๐ฆ, ๐๐ฅ) โ ๐พ holds for each ๐ฅ, ๐ฆ โ ๐พ and any t โ [0,1]. Remark 1. In Definition 5, under certain conditions, the mapping ๐(๐ฆ, ๐๐ฅ) could reduce to ๐(๐ฆ, ๐ฅ). For example when ๐ = 1, then the ๐ โinvex set degenerates an invex set on ๐พ. We next recall the definition of generalized (๐; ๐ , ๐, ๐) โpreinvex function. Definition 6. (Kashuri et al., 2017) Let ๐พ โ โ be an open ๐ โinvex set with respect to the mapping ๐: ๐พ ร ๐พ ร ]0,1] โถ โ and ๐: ๐ผ โถ ๐พ is a continuous function. The function ๐: ๐พ โถ (0, +โ) is said to be generalized (๐; ๐ , ๐, ๐) โpreinvex with respect to ๐, if
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๐(๐๐(๐ฅ) + ๐ก๐(๐(๐ฆ), ๐(๐ฅ), ๐)) โค ๐๐ (๐(๐(๐ฅ)), ๐(๐(๐ฆ)), ๐, ๐ ; ๐ก) holds for some fixed ๐ , ๐ โ ]0,1] and for all ๐ฅ, ๐ฆ โ ๐ผ, ๐ก โ [0,1], where ๐ ๐
๐ ๐
1 (๐(๐ฆ))]๐ ,
[๐(1 โ ๐ก) ๐ (๐(๐ฅ)) + ๐ก ๐ ๐๐ (๐(๐(๐ฅ)), ๐(๐(๐ฆ)), ๐, ๐ ; ๐ก) = { ๐(1โ๐ก)๐ ๐ก๐ [๐ (๐(๐ฅ))] [๐ (๐(๐ฆ))] ,
๐ โ 0; ๐ = 0,
is the weighted power mean of order ๐ for positive numbers ๐ (๐(๐ฅ)) and ๐ (๐(๐ฆ)). Remark 2. In Definition 6, it is worthwhile to note that the class of generalized (๐; ๐ , ๐, ๐) โpreinvex function is a generalization of the class of ๐ โconvex in the second sense function given in Definition 2. For ๐ = 1, we get the notion of generalized (๐ , ๐, ๐) โpreinvex function see (Kashuri et al., 2016). Also, for ๐ = 1 and ๐(๐ฅ) = ๐ฅ, โ๐ฅ โ ๐ผ, we get the notion of generalized (๐ , ๐) โpreinvex function see (Du et al., 2016). Throughout this paper we denote ๐
๐
๐ (๐ฅโ๐๐(๐)) โฑ๐,๐+1 [๐(๐ฅโ๐๐(๐)) ]
๐ผ๐,๐,๐ (๐ฅ; ๐, ๐, ๐, ๐, ๐, ๐) = [
+[
๐ ๐+1 (๐(๐),๐(๐),๐)
] ๐(๐ฅ)
๐ (๐๐(๐) + ๐ (๐(๐), ๐(๐), ๐) โ ๐ฅ)๐ โฑ๐,๐+1 [๐(๐๐(๐) + ๐ (๐(๐), ๐(๐), ๐) โ ๐ฅ)๐ ]
๐ ๐+1 (๐(๐), ๐(๐), ๐)
โ
] ๐(๐ฅ)
๐ ๐ ๐+1 (๐(๐), ๐(๐), ๐)
๐ ๐ ร [(๐ฅ๐,๐,๐ฅโ;๐ ๐)(๐๐(๐)) + (๐ฅ๐,๐,๐ฅ+;๐ ๐)(๐๐(๐) + ๐ (๐(๐), ๐(๐), ๐))].
In this section, in order to prove our main results regarding some generalizations of Ostrowski type inequalities for generalized (๐; ๐ , ๐, ๐) โpreinvex functions via generalized fractional integral operators, we need the following new interesting lemma: Lemma 1. Let ๐: ๐ผ โถ ๐พ be a continuous function. Suppose ๐พ โ โ be an open ๐ โinvex subset with respect to the mapping ๐: ๐พ ร ๐พ ร ]0,1] โถ โ for some fixed ๐ โ ]0,1] and ๐ (๐(๐), ๐(๐), ๐) > 0. Assume that ๐: ๐พ โถ โ is a differentiable function on ๐พ โ . If ๐ โฒ โ ๐ฟ1 [๐๐(๐), ๐๐(๐) + ๐ (๐(๐), ๐(๐), ๐)], then we have the following identity involving generalized fractional integral operators: 1
๐ผ๐,๐,๐ (๐ฅ; ๐, ๐, ๐, ๐, ๐, ๐) = โซ ๐(๐ก) ๐ โฒ (๐๐(๐) + ๐ก๐ (๐(๐), ๐(๐), ๐))๐๐ก, 0
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for each ๐ก โ [0,1], where ๐, ๐ > 0, ๐ โ โ and ๐ฅ โ ๐๐(๐) [; ๐ (๐(๐), ๐(๐), ๐) ๐(๐ก) = ๐ฅ โ ๐๐(๐) ๐ (1 โ ๐ก)๐ โฑ๐,๐+1 [๐๐๐ (๐(๐), ๐(๐), ๐)(1 โ ๐ก)๐ ], ๐ก โ [ , 1] . ๐ (๐(๐), ๐(๐), ๐) { ๐ [๐๐๐ (๐(๐), ๐(๐), ๐)๐ก๐ ], ๐ก ๐ โฑ๐,๐+1
๐ก โ [0,
Proof: Integrating by parts, we get 1
โซ ๐(๐ก) ๐ โฒ (๐๐(๐) + ๐ก๐ (๐(๐), ๐(๐), ๐))๐๐ก 0 ๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
=
๐ [๐๐๐ (๐(๐), ๐(๐), ๐)๐ก๐ ] ๐ โฒ (๐๐(๐) + ๐ก๐ (๐(๐), ๐(๐), ๐))๐๐ก ๐ก ๐ โฑ๐,๐+1
โซ 0 1
+
๐ (1 โ ๐ก)๐ โฑ๐,๐+1 [๐๐๐ (๐(๐), ๐(๐), ๐)(1 โ ๐ก)๐ ] ๐ โฒ (๐๐(๐) + ๐ก๐ (๐(๐), ๐(๐), ๐))๐๐ก
โซ ๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
๐ฅโ๐๐(๐)
๐ (๐๐(๐) + ๐ก๐ (๐(๐), ๐(๐), ๐)) ๐ (๐(๐),๐(๐),๐) ๐ [๐๐๐ (๐(๐), ๐(๐), ๐)๐ก๐ ] = ๐ก ๐ โฑ๐,๐+1 | ๐ (๐(๐), ๐(๐), ๐) 0 ๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
โ๐
๐ [๐๐๐ (๐(๐), ๐(๐), ๐)๐ก๐ ] ๐ก ๐โ1 โฑ๐,๐
โซ 0
๐ (๐๐(๐) + ๐ก๐ (๐(๐), ๐(๐), ๐)) ๐๐ก ๐ (๐(๐), ๐(๐), ๐)
1
+(1 โ ๐ก)
๐
๐ [๐๐๐ (๐(๐), ๐(๐), ๐)(1 โฑ๐,๐+1
โ ๐ก)
๐]
๐ (๐๐(๐) + ๐ก๐ (๐(๐), ๐(๐), ๐)) | ๐ (๐(๐), ๐(๐), ๐)
๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
1
โ๐
(1
โซ ๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
๐ [๐๐๐ (๐(๐), ๐(๐), ๐)(1 โ ๐ก)๐ ] โ ๐ก)๐โ1 โฑ๐,๐ ๐
๐ (๐๐(๐) + ๐ก๐ (๐(๐), ๐(๐), ๐)) ๐๐ก ๐ (๐(๐), ๐(๐), ๐) ๐
๐ (๐ฅ โ ๐๐(๐)) โฑ๐,๐+1 [๐(๐ฅ โ ๐๐(๐)) ] =[ ] ๐(๐ฅ) ๐ ๐+1 (๐(๐), ๐(๐), ๐)
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+[
๐ (๐๐(๐) + ๐ (๐(๐), ๐(๐), ๐) โ ๐ฅ)๐ โฑ๐,๐+1 [๐(๐๐(๐) + ๐ (๐(๐), ๐(๐), ๐) โ ๐ฅ)๐ ]
] ๐(๐ฅ)
๐ ๐+1 (๐(๐), ๐(๐), ๐) โ
๐ ๐ ๐+1 (๐(๐), ๐(๐), ๐)
๐ ๐ ร [(๐ฅ๐,๐,๐ฅโ;๐ ๐)(๐๐(๐)) + (๐ฅ๐,๐,๐ฅ+;๐ ๐)(๐๐(๐) + ๐ (๐(๐), ๐(๐), ๐))].
By using Lemma 1, one can extend to the following results. Theorem 2. Let ๐: ๐ผ โถ ๐ด be a continuous function. Suppose ๐ด โ โ be an open ๐ โinvex subset with respect to the mapping ๐: ๐ด ร ๐ด ร ]0,1] โถ โ for some fixed ๐ , ๐ โ ]0,1] and ๐ (๐(๐), ๐(๐), ๐) > 0. Assume that ๐: ๐ด โถ (0, +โ) is a differentiable function on ๐ดโ . If 0 < ๐ โค 1 and ๐ โฒ is generalized (๐; ๐ , ๐, ๐) โpreinvex function on [๐๐(๐), ๐๐(๐) + ๐ (๐(๐), ๐(๐), ๐)], then the following inequality for generalized fractional integral operators holds: |๐ผ๐,๐,๐ (๐ฅ; ๐, ๐, ๐, ๐, ๐, ๐)| ๐
๐1 [|๐|๐๐ (๐(๐), ๐(๐), ๐)]) ๐๐ (๐(๐)) (โฑ๐,๐+1 { ๐ ๐ ๐ ๐2 +๐ โฒ (๐(๐)) (โฑ๐,๐+1 [|๐|(๐ฅ โ ๐๐(๐)) ]) โฒ
โค
+{
๐๐ โฒ (๐(๐))
๐
๐3 [|๐|(๐๐(๐) (โฑ๐,๐+1 ๐
1 ๐ ๐
} 1 ๐ ๐
+ ๐ (๐(๐), ๐(๐), ๐) โ ๐ฅ)๐ ])
๐
4 [|๐|๐๐ (๐(๐), ๐(๐), ๐)]) +๐ โฒ (๐(๐)) (โฑ๐,๐+1
๐
where ๐, ๐ > 0, ๐ โ โ, ๐ = 0,1,2, โฆ , ๐ฝ(๐ฅ; ๐, ๐) is incompleted beta function and ๐ฅ โ ๐๐(๐) ๐ ๐1 (๐) = ๐ (๐)๐ฝ ( ; ๐ + ๐๐ + 1, + 1) ; ๐ (๐(๐), ๐(๐), ๐) ๐ ๐ฅ โ ๐๐(๐) ๐2 (๐) = ๐ (๐) ( ) ๐ (๐(๐), ๐(๐), ๐)
๐ ๐+ +1 ๐
1
; ๐ ๐ + ๐๐ + ๐ + 1
๐๐(๐) + ๐ (๐(๐), ๐(๐), ๐) โ ๐ฅ ๐3 (๐) = ๐ (๐) ( ) ๐ (๐(๐), ๐(๐), ๐)
๐ ๐+ +1 ๐
1
; ๐ ๐ + ๐๐ + ๐ + 1
๐๐(๐) + ๐ (๐(๐), ๐(๐), ๐) โ ๐ฅ ๐ ๐4 (๐) = ๐ (๐)๐ฝ ( ; ๐ + ๐๐ + 1, + 1). ๐ (๐(๐), ๐(๐), ๐) ๐
} ,
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A. Kashuri and R. Liko
Proof: Let 0 < ๐ โค 1. From Lemma 1, generalized (๐; ๐ , ๐, ๐) โpreinvexity of ๐ โฒ , Minkowski inequality and properties of the modulus, we have
|๐ผ๐,๐,๐ (๐ฅ; ๐, ๐, ๐, ๐, ๐, ๐)| ๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
โค
๐ [|๐|๐๐ (๐(๐), ๐(๐), ๐)๐ก๐ ] |๐ โฒ (๐๐(๐) + ๐ก๐ (๐(๐), ๐(๐), ๐))|๐๐ก ๐ก ๐ โฑ๐,๐+1
โซ 0
1
+
๐ |1 โ ๐ก|๐ โฑ๐,๐+1 [|๐|๐๐ (๐(๐), ๐(๐), ๐)(1 โ ๐ก)๐ ] |๐ โฒ (๐๐(๐)
โซ ๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
+ ๐ก๐ (๐(๐), ๐(๐), ๐))|๐๐ก ๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
โค
โซ
๐
๐
0 1
+
๐ (1 โ ๐ก)๐ โฑ๐,๐+1 [๐๐๐ (๐(๐), ๐(๐), ๐)(1 โ ๐ก)๐ ]
โซ ๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
๐
๐ โฒ
๐ โฒ
1 ๐ ๐
ร [๐(1 โ ๐ก) ๐ (๐(๐)) + ๐ก ๐ (๐(๐)) ] ๐๐ก
๐๐ โฒ (๐(๐))
๐
1 ๐
๐
๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐) ๐
๐ [|๐|๐๐ (๐(๐), ๐(๐), ๐)๐ก๐ ]๐๐ก ๐ก ๐ (1 โ ๐ก)๐ โฑ๐,๐+1
โซ 0
(
โค
๐
๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
+๐ โฒ (๐(๐)) {
1
๐ [|๐|๐๐ (๐(๐), ๐(๐), ๐)๐ก๐ ] [๐(1 โ ๐ก) ๐ ๐ โฒ (๐(๐)) + ๐ก ๐ ๐ โฒ (๐(๐)) ]๐ ๐๐ก ๐ก ๐ โฑ๐,๐+1
๐
โซ
๐
๐ [|๐|๐๐ (๐(๐), ๐(๐), ๐)๐ก๐ ]๐๐ก ๐ก ๐+๐ โฑ๐,๐+1
0
(
)
)
}
1025
AAM: Intern. J., Vol 12, Issue 2 (December 2017)
๐ 1
๐๐ โฒ (๐(๐))
๐
๐
๐ (1 โ ๐ก)๐+๐ โฑ๐,๐+1 [|๐|๐๐ (๐(๐), ๐(๐), ๐)(1 โ ๐ก)๐ ]๐๐ก
โซ ๐ฅโ๐๐(๐) (๐(๐),๐(๐),๐) ๐ (
+
1
+๐ โฒ (๐(๐))
๐
) ๐
+{
) }
1 ๐ ๐ ๐1 ๐ ๐๐ (๐(๐)) (โฑ๐,๐+1 [|๐|๐ (๐(๐), ๐(๐), ๐)]) { } ๐ ๐ ๐ ๐2 +๐ โฒ (๐(๐)) (โฑ๐,๐+1 [|๐|(๐ฅ โ ๐๐(๐)) ]) ๐
โฒ
=
๐
๐ [|๐|๐๐ (๐(๐), ๐(๐), ๐)(1 โ ๐ก)๐ ]๐๐ก ๐ก ๐ (1 โ ๐ก)๐ โฑ๐,๐+1
โซ ๐ฅโ๐๐(๐) (๐(๐),๐(๐),๐) ๐ (
{
1 ๐
๐๐ โฒ (๐(๐))
๐
๐3 [|๐|(๐๐(๐) (โฑ๐,๐+1 ๐
+ ๐ (๐(๐), ๐(๐), ๐) โ ๐ฅ)๐ ]) ๐
๐
4 [|๐|๐๐ (๐(๐), ๐(๐), ๐)]) +๐ โฒ (๐(๐)) (โฑ๐,๐+1
1 ๐ ๐
} .
โ
This completes the proof of the theorem. Corollary 1.
Under the same conditions as in Theorem 2, if we choose ๐ = ๐ = 1, ๐ (๐(๐), ๐(๐), ๐) = ๐(๐) โ ๐๐(๐) and ๐(๐ฅ) = ๐ฅ, we get
| |
[
๐ ๐ (๐ฅ โ ๐)๐ โฑ๐,๐+1 [๐(๐ฅ โ ๐)๐ ] + (๐ โ ๐ฅ)๐ โฑ๐,๐+1 [๐(๐ โ ๐ฅ)๐ ]
(๐ โ ๐)๐+1 โ
โค
๐ ๐ ๐ [(๐ฅ๐,๐,๐ฅโ;๐ ๐)(๐) + (๐ฅ๐,๐,๐ฅ+;๐ ๐)(๐)] (๐ โ ๐)๐+1
๐1โ โฒ (๐)๐ [|๐|(๐ ๐ (โฑ { ๐,๐+1
+ {๐
โฒ (๐)๐
๐3โ [|๐|(๐ (โฑ๐,๐+1
๐
1 ๐ ๐
๐2โ
โ ๐)๐ ]) + ๐ โฒ (๐)๐ (โฑ [|๐|(๐ฅ โ ๐)๐ ]) } ๐,๐+1
โ ๐ฅ)
๐
๐ ])
+๐
โฒ (๐)๐
๐4โ [|๐|(๐ (โฑ๐,๐+1
โ ๐)
where ๐1โ (๐) = ๐ (๐)๐ฝ (
๐ฅโ๐ 1 ; ๐ + ๐๐ + 1, + 1) ; ๐โ๐ ๐ 1
๐2โ (๐)
] ๐(๐ฅ) | |
๐ฅ โ ๐ ๐+๐ +1 1 = ๐ (๐) ( ) ; 1 ๐โ๐ ๐ + ๐๐ + ๐ + 1
1 ๐ ๐
๐ ])
} ,
(9)
1026
A. Kashuri and R. Liko 1
๐ โ ๐ฅ ๐+๐ +1 1 โ (๐) ๐3 = ๐ (๐) ( ) ; 1 ๐โ๐ ๐ + ๐๐ + ๐ + 1 ๐โ๐ฅ 1 ๐4โ (๐) = ๐ (๐)๐ฝ ( ; ๐ + ๐๐ + 1, + 1). ๐โ๐ ๐ Corollary 2. If we choose ๐ = ๐ (0) = 1, ๐ = 0 in Corollary 1, the inequality (9) reduces to inequality (2.1) of see (Yildiz et al., 2016; Theorem 2.1). Theorem 3. Let ๐: ๐ผ โถ ๐ด be a continuous function. Suppose ๐ด โ โ be an open ๐ โinvex subset with respect to the mapping ๐: ๐ด ร ๐ด ร ]0,1] โถ โ for some fixed ๐ , ๐ โ ]0,1] and ๐ (๐(๐), ๐(๐), ๐) > 0. Assume that ๐: ๐ด โถ (0, +โ) is a differentiable function on ๐ดโ . If 0 < ๐ โค 1 and ๐ โฒ๐ is generalized (๐; ๐ , ๐, ๐) โpreinvex function on [๐๐(๐), ๐๐(๐) + ๐ (๐(๐), ๐(๐), ๐)], ๐ > 1, ๐โ1 + ๐ โ1 = 1, then the following inequality for generalized fractional integral operators holds: 1
๐ ๐ |๐ผ๐,๐,๐ (๐ฅ; ๐, ๐, ๐, ๐, ๐, ๐)| โค ( ) ๐ +๐
[
๐ ๐ [๐๐+1 (๐(๐), ๐(๐), ๐)
โ (๐๐(๐) + ๐ (๐(๐), ๐(๐), ๐) โ +(๐ฅ โ ๐๐(๐))
ร (๐ฅ โ ๐๐(๐))
ร
1 ๐ +1 1 ๐+ + ๐ ๐๐ ๐ (๐(๐), ๐(๐), ๐)
๐+
๐ +1 โฒ
๐ (๐(๐))
1 โ ๐โฑ ๐ ๐,๐+1 [|๐|(๐ฅ
๐
+ [๐ ๐+1 (๐(๐), ๐(๐), ๐) โ (๐ฅ โ
{ร (๐๐(๐) + ๐ (๐(๐), ๐(๐), ๐) โ ๐ฅ)
๐+
โฒ
๐ (๐(๐))
๐๐
1 ๐๐ ๐๐
]
๐
โ ๐๐(๐)) ]
๐(๐๐(๐) + ๐ (๐(๐), ๐(๐), ๐) โ ๐ฅ) +[
๐ ๐ +1 ๐ ๐ฅ) ]
๐ +1 โฒ
๐ (๐(๐))
๐ ๐ +1 ๐ ๐๐(๐)) ]
1 โ ๐โฑ ๐ ๐,๐+1 [|๐|(๐๐(๐)
1 ๐๐
๐๐
๐ โฒ (๐(๐))
, (10)
๐๐ ]
+ ๐ (๐(๐), ๐(๐), ๐) โ ๐ฅ)๐ ]}
where ๐, ๐ > 0, ๐ โ โ, ๐ = 0,1,2, โฆ, and 1
๐
โ (๐)
๐ 1 = ๐ (๐) ( ) . ๐(๐ + ๐๐) + 1
Proof: Suppose that ๐ > 1 and 0 < ๐ โค 1. From Lemma 1, generalized (๐; ๐ , ๐, ๐) โpreinvexity of ๐ โฒ๐ , Hรถlder inequality, Minkowski inequality and properties of the modulus, we have |๐ผ๐,๐,๐ (๐ฅ; ๐, ๐, ๐, ๐, ๐, ๐)|
1027
AAM: Intern. J., Vol 12, Issue 2 (December 2017) ๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
โค
๐ [|๐|๐๐ (๐(๐), ๐(๐), ๐)๐ก๐ ] |๐ โฒ (๐๐(๐) + ๐ก๐ (๐(๐), ๐(๐), ๐))|๐๐ก ๐ก ๐ โฑ๐,๐+1
โซ 0 1
+
โซ
๐ |1 โ ๐ก| ๐ โฑ๐,๐+1 [|๐|๐๐ (๐(๐), ๐(๐), ๐)(1 โ ๐ก)๐ ] |๐ โฒ (๐๐(๐) + ๐ก๐ (๐(๐), ๐(๐), ๐))|
๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐) +โ
โคโ ๐=0
๐ (๐)|๐|๐ ๐๐๐ (๐(๐), ๐(๐), ๐) ๐ค(๐ + ๐๐ + 1)
1 ๐
๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
๐ก ๐(๐+๐๐) ๐๐ก
โซ
0
) (
)
1 ๐
1
+
โซ
๐
(๐ โฒ (๐๐(๐) + ๐ก๐ (๐(๐), ๐(๐), ๐))) ๐๐ก
โซ
0
(
ร
1 ๐
๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
1
(1 โ ๐ก)
๐(๐+๐๐)
๐๐ก
๐
(๐ โฒ (๐๐(๐) + ๐ก๐ (๐(๐), ๐(๐), ๐))) ๐๐ก
โซ
๐ฅโ๐๐(๐)
1 ๐
๐ฅโ๐๐(๐)
{ (๐ (๐(๐),๐(๐),๐)
) (๐ (๐(๐),๐(๐),๐)
) }
+โ
๐ (๐)|๐|๐ ๐๐๐ (๐(๐), ๐(๐), ๐) โคโ ๐ค(๐ + ๐๐ + 1) ๐=0 1 ๐
๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
โซ
๐ก
๐(๐+๐๐)
๐๐ก
{
โซ
[๐(1 โ ๐ก) ๐ (๐(๐))
๐๐
1 ๐๐ ๐
๐ โฒ
+ ๐ก ๐ (๐(๐)) ] ๐๐ก
0
) ( 1
+
๐ โฒ
โซ
0
(
ร
1 ๐
๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
)
1 ๐ 1
(1 โ ๐ก)
๐(๐+๐๐)
๐๐ก
๐ฅโ๐๐(๐) (๐(๐),๐(๐),๐) ๐ (
โซ )
+โ
โคโ ๐=0
๐ โฒ
[๐(1 โ ๐ก) ๐ (๐(๐))
๐ฅโ๐๐(๐) (๐(๐),๐(๐),๐) ๐ (
๐ (๐)|๐|๐ ๐๐๐ (๐(๐), ๐(๐), ๐) ๐ค(๐ + ๐๐ + 1)
๐๐
๐ โฒ
1 ๐
1 ๐๐ ๐
+ ๐ก ๐ (๐(๐)) ] ๐๐ก ) }
1028
A. Kashuri and R. Liko 1 ๐
๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
๐ก ๐(๐+๐๐) ๐๐ก
โซ 0
(
) ๐
๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
ร ๐๐ โฒ (๐(๐))
๐๐
๐
+ ๐ โฒ (๐(๐))
(1 โ ๐ก)๐ ๐๐ก
โซ
๐๐
ร
๐
โซ
0
[
1 ๐ ๐๐
๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
๐ก ๐ ๐๐ก
0
(
)
) ]
(
1 ๐
1
+
(1 โ ๐ก)๐(๐+๐๐) ๐๐ก
โซ ๐ฅโ๐๐(๐)
(๐ (๐(๐),๐(๐),๐)
)
1 ๐ ๐๐
๐ 1
ร ๐๐ โฒ (๐(๐))
๐๐
(1 โ
โซ ๐ฅโ๐๐(๐) (๐(๐),๐(๐),๐) ๐ (
{ [
1
๐ ๐ก)๐ ๐๐ก
+ ๐ โฒ (๐(๐))
1
+(๐ฅ โ ๐๐(๐)) ร
๐ +1 โฒ
๐ (๐(๐))
1 ๐+ ๐โ ๐๐(๐)) ๐ โฑ๐,๐+1 [|๐|(๐ฅ
๐ + [๐๐+1 (๐(๐), ๐(๐), ๐)
{ร (๐๐(๐) + ๐ (๐(๐), ๐(๐), ๐) โ ๐ฅ) This completes the proof of the theorem.
๐+
โ (๐ฅ โ
๐ ๐ ๐ฅ)๐+1 ]
๐ โฒ (๐(๐))
๐๐
1 ๐๐ ๐๐
]
๐
โ ๐๐(๐)) ]
๐(๐๐(๐) + ๐ (๐(๐), ๐(๐), ๐) โ ๐ฅ) +[
) ] }
๐ +1 1 ๐+ + ๐ ๐๐ ๐ (๐(๐), ๐(๐), ๐)
โ (๐๐(๐) + ๐ (๐(๐), ๐(๐), ๐) โ
ร (๐ฅ โ
๐ก ๐ ๐๐ก
๐ฅโ๐๐(๐) (๐(๐),๐(๐),๐) ๐ (
1
[
๐
โซ
)
๐ ๐ =( ) ๐ +๐ ๐ ๐ [๐๐+1 (๐(๐), ๐(๐), ๐)
๐๐
๐ +1 โฒ
๐ (๐(๐))
๐ ๐ +1 ๐ ๐๐(๐)) ]
1 โ ๐โฑ ๐ ๐,๐+1 [|๐|(๐๐(๐)
1 ๐๐
๐๐
๐ โฒ (๐(๐))
.
๐๐ ]
+ ๐ (๐(๐), ๐(๐), ๐) โ ๐ฅ)๐ ]} โ
Corollary 3. Under the same conditions as in Theorem 3, if we choose ๐ = ๐ = 1, ๐ (๐(๐), ๐(๐), ๐) = ๐(๐) โ ๐๐(๐) and ๐(๐ฅ) = ๐ฅ, we get
1029
AAM: Intern. J., Vol 12, Issue 2 (December 2017)
| |
[
๐ ๐ (๐ฅ โ ๐)๐ โฑ๐,๐+1 [๐(๐ฅ โ ๐)๐ ] + (๐ โ ๐ฅ)๐ โฑ๐,๐+1 [๐(๐ โ ๐ฅ)๐ ]
(๐ โ ๐)๐+1 โ
๐ ๐ ๐ [(๐ฅ๐,๐,๐ฅโ;๐ ๐)(๐) + (๐ฅ๐,๐,๐ฅ+;๐ ๐)(๐)] (๐ โ ๐)๐+1 1
๐ ๐ โค( ) ๐+1
[[(๐ โ
1 ๐)๐ +1
โ (๐ โ
1 (๐ โ ๐)
๐ 1 +1 ๐ ๐ฅ) ]
ร (๐ฅ โ ๐)
ร
๐+
ร (๐ โ ๐ฅ)
๐+
๐+
2 1 + ๐๐ ๐
๐ โฒ (๐)๐๐ + +(๐ฅ โ ๐)2 ๐ โฒ (๐)๐๐ ]
1 โ ๐โฑ๐ ๐,๐+1 [|๐|(๐ฅ
2 โฒ ๐๐ + [(๐ โ ๐ฅ) ๐ (๐) + [(๐ โ
{
] ๐(๐ฅ) | |
1 ๐)๐ +1
โ ๐)๐ ]
โ (๐ฅ โ
1 โ ๐โฑ๐ ๐,๐+1 [|๐|(๐
1 ๐๐
๐ 1 ๐)๐ +1 ]
.
1 ๐๐
(11)
๐ โฒ (๐)๐๐ ]
โ ๐ฅ)๐ ]
}
Theorem 4. Let ๐: ๐ผ โถ ๐ด be a continuous function. Suppose ๐ด โ โ be an open ๐ โinvex subset with respect to the mapping ๐: ๐ด ร ๐ด ร ]0,1] โถ โ for some fixed ๐ , ๐ โ ]0,1] and ๐ (๐(๐), ๐(๐), ๐) > 0. Assume that ๐: ๐ด โถ (0, +โ) is a differentiable function on ๐ดโ . If 0 < ๐ โค 1 and ๐ โฒ๐ is generalized (๐; ๐ , ๐, ๐) โpreinvex function on [๐๐(๐), ๐๐(๐) + ๐ (๐(๐), ๐(๐), ๐)], ๐ โฅ 1, then the following inequality for generalized fractional integral operators holds: |๐ผ๐,๐,๐ (๐ฅ; ๐, ๐, ๐, ๐, ๐, ๐)| โค
๐1 (โฑ๐,๐+1 [|๐|(๐ฅ
๐
1โ
โ ๐๐(๐)) ])
1 ๐ 1 ๐
๐
โฑ๐,๐+1 [|๐|๐๐ (๐(๐), ๐(๐), ๐)]
ร
1 ๐
๐ฅ โ ๐๐(๐) ๐ ๐๐ ๐๐ โฒ (๐(๐)) ๐ฝ๐ ( ; ๐๐ + ๐๐ + 1, + 1) ๐ (๐(๐), ๐(๐), ๐) ๐ +๐ โฒ (๐(๐)) [{
๐๐
๐ฅ โ ๐๐(๐) [( ) ๐ (๐(๐), ๐(๐), ๐)
๐2 [|๐|(๐๐(๐) + (โฑ๐,๐+1
๐
๐ ๐๐+๐๐+ +1 ๐
1
] ๐ ๐๐ + ๐๐ + ๐ + 1 } ]
+ ๐ (๐(๐), ๐(๐), ๐) โ ๐ฅ)
1 1โ ๐ ]) ๐
1030
A. Kashuri and R. Liko 1 ๐
๐
โฑ๐,๐+1 [|๐|๐๐ (๐(๐), ๐(๐), ๐)] ๐๐ โฒ (๐(๐))
ร
[{
๐๐
[(
๐๐(๐) + ๐ (๐(๐), ๐(๐), ๐) โ ๐ฅ ) ๐ (๐(๐), ๐(๐), ๐)
๐
๐ ๐๐+๐๐+ +1 ๐
1
1 ๐
] ๐ ๐๐ + ๐๐ + ๐ + 1
๐๐(๐) + ๐ (๐(๐), ๐(๐), ๐) โ ๐ฅ ๐ ๐๐ +๐ โฒ (๐(๐)) ๐ฝ ๐ ( ; ๐๐ + ๐๐ + 1, + 1) ๐ (๐(๐), ๐(๐), ๐) ๐
, (12)
} ]
where ๐, ๐ > 0, ๐ โ โ, ๐ = 0,1,2, โฆ, and ๐ฅ โ ๐๐(๐) 1 ๐1 (๐) = ๐ (๐) ( ) ; ๐ (๐(๐), ๐(๐), ๐) ๐๐ + 1 ๐2 (๐) = ๐ (๐) (
๐๐(๐) + ๐ (๐(๐), ๐(๐), ๐) โ ๐ฅ 1 ) . ๐ (๐(๐), ๐(๐), ๐) ๐๐ + 1
Proof: Suppose that ๐ โฅ 1 and 0 < ๐ โค 1. From Lemma 1, generalized (๐; ๐ , ๐, ๐) โpreinvexity of ๐ โฒ๐ , the well-known power mean inequality, Minkowski inequality and properties of the modulus, we have |๐ผ๐,๐,๐ (๐ฅ; ๐, ๐, ๐, ๐, ๐, ๐)| ๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
โค
๐ [|๐|๐๐ (๐(๐), ๐(๐), ๐)๐ก๐ ] |๐ โฒ (๐๐(๐) + ๐ก๐ (๐(๐), ๐(๐), ๐))|๐๐ก ๐ก ๐ โฑ๐,๐+1
โซ 0 1
+
โซ
๐ |1 โ ๐ก| ๐ โฑ๐,๐+1 [|๐|๐๐ (๐(๐), ๐(๐), ๐)(1 โ ๐ก)๐ ] |๐ โฒ (๐๐(๐) + ๐ก๐ (๐(๐), ๐(๐), ๐))|
๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐) 1โ
๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
โค
โซ
1 ๐
๐ [|๐|๐๐ (๐(๐), ๐(๐), ๐)๐ก๐ ] ๐๐ก โฑ๐,๐+1
0
(
) 1 ๐
๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
ร
โซ
๐
๐ [|๐|๐๐ (๐(๐), ๐(๐), ๐)๐ก๐ ] (๐ โฒ (๐๐(๐) + ๐ก๐ (๐(๐), ๐(๐), ๐))) ๐๐ก ๐ก ๐๐ โฑ๐,๐+1
0
(
)
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1โ
1 ๐
1 ๐ [|๐|๐๐ (๐(๐), ๐(๐), ๐)(1 โ ๐ก)๐ ] ๐๐ก โฑ๐,๐+1
โซ ๐ฅโ๐๐(๐)
(๐ (๐(๐),๐(๐),๐)
) 1 ๐
1 ๐ (1 โ ๐ก) ๐๐ โฑ๐,๐+1 [|๐|๐๐ (๐(๐), ๐(๐), ๐)(1 โ ๐ก)๐ ]
โซ ร
๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐) ๐
ร (๐ โฒ (๐๐(๐) + ๐ก๐ (๐(๐), ๐(๐), ๐))) ๐๐ก
(
+โ
๐ (๐)|๐|๐ ๐๐๐ (๐(๐), ๐(๐), ๐) โ ๐ค(๐ + ๐๐ + 1)
โค
1โ
๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
0
(
) +โ
โ
๐
1 ๐
(๐)|๐|๐ ๐๐ (๐(๐),
๐ ๐(๐), ๐) ๐ค(๐ + ๐๐ + 1)
๐=0 ๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
ร
1 ๐
๐ก๐๐ ๐๐ก
โซ
๐=0
)
ร
โซ
[
๐ก
๐๐+๐๐
๐ โฒ
[๐(1 โ ๐ก) ๐ (๐(๐))
๐๐
๐ โฒ
1 ๐๐ ๐
+ ๐ก ๐ (๐(๐)) ] ๐๐ก ]
0 1โ 1
+โ
๐ (๐)|๐|๐ ๐๐๐ (๐(๐), ๐(๐), ๐) + โ ๐ค(๐ + ๐๐ + 1) ๐=0
( +โ
โ
๐
(1 โ ๐ก)๐๐ ๐๐ก
โซ ๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
) 1 ๐
(๐)|๐|๐ ๐๐ (๐(๐),
๐=0
๐ ๐(๐), ๐) ๐ค(๐ + ๐๐ + 1)
1
ร ร [
โซ
(1 โ ๐ก)
1 ๐
๐๐+๐๐
๐ โฒ
[๐(1 โ ๐ก) ๐ (๐(๐))
๐๐
๐ โฒ
1 ๐๐ ๐
+ ๐ก ๐ (๐(๐)) ] ๐๐ก
๐ฅโ๐๐(๐) ๐ (๐(๐),๐(๐),๐)
] โค
๐1 (โฑ๐,๐+1 [|๐|(๐ฅ
๐
1โ
โ ๐๐(๐)) ])
1 ๐
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A. Kashuri and R. Liko 1 ๐
๐
โฑ๐,๐+1 [|๐|๐๐ (๐(๐), ๐(๐), ๐)] 1 ๐
๐ฅ โ ๐๐(๐) ๐ ๐๐ โฒ (๐(๐)) ๐ฝ ( ; ๐๐ + ๐๐ + 1, + 1) ๐ (๐(๐), ๐(๐), ๐) ๐ ๐๐ ๐
ร
๐ฅ โ ๐๐(๐) ๐๐ +๐ โฒ (๐(๐)) [( ) ๐ (๐(๐), ๐(๐), ๐) [{ ๐2 [|๐|(๐๐(๐) + (โฑ๐,๐+1
๐
๐ ๐๐+๐๐+ +1 ๐
1
] ๐ ๐๐ + ๐๐ + ๐ + 1 } ] 1โ
+ ๐ (๐(๐), ๐(๐), ๐) โ ๐ฅ)๐ ])
1 ๐ 1 ๐
๐ โฑ๐,๐+1 [|๐|๐๐ (๐(๐), ๐(๐), ๐)]
๐๐ โฒ (๐(๐))
ร
[{
๐๐
[(
๐๐(๐) + ๐ (๐(๐), ๐(๐), ๐) โ ๐ฅ ) ๐ (๐(๐), ๐(๐), ๐)
๐
๐ ๐๐+๐๐+ +1 ๐
1
1 ๐
] ๐ ๐๐ + ๐๐ + ๐ + 1
๐๐(๐) + ๐ (๐(๐), ๐(๐), ๐) โ ๐ฅ ๐ ๐๐ +๐ โฒ (๐(๐)) ๐ฝ ๐ ( ; ๐๐ + ๐๐ + 1, + 1) ๐ (๐(๐), ๐(๐), ๐) ๐
.
} ] โ
This completes the proof of the theorem. Corollary 4.
Under the same conditions as in Theorem 4, if we choose ๐ = ๐ = ๐ = 1, ๐ (๐(๐), ๐(๐), ๐) = ๐(๐) โ ๐๐(๐) and ๐(๐ฅ) = ๐ฅ, we get
| |
[
๐ ๐ (๐ฅ โ ๐)๐ โฑ๐,๐+1 [๐(๐ฅ โ ๐)๐ ] + (๐ โ ๐ฅ)๐ โฑ๐,๐+1 [๐(๐ โ ๐ฅ)๐ ]
(๐ โ ๐)๐+1 โ
๐ ๐ ๐ [(๐ฅ๐,๐,๐ฅโ;๐ ๐)(๐) + (๐ฅ๐,๐,๐ฅ+;๐ ๐)(๐)] (๐ โ ๐)๐+1 โค
ร
๐1โ [|๐|(๐ฅ (โฑ๐,๐+1
๐2โ [|๐|(๐ [๐ โฒ (๐)๐ โฑ๐,๐+1
โ ๐)๐ ] +
๐4โ [|๐|(๐ + (โฑ๐,๐+1
โ ๐)
1 1โ ๐ ]) ๐
๐3โ [|๐|(๐ฅ ๐ โฒ (๐)๐ โฑ๐,๐+1
โ ๐ฅ)
] ๐(๐ฅ) | |
โ ๐)๐ ]]
1 ๐
1 1โ ๐ ]) ๐ 1
ร [๐
where
โฒ (๐)๐
๐5โ [|๐|(๐ โฑ๐,๐+1
โ ๐ฅ)
๐]
+๐
โฒ (๐)๐
๐6โ [|๐|(๐ โฑ๐,๐+1
โ ๐)
๐ ]]๐
,
(13)
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AAM: Intern. J., Vol 12, Issue 2 (December 2017)
๐1โ (๐) = ๐ (๐) ( ๐2โ (๐) = ๐ (๐)๐ฝ (
๐ฅโ๐ 1 ) ; ๐ โ ๐ ๐๐ + 1
๐ฅโ๐ ; ๐๐ + ๐๐ + 1, 2) ; ๐โ๐
๐ฅ โ ๐ ๐๐+2 1 ) ; ๐โ๐ ๐๐ + ๐๐ + 2
๐3โ (๐) = ๐ (๐) (
๐โ๐ฅ 1 ๐4โ (๐) = ๐ (๐) ( ) ; ๐ โ ๐ ๐๐ + 1 ๐ โ ๐ฅ ๐๐+2 1 ๐5โ (๐) = ๐ (๐) ( ) ; ๐โ๐ ๐๐ + ๐๐ + 2 ๐6โ (๐) = ๐ (๐)๐ฝ (
๐โ๐ฅ ; ๐๐ + ๐๐ + 1, 2). ๐โ๐
Corollary 5. If we choose ๐ (0) = 1, ๐ = 0 in Corollary 4, the inequality (13) reduces to inequality (2.4) of see (Yildiz et al., 2016; Theorem 2.3).
3. Conclusion In the present paper, the notion of generalized (๐; ๐ , ๐, ๐) โpreinvex function was applied to establish some new generalizations of Ostrowski type inequalities via fractional integral operators. These results not only extended the results appeared in the literature but also provided new estimates on these type. Motivated by this new interesting class of generalized (๐; ๐ , ๐, ๐) โpreinvex functions we can indeed see to be vital for fellow researchers and scientists working in the same domain. We conclude that our methods considered here may be a stimulant for further investigations concerning Ostrowski, HermiteโHadamard and Simpson type integral inequalities for various kinds of preinvex functions involving classical integrals, Riemann-Liouville fractional integrals, ๐ โfractional integrals, local fractional integrals, fractional integral operators, ๐ โcalculus, (๐, ๐) โcalculus, time scale calculus and conformable fractional integrals.
Acknowledgments The authors would like to thank the reviewers for their valuable suggestions and comments to improve the presentation of this paper.
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Biographical Note of Authors: The authors are Assistant Professors of Department of Mathematics, Faculty of Technical Science, University โIsmail Qemaliโ, Vlora, Albania. Their research interests include Mathematical Inequalities, Fractional Calculus, Approximation Theory and Applied Mathematics. They have published more than 15 research articles in reputed international journals.