Filomat 3:5 6), 43 5 DOI.98/FIL6543W Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: htt://www.mf.ni.ac.rs/filomat Hermite-Hadamard Ineualities Involving Riemann-Liouville Fractional Integrals via s-convex Functions and Alications to Secial Means JinRong Wang a,b, Xuezhu Li a, Yong Zhou c a Deartment of Mathematics, Guizhou University, Guiyang, Guizhou 555, P.R. China b School of Mathematics and Comuter Science, Guizhou Normal College, Guiyang, Guizhou 558, P.R. China c Deartment of Mathematics, Xiangtan University, Xiangtan, Hunan 45, P.R. China Abstract. In this aer, we establish some new Hermite-Hadamard tye ineualities involving Riemann- Liouville fractional integrals via s-convex functions in the second sense. Meanwhile, we resent many useful estimates on these tyes of new Hermite-Hadamard tye ineualities. Some alications to secial means of real numbers are given.. Introduction It is well known that the classical Hermite-Hadamard tye ineuality rovides a lower and an uer estimations for the integral average of any convex function defined on a comact interval, involving the midoint and the endoints of the domain. This interesting ineuality was firstly discovered by Hermite in 88 in the journal Mathesis see Mitrinović and Lacković []). However, this beautiful result was nowhere mentioned in the mathematical literature and was not widely known as Hermite s result see Pečarić et al. [7]). For more recent results which generalize, imrove, and extend this classical Hermite-Hadamard ineuality, one can see Dragomir [5], Sarikaya et al. [9], Xiao et al. [7], Bessenyei [4], Tseng et al. [4], Niculescu [3], Bai et al. [3], Li and Qi [], Tunç [5], Srivastava et al. [3], and references therein. In articular, let us note that Professor Srivastava et al. resent some interesting refinements and extensions of the Hermite-Hadamard ineualities in n variables see Theorems 6-, [3]) and comare with some various ineualities in earlier works. Fractional calculus have recently roved to be a owerful tool for the study of dynamical roerties of many interesting systems in hysics, chemistry, and engineering. It draws a great alication in nonlinear oscillations of earthuakes, many hysical henomena such as seeage flow in orous media and in fluid dynamic traffic model. For more recent develoment on fractional calculus, one can see the monograhs of Kilbas et al. [9], Lakshmikantham et al. [], and Podlubny [8]. Mathematics Subject Classification. Primary 6A33; Secondary 6A5, 6D5. Keywords. Hermite-Hadamard ineualities, s-convex functions, Riemann-Liouville fractional integrals, Integral ineualities. Received: 9 March 4; Acceted: 6 June 4 Communicated by Prof. Hari M. Srivastava The first and second author s work was artially suorted by Training Object of High Level and Innovative Talents of Guizhou Province 5GZ9578), Unite Foundation of Guizhou Province [5]764) and Outstanding Scientific and Technological Innovation Talent Award of Education Deartment of Guizhou Province [4]4). The third authors work was suorted by National Natural Science Foundation of P.R. China 739). Email addresses: sci.jrwang@gzu.edu.cn JinRong Wang), xzleemath@6.com Xuezhu Li), yzhou@xtu.edu.cn Yong Zhou)
J. Wang, X. Li, Y. Zhou / Filomat 3:5 6), 43 5 44 Due to the widely alication of Hermite-Hadamard tye ineuality and fractional calculus, it is natural to offer to study Hermite-Hadamard tye ineualities involving fractional integrals. It is remarkable that Sarikaya et al. [] begin to study ineualities of Hermite-Hadamard tye involving Riemann-Liouville fractional integrals and give the following new Hermite-Hadamard tye ineualities. Theorem.. Theorem, []) Let f : [a, b] R be a ositive function with a < b and f L[a, b]. If f is a convex function on [a, b], then the following ineuality for fractional integrals hold f ) a + b Γα + ) ) α [Jα a + f b) + J α b f a) + f b). Here, the symbol J α a + f and J α b f denote the left-sided and right-sided Riemann-Liouville fractional integrals of the order α R + := [, ) are defined by J α a f )x) = x x t) α f t)dt, a < x b), J α + Γα) b f )x) = b t x) α f t)dt, a < x b), a Γα) x resectively. Here Γ ) is the Gamma function. Lemma.. Lemma, []) Let f : [a, b] R be a differentiable maing on a, b) with a < b. If f L[a, b], then the following euality for fractional integrals holds f a) + f b) Γα + ) ) α [Jα a = [ t) α t α ] f ta + t)b)dt. Theorem.3. Theorem 3, []) Let f : [a, b] R be a differentiable maing on a, b) with a < b. If f is convex on [a, b], then the following euality for fractional integrals holds f a) + f b) Γα + ) ) α [Jα a f b) + J α + b ) α + ) α f a) + f b). In addition to the classical convex functions, Hudzik and Maligranda [8] introduced the definition of s-convex functions in the second sense as following. Definition.4. A function f : I R + R + is said to be s-convex on I if the ineuality f λx + λ)y) λ s f x) + λ) s f y) holds for all x, y I and λ [, ] and for some fixed s, ]. Alying the tool of s-convex functions in the second sense, Set [] gives some new ineualities of Ostrowski tye involving fractional integrals. The author generalizes the interesting and useful Ostrowski ineuality see [4]) and other numerous extensions and variants of Ostrowski ineualities see for examle [, 7, 5, ]). The authors [, 6] study the ineualities of Hermite-Hadamard tye via s-convex functions in the second sense, to the best of our knowledge, Hadamard tye ineualities involving Riemann-Liouville fractional integrals via s-convex functions have not been studied extensively. Motivated by the recent results given in [, 6,,, 6], we will establish here some new Hermite-Hadamard tye ineualities involving Riemann-Liouville fractional integrals via s-convex functions in the second sense. Meanwhile, we resent many useful estimates on these tyes of new Hermite-Hadamard tye ineualities for Riemann-Liouville fractional integrals.. Hermite-Hadamard Ineuality for Fractional Integrals via s-convex Functions An imortant Hermite-Hadamard ineuality involving Riemann-Liouville fractional integrals with α ) can be reresented as follows.
J. Wang, X. Li, Y. Zhou / Filomat 3:5 6), 43 5 45 Theorem.. Let α and f : [a, b] R be a ositive function with a < b and f L[a, b]. If f is a s-convex function on [a, b], then the following ineuality for fractional integrals hold ) a + b s Γα + ) f ) α [Jα a f b) + J α [ f a) + f b)] [ + b α + s + )] α + s α+s. ) Proof. Since f is a s-convex function on [a, b], we have for x, y [a, b] with λ = f ) x+y f x) + s f y), s i.e., with x = ta + t)b, y = t)a + tb, ) a + b s f f ta + t)b) + f t)a + tb). ) Multilying both sides of ) by t α, then integrating the resulting ineuality with resect to t over [, ], we obtain ) a + b s Γα + ) f ) α [Jα a and the first ineuality in ) is roved. Because f is a s-convex function, we have f ta + t)b) + f t)a + tb) [t s + t) s ][ f a) + f b)]. 3) Then multilying both sides of 3) by t α and integrating the resulting ineuality with resect to t over [, ], we obtain the right-sided ineuality in ). The roof is comleted. Remark.. One can follow the same ideas to construct fractional version F ) n, F ) n, F 3) n see 3.4)-3.6), [3]) try to extend to study fractional Hermite-Hadamard ineualities in n variables based on these fundamental results. We shall study such interesting roblems in the forthcoming works. 3. Hermite-Hadamard Tye Ineualities for Fractional Integrals via s-convex Functions Now we are ready to state the first result in this section. Theorem 3.. Let f : [a, b] R be a differentiable maing on a, b) with a < b, such that f L[a, b]. If f is s-convex on [a, b], for some fixed s, ], then the following ineuality for fractional integrals holds f a) + f b) Γα + ) ) α [Jα a f b) + J α + b ) α + s + α+s f a) + f b) ). Proof. Using Lemma. and the s-convexity of f, we have f a) + f b) Γα + ) ) α [Jα a { [ t) α t α ][t s f a) + t) s f b) ]dt + := K + K ). Calculating K and K, we have K f a) t) α+s ds t α+s ds + f b) = f a) + f b) α + s + } [t α t) α ][t s f a) + t) s f b) ]dt t) α+s ds t α+s ds α+s ), 5) 4)
and K f a) = f a) + f b) α + s + J. Wang, X. Li, Y. Zhou / Filomat 3:5 6), 43 5 46 t α+s ds t) α+s ds + f b) t α+s ds t) α+s ds α+s ). 6) Thus if we use 5) and 6) in 4), we obtain the result. The roof is comleted. The second theorem gives a new uer bound of the left-hadamard ineuality for s-convex maings. Theorem 3.. Let f : [a, b] R be a differentiable maing on a, b) with a < b, such that f L[a, b]. If f > ) is s-convex on [a, b], for some fixed s, ], then the following ineuality for fractional integrals holds f a) + f b) Γα + ) ) α [Jα a ) ) ) ) α + α s + ) s+ where =. [ f a) + s+ ) f b) ) + s+ ) f a) + f b) ) ], 7) Proof. Using Lemma. and Hölder ineuality and the s-convexity of f > ), we have f a) + f b) Γα + ) ) α [Jα a )) { α + α t s f a) + t) s f b) )dt + We note that and t s f a) + t) s f b) dt = t s f a) + t) s f b) )dt = s + t s f a) + t) s f b) )dt }. s + ) f a) + ) f b) 8) s+) s + s+ s+ ) f a) + Note that 8) and 9), we get our result. The roof is comleted. It is not difficult to see that Theorem 3. can be extended to the following result. s + ) s+) f b). 9) Corollary 3.3. Let f : [a, b] R be a differentiable maing on a, b) with a < b, such that f L[a, b]. If f, > ) is s-convex on [a, b], for some fixed s, ], then the following ineuality for fractional integrals holds f a) + f b) Γα + ) ) α [Jα a ) ) ) ) α + α ) + s+ ) s + ) s+ f a) + f b) ), where =. Proof. We consider ineuality 7), and we let a = f a), b = s+ ) f b), a = s+ ) f a), b = f b). Here, < < for >. Using the fact n i= a i + b i ) r n i= ar i + n i= br i for < r <, a, a,, a n > and b, b,, b n >, we obtain the reuired result. This comletes the roof. The following theorem is the third main result in this section.
J. Wang, X. Li, Y. Zhou / Filomat 3:5 6), 43 5 47 Theorem 3.4. Let f : [a, b] R be a differentiable maing on a, b) with a < b,such that f L[a, b]. If f, > ) is s-convex on [a, b], for some fixed s, ], then the following ineuality for fractional integrals holds f a) + f b) Γα + ) ) α [Jα a ) ) ) ) α + α ) f α + s + α+s a) + f b) ). ) Proof. Using Lemma. and the ower mean ineuality, we have f a) + f b) Γα + ) ) α [Jα a { + α + [ t) α t α ] f ta + t)b) dt + ) α ) { [t α t) α ] f ta + t)b) dt Because the s-convex of f, > ), we have [ t) α t α ] f ta + t)b) dt and similarly t) α t s dt f a) + α + s + } [t α t) α ] f ta + t)b) dt [ t) α t α ] f ta + t)b) dt }. t) α+s dt f b) t α+s dt f a) t α t) s dt f b) α+s ) f a) + f b) ), ) [t α t) α ] f ta + t)b) dt ) α + s + α+s f a) + f b) ). ) A combination of ) and ), we get f a) + f b) Γα + ) ) α [Jα a ) ) ) ) α + α ) f α + s + α+s a) + f b) ). This comletes the roof. Corollary 3.5. Let f be as in Theorem 3.4, then the following ineuality holds f a) + f b) Γα + ) ) α [Jα a ) ) ) ) α + α ) f α + s + α+s a) + f b) ). Proof. Using the techniue in the roof of Corollary 3.3, by considering ineuality ), one can obtain the result. To end this section, we give the following Hermite-Hadamard tye ineuality for concave maing.
J. Wang, X. Li, Y. Zhou / Filomat 3:5 6), 43 5 48 Theorem 3.6. Let f : [a, b] R be a differentiable maing on a, b) with a < b, such that f L[a, b]. If f > ) is concave on [a, b], for some fixed s, ], then the following ineuality for fractional integrals holds f a) + f b) Γα + ) ) α [Jα a ) ) ) [ ) ) a + 3b α + α f 4 + ] 3a + b f. 4 Proof. Using Theorem 3. and Hölder ineuality and the s-convex of f, > ), we have f a) + f b) Γα + ) ) α [Jα a )) { α + α f ta + t)b) dt + f ta + t)b) dt where =. We note that f is concave on [a, b], and use the Jensen integral ineuality, we have f ta + t)b dt t ta + t)b)dt dt f ) a + 3b t dt f, 4 and analogously f ta + t)b) dt t dt Combining all obtained ineualities, we get the reuired result. f ta + t)b)dt t dt ) 3a + b f. 4 }, 4. Alications to Some Secial Means Consider the following secial means see Pearce and Pečarić [6]) for arbitrary real numbers α, β, α β as follows: Hα, β) = α + β, α, β R \ {}, Aα, β) = α + β, α, β R, Lα, β) = β α, α β, αβ, ln β ln α L n α, β) = [ β n+ α n+ ] n, n Z \ {, }, α, β R, α β. n + )β α) Now, using the obtained results in Section 3, we give some alications to secial means of real numbers. Proosition 4.. Let a, b R +, a < b, s, ] and >. Then ) b a) s+ Aa, b ), ) +s ) Aa, b ) L a, b) where =. ) + ) + s+ ) Aa, b ), ) 4 ) s+ ) s+) s+ ) ) Aa, b ), +s 3)
J. Wang, X. Li, Y. Zhou / Filomat 3:5 6), 43 5 49 Proof. Alying Theorem 3., Corollary 3.3, and Corollary 3.5 resectively, for f x) = x and α =, one can obtain the result immediately. Proosition 4.. Let a, b R +, a < b, n Z, n, s, ] and >. Then ) n b a) s+ Aa n, b n ), +s ) ) Aa n, b n ) L n na, b) n ) + ) + s+ ) Aa n, b n ), n ) 4 ) s+ s+) s+ ) ) ) Aa n, b n ), +s 4) where =. Proof. Alying Theorem 3., Corollary 3.3, and Corollary 3.5 resectively, for f x) = x n and α =, one can obtain the result immediately. Proosition 4.3. Let a, b R +, a < b), a > b. For n Z, n, s, ] and >, we have ) b a) abs+) H a, b ), ) +s ) b a ab + H b, a ) L b, a ) ) + s+ ) H a, b ), b a) ab 4) s+ n b a) abs+) n b a) ab + ) s+) s+ ) ) H a, b ), +s ) H a n, b n ), ) +s ) H b n, a n ) L n nb, a ) ) + s+ ) H a n, b n ), n b a) ab 4) s+ ) s+) s+ ) ) H a n, b n ), +s 5) 6) where =. Proof. Making the substitutions a b, b a in the ineualities 3) and 4), one can obtain desired ineualities 5) and 6) resectively. 5. Acknowledgements The authors thank the referees for their careful reading of the manuscrit and insightful comments, which heled to imrove the uality of the aer. We would also like to acknowledge the valuable comments and suggestions from the editors, which vastly contributed to imrove the resentation of the aer. References [] M. Alomari, M. Darus, S. S. Dragomir, P. Cerone, Ostrowski tye ineualities for functions whose derivatives are s-convex in the second sense, Al. Math. Lett., 3) 7 76. [] M. Avci, H. Kavurmaci, M.E. Ödemir, New ineualities of Hermite-Hadamard tye via s-convex functions in the second sense with alications, Al. Math. Comut., 7) 57 576. [3] R. Bai, F. Qi, B. Xi, Hermite-Hadamard tye ineualities for the m- and α, m)-logarithmically convex functions, Filomat, 73) 7. [4] M. Bessenyei, The Hermite-Hadamard ineuality in Beckenbach s setting, J. Math. Anal. Al., 364) 366 383. [5] S.S. Dragomir, Some ineualities for convex functions of selfadjoint oerators in Hilbert saces, Filomat, 39) 8 9.
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