Research Article Applying GG-Convex Function to Hermite-Hadamard Inequalities Involving Hadamard Fractional Integrals

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1 International Journal of Mathematics and Mathematical Sciences Volume 4, Article ID 3635, pages Research Article Applying GG-Convex Function to Hermite-Hadamard Inequalities Involving Hadamard Fractional Integrals Zhi Zhang, JinRong Wang,, and JianHua Deng Department of Mathematics, Guizhou University, Guiyang, Guizhou 555, China School of Mathematics and Computer Science, Guizhou Normal College, Guiyang, Guizhou 558, China Correspondence should be addressed to JinRong Wang; Received February 4; Revised 7 June 4; Accepted June 4; Published 4 July 4 AcademicEditor:HariM.Srivastava Copyright 4 Zhi Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. By virtue of fractional integral identities, incomplete beta function, useful series, and inequalities, we apply the concept of GG-convex function to derive new type Hermite-Hadamard inequalities involving Hadamard fractional integrals. Finally, some applications to special means of real numbers are demonstrated.. Introduction Fractional calculus played an important role in various fields such as electricity, biology, economics, and signal and image processing [ 8]. ThefractionalHermite-Hadamard inequality gives a lower and an upper estimation for both righthand and left-hand integrals average of any convex function defined on a compact interval, involving the midpoint and the endpoints of the domain. As we know, Set [9] firstly studied fractional Ostrowski inequalities involving Riemann-Liouville fractional integrals. Then, Sarikaya et al. [] studied Hermite-Hadamard type inequalities involving Riemann-Liouville fractional integrals. Further, our group go on studying fractional version Hermite-Hadamard inequality involving Riemann-Liouville and Hadamard fractional integrals for all kinds of functions [ 9]. Recently, Wang et al. [6, 7] established the following two powerful fractional integral identities involving Hadamard fractional integrals. Lemma (see [, Lemma 3.]. Let f : [a,b] R be a differentiable mapping on (a, b. Iff L[a,b], then the following equality for fractional integrals holds: f (a f(b = Γ (α ( α [ H J α a f (b H Jα bf (a] [(t α t α ]a t b t f (a t b t dt, ( where the symbols H J α a f and H Jα bf are defined by ( H J α a f (x = ( H J α bf (x = Γ (α x a Γ (α b where Γ( is the Gamma function. x (ln x t α f (t dt t, ( <a<xb, (ln t x α f (t dt t, ( <ax<b, Lemma (see [, Lemma.].Let f:[a,b] R be a differentiable mapping on (a, b with <a<b.iff L[a,b], then the following equality for fractional integrals holds: (

2 International Journal of Mathematics and Mathematical Sciences Γ (α ( α [ H J α a f (b H Jα bf (a]f(ab = ba kf (ta (t b dt [(t α t α ]a t b t f (a t b t dt, (3 where {, t < k=, { (4, { t<. Remark 3. It is remarkable that Professor Srivastava et al. [] give some further refinements and extensions of the Hermite- Hadamard inequalities in n variables. In the forthcoming works, we will try to extend to study fractional version Hermite-Hadamard inequalities in n variables based on such fundamental results. Next, we recall the following basic concepts and results in our previous papers. Definition 4 (see [, ]. Let f:i R R.Afunction f is said to be GG-convex on I if, for every x, y I and λ [, ],onehas f(x λ y λ [f(x] λ [f (y] λ. (5 Remark 5. By the arithmetic-geometric mean inequality, we have [f (x] λ [f (y] λ λf(x (λ f (y. (6 Linking (5 and(6, we obtain f (x λ y λ λf(x (λ f (y, (7 which appears in the standard definition of GA-convex function []. So GG-convex function is GA-convex function. Lemma 6 (see [9,Lemma.5]. For t [, ], x, y >,one has tx (t y y t x t. (8 Lemma 7 (see [3,Lemma.]. For α>and k>,onehas I (α = t α k t dt = k( i (ln k i <, (9 i= (α i where (α i = α(α (α (α i. Lemma 8 (see [3, Lemma.]. For α>and k>,z>, one has J (α, k = (t α k t dt = z H (α, k, z = i= (ln k i <, (α i t α k t dt = z α k z (z ln k i <. i= (α i ( Lemma 9 (see [3, Lemma.3]. For α>and k>,> z>,onehas z R (α, k, z = (t α k t dt = i= (ln k i (α i ( k z (z αi. ( Inthepresentpaper,wewillusetheaboveconceptsand lemmas to derive some new fractional Hermite-Hadamard inequalities involving Hadamard fractional integrals.. Main Results Based on Lemma Now we are ready to state the following main results in this section. Theorem. Let f:[,b] R be a differentiable mapping. If f is measurable and f is GG-convex on [a, b] for some fixed α (, and t [,], a<b, then the following integrals hold: f (a f(b {( α Γ (α ( α [ H J α a f (b H Jα b f (a] [ b f α b f (a a f b f ] α a f (a a f b f (a α3 a f (a b f (a a (α3 α3 f (a (α(α(α3 b f (a a f a f (a (α(α3 a ( f (a f b f (a α3 (α(α3 b f (a a f α (α a f (a (α α a f (a (α }. α (

3 International Journal of Mathematics and Mathematical Sciences 3 Proof. Noting Definition 4 and Lemmas and 6,wehave f (a f(b = Γ (α ( α [ H J α a f (b H Jα b f (a] a t b t [(t α t α ]f (a t b t dt a t b t (tα t α f (a t b t dt a t b t [t α (t α ] / f (a t b t dt / a t b t [(t α t α ] f (a t b t dt [at b (t][t α (t α ] / f (a t f t dt / [at b (t] [(t α t α ] f (a t f t dt [at b (t][t α (t α ] / [ f (a t f (t]dt / [at b (t] [(t α t α ] [ f (a t f (t]dt { a f (a α3 b f (a α b f (a α3 a f α a f α3 b f α b f α (α3 t α dt / (α t α dt / (α3 t α dt / (α t α dt / (α3 t α dt / (α t α dt / (α t α dt / { a f (a α3 b f α3 a f (a (α3 t α dt / / b f (a / a f b f α3 / (t α t dt t(t α dt t(t α dt (α3(t α dt} / / {a f (a t (t α dt / b f (a t(t α dt / a f t(t α dt b f α3 a f (a α3 b f (a α b f (a α3 a f α a f α3 b f α b f α b f α3 ( α3 / (α3(t α dt / (α3 t α dt / (α t α dt / (α3 t α dt / (α t α dt / (α3 t α dt / (α t α dt / (α t α dt / (α3 t α dt}

4 4 International Journal of Mathematics and Mathematical Sciences b f (a α a f α b f α ( ( α b α a f (a α3 f α3 ( b f α α b f α3 ( α3 a f (a b f (a / a f b f α3 / t(t α dt t(t α dt ( α3 ( α3 / (α3(t α dt} / / {a f (a t (t α dt / b f (a t(t α dt / a f t(t α dt b f α3 ( a f (a α3 α3 b f (a α a f α b f α b f (a α3 b f (a α α3 a f α α3 b f α α α3 } α3 α3 α { (ab [ f (a f ] α3 b f α ( α ( α α3 (t α t dt ( α3 b[ f (a f ](ab f α α a f (a (t α t dt / a f (a (t α t dt b f (a t(t α dt / b f (a t(t α dt a f t(t α dt / a f t(t α dt} / {a f (a t (t α dt / b f (a t(t α dt / a f t(t α dt b f α3 ( a f α3 (α3 α3 b f (a a f b f (α α {( α [ b f α b f (α α } b f (a a f b f ] α a f (a a f b f (a α3 a f (a b f (a (α3 α3 a f (a (α(α(α3 b f (a a f a f (a (α(α3

5 International Journal of Mathematics and Mathematical Sciences 5 The proof is done. a ( f (a f b f (a α3 (α(α3 b f (a a f α (α a f (a (α α a f (a (α }. α (3 Theorem. Let f:[,b] R be a differentiable mapping. If f q is measurable and f q, (q > is GG-convex on [a, b] for some fired α (, and t [, ], a<b, then the following integrals hold: f (a f(b Γ (α ( α [ H J α a f (b H Jα b f (a] ( (a /q ap b p pα where /p /q =. b p pα b p a p pα (pα b p a p pα (pα a p pα (pα a p b p pα (pα (pα a p (pα (pα Proof. By using Definition 4 and Lemmas and 6,wehave f (a f(b /p, Γ (α ( α [ H J α a f (b H Jα b f (a] ( a pt b p(t (tα t α p /p dt /q f (a qt(t dt [ta p (t b p ] (tα t α p /p dt /q [t (a (t ]dt (4 [ta p (t b p ](t pa (t pa dt / / /p [ta p (t b p ][(t pα t pα ]dt (a ( (a /q a p / b p / a p / /q t pα dt a p t(t pα dt / t pα (t dt b p (t pα dt / t(t pα dt a p / t pα dt b p / (t pα dt b p / /p t pα (t dt ( (a /q pα ap ap t(t pα dt a p / t(t pα b p dt pα b p pα pα b p pα b p pα pα a p / t(t pα dt a p pα pα b p pα pα b p pα b p / /p t pα (t dt ( (a /q ap b p pα b p pα b p a p pα (pα

6 6 International Journal of Mathematics and Mathematical Sciences The proof is done. b p a p pα (pα a p pα (pα a p b p pα (pα (pα a p /p (pα (pα. (5 Theorem. Let f:[,b] R be a differentiable mapping. If f is measurable and f is GG-convex on [a, b] for some fired α (,, t [, ] and k>, a<b, then the following integrals hold: f (a f(b b f { a f (a { b { f i= i= Γ (α ( α [ H J α a f (b H Jα b f (a] ( i [ln (a f (a /b f ]i (α i [ln (a f (a /b f ]i (α i (/α (a f (a /b f (b / α (/α (a f (a /b f (b / } ln (a f (a /b f }. } (6 Proof. By using Definition 4 and Lemmas, 7, 8, and9, we have f (a f(b Γ (α ( α [ H J α a f (b H Jα b f (a] a t b t [t α (t α ] / f (a t b t dt / a t b t [(t α t α ] f (a t b t dt b f / b f / b f { { { a t b t [t α (t α ] / f (a t f t dt / a t b t [(t α t α ] f (a t f t dt ( a f t (a [t α (t α ]dt b f ( a f t (a [(t α t α ]dt b f ( a f t (a t α dt b f / / b f { a f (a { b { f i= i= ( α i= ( a f t (a t α dt b f ( a f t (a (t α dt b f ( a f t (a (t α dt } b f (b } } ( i [ln (a f (a /b f ]i (α i [ln (a f (a /b f ]i (α i ( a f / (a b f [ (/ ln (a f (a /b f ]i (α i

7 International Journal of Mathematics and Mathematical Sciences 7 i= b f [ln (a f (a /b f ]i (α i [ ( αi ( a f (a b [ f { a f (a { b { f i= i= / } ] } ]} ( i [ln (a f (a /b f ]i (α i [ln (a f (a /b f ]i (α i ( α ( a f (a b f i= / [ (/ ln (a f (a /b f ]i (α i (/α (a f (a /b f / ln (a f (a /b f i= [ln (a f (a /b f ]i (α i ( α ( a f (a b f i= b f { a f (a { b { f i= i= / [(/ ln (a f (a /b f ]i } (α } i } ( i [ln (a f (a /b f ]i (α i [ln (a f (a /b f ]i (α i The proof is done. (/α (a f (a /b f (b / α (/α (a f (a /b f (b / } ln (a f (a /b f }. } (7 Theorem 3. Let f:[,b] R be a differentiable mapping. If f q is measurable and f q, (q > is GG-convex on [a, b], for some fired α (,, t [,],andk>, a<b, then the following integrals hold: f (a f(b Γ (α ( α [ H J α a f (b H Jα b f (a] a p ( i [(/ ln (a/b p ] i i= (pα i b p [ln (a/b p ] i (/ pα (ab p/ i= (pα i pα [ln ( a b p ] [( /p pα (ab p/ b p ] f q (a ln ( f (a / f [ q f ] where /p /q =. /q, (8 Proof. By using Definition 4 and Lemmas, 7, 8, and9, we have f (a f(b Γ (α ( α [ H J α a f (b H Jα b f (a] /p ( (a t b t (tα t α p dt /q f (a t b t q dt (b p ( a / b pt t pα dt b p ( a / b pt (t pα dt b p / ( a b pt (t pα dt b p / ( a /p b pt t pα dt

8 8 International Journal of Mathematics and Mathematical Sciences ln ( f (a / f q ln ( f (a [ f (a ] dt f f q qt /q (b p ( a b p ( i [ln (a/b p ] i i= b p [ln (a/b p ] i i= (pα i b p / ( a b pt t pα dt (pα i b p / ( a /p b pt (t pα dt f q (a ln ( f (a / f [ q f ] b p ( a b p ( i [ln (a/b p ] i i= b p [ln (a/b p ] i i= (pα i ( pα b p ( a b p/ (pα i i= b p [ln (a/b p ] i i= (pα i /q [ (/ ln (a/b p ] i (pα i [( a b p/ ( pαi] f q (a ln ( f (a / f [ q f ] a p ( i [ln (a/b p ] i i= (pα i b p [ln (a/b p ] i i= (pα i ( pα(ab p/ [ (/ ln (a/b p ] i i= (pα i /p /q b p [ln (a/b p ] i i= (pα i b p ( a b p/ ( pαi [ln (a/b p ] i (pα i i= f q (a ln ( f (a / f [ q f ] a p ( i [ln (a/b p ] i i= (pα i b p i= ( pα(ab p/ [ (/ ln (a/b p ] i i= (pα i /q /p [ln (a/b p ] i (pα i b p [ln (a/b p ] i b p [ln (a/b p ] i= (pα i b p ( a b p/ ( pαi [ln (a/b p ] i (pα i i= f q (a ln ( f (a / f [ q f ] a p ( i [ln (a/b p ] i i= (pα i b p i= ( pα(ab p/ [ (/ ln (a/b p ] i i= (pα i ( pα(ab p/ ( i i= [ln (a/b p ] i (pα i b p [ln ( a /p b p ] /q /p [ln (a/b p ] i (pα i f q (a ln ( f (a / f [ q f ] a p ( i [ln (a/b p ] i i= (pα i b p i= ( pα(ab p/ i= [ (/ ln (a/b p ] i (pα i /q [ln (a/b p ] i (pα i

9 International Journal of Mathematics and Mathematical Sciences 9 ( pα(ab p/ [(/ ln (a/b p ] i i= (pα i ( pα(ab p/ [ln ( a b p ] b p [ln ( a /p b p ] f q (a ln ( f (a / f [ q f ] a p ( i [ln (a/b p ] i i= (pα i ( pα (ab p/ pα b p i= /q [ln (a/b p ] i (pα i [ln ( a b p ] [( /p pα (ab p/ b p ] f q (a ln ( f (a / f [ q f ] a p ( i [(/ ln (a/b p ] i i= (pα i b p [ln (a/b p ] i (/ pα (ab p/ i= (pα i pα /q [ln ( a b p ] [( /p pα (ab p/ b p ] f q (a ln ( f (a / f [ q f ] The proof is done. 3. Main Results Based on Lemma /q. (9 Theorem 4. Let f:[,b] R be a differentiable mapping. If f is measurable and f is GG-convex on [a, b],forsome fired α (, and t [,], a<b, then the following integrals hold: Γ (α ( α [ H J α a f (b H Jα bf (a]f(ab f f (a {( α [ b f α b f (a a f b f ] α a f (a a f b f (a α3 a f (a b f (a a (α3 α3 f (a (α(α(α3 b f (a a f a f (a (α(α3 a ( f (a f b f (a α3 (α(α3 b f (a a f α (α a f (a (α α a f (a (α }. α ( Proof. By using Definition 4 and Lemmas and 6,wehave Γ (α ( α [ H J α a f (b H Jα bf (a]f(ab ba kf (ta (t b dt ( f (a f a t b t (tα t α f (a t b t dt { a t b t [t α (t α ] / f (a t b t dt / a t b t [(t α t α ] f (a t b t dt}

10 International Journal of Mathematics and Mathematical Sciences ( f (a f { t α [at b (t] / [t f (a (t f ]dt (t α [at b (t] / [t f (a (t f ]dt} / { (t α [at b (t] [t f (a (t f ]dt / t α [at b (t] ( f (a f { f (a [t f (a (t f ]dt} / f t α [at b (t] dt / f (a / f / { f (a / (t t α [at b (t] dt (t α t [at b (t] dt (t α [at b (t] dt} t(t α [at b (t] dt / f (t α [at b (t] dt / f (a t α [at b (t] dt / f (t t α [at b (t] dt} f f (a { a f (a α3 b f (a α3 a f α3 b f α ( a f (a α3 b ( ( ( / b f (a / a f b f α3 / α3 a α3 b α b (t α t dt t(t α dt t(t α dt f (a α f α f α (α3(t α dt} / / {a f (a t (t α dt / b f (a t(t α dt / a f t(t α dt b f α3 b f (a α a f α ( a α3 b f (a α α3 b f α α ( α ( α ( α f α3 ( α3 f (a α3 α3 α3 a f b f α α3 α3 α b f (a α α3 α3 }

11 International Journal of Mathematics and Mathematical Sciences f f (a { (ab [ f (a f ] α3 b f α ( α ( α3 b[ f (a f ](ab f α α a f (a (t α t dt / a f (a (t α t dt b f (a t(t α dt / b f (a t(t α dt a f t(t α dt / a f t(t α } / {a f (a t (t α dt / b f (a t(t α dt / a f t(t α dt b f α3 ( a f α3 (α3 α3 b f (a a f b f (α α b f (α α } f f (a { (ab [ f (a f ] α3 b f α ( α ( α3 b[ f (a f ](ab f α α a f (a t (t α dt / a f (a t (t α dt b f (a t(t α dt / b f (a t(t α dt a f t(t α dt / a f t(t α dt} f f (a {( α / {a f (a t (t α dt / b f (a t(t α dt / a f t(t α dt b f α3 ( a f α3 (α3 α3 b f (a a f b f (α α b f (α α }

12 International Journal of Mathematics and Mathematical Sciences [ b f α The proof is done. b f (a a f b f ] α a f (a a f b f (a α3 a f (a b f (a a (α3 α3 f (a (α(α(α3 b f (a a f a f (a (α(α3 a ( f (a f b f (a α3 (α(α3 b f (a a f α (α a f (a (α α a f (a (α }. α ( Theorem 5. Let f:[,b] R be a differentiable mapping. If f q is measurable and f q, (q > is GG-convex on [a, b], for some fired α (, and t [, ], a<b, then the following integrals hold: Γ (α ( α [ H J α a f (b H Jα bf (a]f(ab f f (a (a (a /q (a ap b p pα where /p /q =. b p pα b p a p pα (pα b p a p pα (pα a p pα (pα a p b p pα (pα (pα a p /p (pα (pα, ( Proof. By using Definition 4 and Lemmas and 6,wehave Γ (α ( α [ H J α a f (b H Jα bf (a]f(ab ba ba (ba kf (ta (t b dt a t b t (tα t α f (a t b t dt k f (at b (t dt a t b t (tα t α f (a t b t dt f (at b (t dt ( f (a f a t b t (tα t α f (a t b t dt /p ( (a t b t (tα t α p dt /q f (a t b t q dt ( f (a f ( a pt b p(t (tα t α p /p dt /q f (a qt(t dt ( f (a f [ta p (t b p ] (tα t α p /p dt /q [t (a (t ]dt ( f (a f [ta p (t b p ](t pa (t pa dt /

13 International Journal of Mathematics and Mathematical Sciences 3 / /p [ta p (t b p ][(t pα t pα ]dt (a ( f (a f /q b p a p pα (pα a p pα (pα a p b p pα (pα (pα a p (pα (pα /p. (3 ( (a /q a p / b p / a p / b p / ( f (a f t pα dt a p t(t pα dt / t pα (t dt b p (t pα dt / t(t pα dt a p / t pα dt (t pα b p / /p t pα ( tdt ( (a /q pα ap ap t(t pα dt a p / t(t pα b p dt pα b p pα pα b p pα b p / pα pα ap t(t pα dt a p pα pα b p pα pα b p pα b p / /p t pα ( tdt f f (a ( (a /q ap b p pα b p pα b p a p pα (pα The proof is done. Theorem 6. Let f:[,b] R be a differentiable mapping. If f is measurable and f is GG-convex on [a, b],forsome fired α (, and t [,], a<b, then the following integrals hold: Γ (α ( α [ H J α a f (b H Jα bf (a]f(ab ( f f (a b f { a f (a { ( i [ln (a f (a /b f (b ]i b { f i= (α i i= [ln (a f (a /b f ]i (α i (/α (a f (a /b f / α (/α (a f (a /b f (b / } ln (a f (a /b f }. } (4 Proof. By using Definition 4 and Lemmas, 7, 8, and9, we have Γ (α ( α [ H J α a f (b H Jα bf (a]f(ab ba kf (ta (t b dt a t b t (tα t α f (a t b t dt

14 4 International Journal of Mathematics and Mathematical Sciences ( f (a f { a t b t [t α (t α ] / f (a t b t dt / a t b t [(t α t α ] f (a t b t dt} ( f (a f a t b t [t α (t α ] / f (a t f t dt / a t b t [(t α t α ] f (a t f t dt ( f (a f b f / b f / ( a f t (a [t α (t α ]dt b f ( a f t (a [(t α t α ]dt b f ( f f (a { { { b f ( a f t (a t α dt b f / / ( a f t (a (t α dt b f ( a f t (a t α dt b f ( a f t (a (t α dt } b f (b } } ( f f (a b f { a f (a { ( i [ln (a f (a /b f (b ]i b { f i= (α i i= ( α i= i= [ln (a f (a /b f ]i (α i ( a f / (a b f [ (/ ln (a f (a /b f ]i (α i [ln (a f (a /b f ]i (α i [ ( αi ( a f / (a } ] b [ f (b } ] } ( f f (a b f { a f (a { ( i [ln (a f (a /b f (b ]i b { f i= (α i i= The proof is done. [ln (a f (a /b f ]i (α i (/α (a f (a /b f / α (/α (a f (a /b f (b / } ln (a f (a /b f }. } (5 Theorem 7. Let f:[,b] R be a differentiable mapping. If f q is measurable and f q, (q > is GG-convex on [a, b],

15 International Journal of Mathematics and Mathematical Sciences 5 for some fired α (, and t [, ], a<b, then the following integrals hold: Γ (α ( α [ H J α a f (b H Jα bf (a]f(ab ( f f (a a p ( i [(/ ln (a/b p ] i i= (pα i b p [ln (a/b p ] i (/ pα (ab p/ i= (pα i pα [ln ( a b p ] [( pα /p (ab p/ b p ] f q (a ln ( f (a / f [ q f ] where /p /q =. /q, (6 Proof. By using Definition 4 and Lemmas, 7, 8, and9, we have Γ (α ( α [ H J α a f (b H Jα bf (a]f(ab ba ba ba kf (ta (t b dt a t b t (tα t α f (a t b t dt k f (ta (t b dt a t b t (tα t α f (a t b t dt k f (ta (t b dt /p ( (a t b t (tα t α p dt /q f (a t b t q dt ( f (a f b p / ( a b pt t pα dt b p ( a / b pt (t pα dt b p / ( a b pt (t pα dt b p / ( a /p b pt t pα dt f q ( f (a f b p b p qt [ f (a ] dt f /q ( a b pt t pα dt b p / ( a b pt t pα dt ( a b pt (t pα / dt ( a b pt (t pα dt b p / ( a b pt (t pα dt b p / ( a /p b pt t pα dt f q (a ln ( f (a / f [ q f ] ( f (a f b p ( a b p ( i [ln (a/b p ] i i= b p [ln (a/b p ] i i= (pα i (pα i b p / ( a /p b pt (t pα dt /q b p / ( a b pt t pα dt f q (a ln ( f (a / f [ q f ] ( f (a f b p ( a b p ( i [ln (a/b p ] i i= (pα i /q

16 6 International Journal of Mathematics and Mathematical Sciences b p [ln (a/b p ] i i= (pα i ( pα b p ( a b p/ i= [ (/ ln (a/b p ] i (pα i b p [ln (a/b p ] i [ ( a p/ /p pαi ( i= (pα i b ] f q (a ln ( f (a / f [ q f ] ( f (a f a p ( i [ln (a/b p ] i i= (pα i b p i= ( pα(ab p/ [ (/ ln (a/b p ] i i= (pα i b p [ln (a/b p ] i i= (pα i /q [ln (a/b p ] i (pα i b p ( a b p/ ( pαi [ln (a/b p ] i (pα i i= f q (a ln ( f (a / f [ q f ] ( f (a f a p ( i [ln (a/b p ] i i= (pα i b p i= ( pα(ab p/ [ (/ ln (a/b p ] i i= (pα i /q /p [ln (a/b p ] i (pα i b p [ln (a/b p ] i b p [ln ( a i= (pα i b p ] b p ( a b p/ ( pαi [ln (a/b p ] i (pα i i= ln ( f (a / [ f f (a /f q /p q ] /q ( f f (a a p ( i [ln (a/b p ] i i= (pα i b p i= ( pα(ab p/ [ (/ ln (a/b p ] i i= (pα i ( pα(ab p/ [(/ ln (a/b p ] i i= (pα i ( pα(ab p/ [ln ( a b p ] b p [ln ( a /p b p ] [ln (a/b p ] i (pα i f q (a ln ( f (a / f [ q f ] ( f f (a a p ( i [(/ ln (a/b p ] i i= (pα i b p [ln (a/b p ] i (/ pα (ab p/ i= (pα i pα /q [ln ( a b p ] [( /p pα (ab p/ b p ] f q (a ln ( f (a / f [ q f ] The proof is done. 4. Applications to Special Means /q. (7 Consider the following special means (see [3] forarbitrary real numbers x, y, x =yas follows: (M A(x, y = (x y/, x, y R; (M H(x, y = /(/x /y, x, y R \{}; (M 3 G(x, y = xy; (M 4 L(x, y = (y x/(ln y ln x, x = y,xy =;

17 International Journal of Mathematics and Mathematical Sciences 7 (M 5 L n (x, y = [(y n x n /(n (y x] /n,n Z \ {, }, x, y R,x=y. We give some applications to special means of real numbers. Proposition 8. Let a, b R \{}, a<b, x [,b]. Then, A (a, b L(a, b A (a, b L(a, b L (a, b A(a, b L (a, b A(a, b 63a b 9 ; ( /q ap b p p bp p bp a p p (p b p a p p (p a p (p (p a p /p p (p (p ; 63a b 9 ; ( /q ap b p p A (a, b L(a, b b [ a/b ln (a/b By using Lemmas 7, 8,and9,wehave A (a, b L(a, b b bp p bp a p p (p b p a p p (p a p (p (p a p /p p (p (p ; (a/b/ a/b (a/b (ln (a/b ]. I[p]H(p,( a b p, J(p,( a b p br (p, ( a b p, /p; (8 L (a, b A(a, b b [ a/b ln (a/b L (a, b A(a, b b (a/b/ a/b (a/b (ln (a/b ]; I[p]H(p,( a b p, J(p,( a b p br (p, ( a b p, /p. (9 Proof. Applying Theorems,, 4, 5,, 3, 6, and7, for f(x = x and α=,onecanobtaintheresultsimmediately. Proposition 9. Let a, b R \{}, a<b, x [,b], n.then, A(an,b n L n n (a, b L (a, b A(an,b n L n n (a, b L (a, b ap b p p na n nba n 9nab n nb n ; 96 ( nq a q(n n q b q(n /q bp p p b p p p b p a p p p3 p3 a p p(p a p /p (p (p ; Ln n (a, b L (a, b An (x, y nbn na n na n nba n 9nab n nb n ; 96

18 8 International Journal of Mathematics and Mathematical Sciences Ln n (a, b L (a, b An (x, y nbn na n ap b p p ( nq a q(n n q b q(n /q bp p p b p p p b p a p p p3 p3 a p p(p a p /p (p (p ; A(an,b n L n n (a, b L (a, b nb n an /b n ln (a n /b n 4(an /b n (a n /b n (ln (a n /b n. By using Lemmas 7, 8,and9,wehave A(an,b n L n n (a, b L (a, b b (I[p]H(p,( a b p, J(p,( a b p br (p, ( a b p, /p nbn q ln (a n /b n q [( an q b n ] Ln n (a, b L (a, b An (x, y nbn na n nb n /q an /b n ln (a n /b n 4(an /b n (a n /b n (ln (a n /b n ; ; (3 Ln n (a, b L (a, b An (x, y nbn na n b (I[p]H(p,( a b p, J(p,( a b p br (p, ( a b p, /p nbn q ln (a n /b n q [( an q b n ] /q. (3 Proof. Applying Theorems,, 4, 5,, 3, 6, and7 for f(x = x n and α=,onecanobtaintheresultsimmediately. Proposition. Let a, b R \{}, a<b, x [,b], n.then, H (a, b G (a, b L (a, b (45 (/a 6 (a/b 3(b/a 3 (/a 4(/a (9 ; H (a, b G (a, b L (a, b ap b p p ( aq b q /q bp p a p 3b p p p b p a p p p p(p a p b p /p p (p (p ; G (a, b L (a, b A (a, b b a a b (45 (/a 6 (a/b 3(b/a 3 (/a 4(/a (9 ;

19 International Journal of Mathematics and Mathematical Sciences 9 G (a, b L (a, b A (a, b b a a b ap b p p ( aq b q /q bp p a p 3b p p p b p a p p p p(p a p b p p (p (p H (a, b G (a, b L (a, b b By using Lemmas 7, 8,and9,wehave /p ( b/a ln (b/a 4(b/a/ (b/a (ln (b/a. H (a, b G (a, b L (a, b b (I[p]H(p,( a b p, J(p,( a b p br (p, ( a b p, /p /b q b q ln b /a q [a ] G (a, b L (a, b A (a, b b a a b b G (a, b L (a, b A (a, b b a a b /q ; ; (3 ( b/a ln (b/a 4(b/a/ (b/a (ln (b/a ; b (I[p]H(p,( a b p, J(p,( a b p /b q ln b /a q [ b /a br (p, ( a b p, /p q ] /q. (33 Proof. Applying Theorems,, 4, 5,, 3, 6, and7 for f(x = /x and α=,onecanobtaintheresultsimmediately. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments The authors thank the referees for their careful reading of the paper and insightful comments, which helped to improve the quality of the paper. They would also like to acknowledge the valuable comments and suggestions from the editors, which vastly contributed to improvement of the presentation of the paper. This work was supported in part by Key Project on the Reforms of Teaching Contents and Course System of Guizhou Normal College, Doctor Project of Guizhou Normal College (3BS, Guizhou Province Education Planning Project (3A6, and Key Support Subject (Applied Mathematics. References [] D. Baleanu, J. A. T. Machado, and A. C. J. Luo, Fractional Dynamics and Control, Springer, New York, NY, USA,. [] K. Diethelm, The Analysis of Fractional Differential Equations, vol. 4 of Lecture Notes in Mathematics, Springer, Berlin, Germany,. [3] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol.4, Elsevier Science B.V., Amsterdam, The Netherlands, 6. [4] V.Lakshmikantham,S.Leela,andJ.V.Devi,Theory of Fractional Dynamic Systems, Cambridge Scientific, 9. [5] K.S.MillerandB.Ross,An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 993. [6] M. W. Michalski, Derivatives of noninteger order and their applications, Dissertationes Mathematicae, vol. 38, 47pages, 993. [7] I. Podlubny, Fractional Differential Equations, Academic Press, 999. [8]V.E.Tarasov,Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer,. [9] E.Set, NewinequalitiesofOstrowskitypeformappingswhose derivatives are s-convex in the second sense via fractional integrals, Computers & Mathematics with Applications,vol.63, no. 7, pp ,.

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