On Hermite-Hadamard-Fejér type inequalities for convex functions via fractional integrals

Transkript

1 Mthemtic Morvic Vol. 1, No. 1 (017), On Hermite-Hdmrd-Fejér type inequlities for convex functions vi frctionl integrls Adullh Akkurt nd Hüsey ın Yıldırım Astrct. In this pper, we hve estlished some generlized integrl inequlities of Hermite-Hdmrd-Fejér type for generlized frctionl integrls. The results presented here would provide generliztions of those given in erlier works. 1. Introduction Let f : I R R e convex function define on n intervl I of rel numers, nd, I with <. Then the following inequlities hold: (1) f ( ) 1 f (x) dx f () f (). It ws first discovered y Hermite in 1881 in the Journl Mthesis. This inequlity (1) ws nowhere mentioned in the mthemticl literture untill In 4, Beckench, leding expert on the theory of convex functions, wrote tht the inequlity (1) ws proved y Hdmrd in In 1974, Mitrinovič found Hermite nd Hdmrd s note in Mthesis. Tht is why, the inequlity (1) ws known s Hermite-Hdmrd inequlity. We note tht Hermite-Hdmrd s inequlity my e regrded s refinements of the concept of convexity nd it follows esily from Jensen s inequlity. This inequlity (1) hs een received renewed ttention in recent yers nd remrkle vriety of refinements nd generliztions hve een found in 4-16, 18, 0. The most well known inequlities connected with the integrl men of convex functions re Hermite-Hdmrd inequlities or its weighted versions, the so-clled Hermite-Hdmrd-Fejěr inequlity. In 10, Fejér estlished the following Fejér inequlity which is the weighted generliztion of Hermite-Hdmrd inequlities (1): 010 Mthemtics Suject Clssifiction. Primry: 6A33, 6A51; Secondry: 6D15. Key words nd phrses. Integrl inequlities, Frctionl integrls, Hermite-Hdmrd- Fejér Inequlity. 105 c 017 Mthemtic Morvic

2 106 On Hermite-Hdmrd-Fejér type inequlities for convex... Theorem 1.1. Let f : I R e convex on I nd let, I with <. Then the inequlity () f ( ) g(x)dx 1 f() f() f(t)g(x)dt g(x)dx. holds, where f :, R is nonnegtive, integrle, nd symmetric to. In 11, Sriky et l. represented Hermite Hdmrd s inequlities in frctionl integrl forms s follows. Theorem 1.. Let f :, R e positive function with 0 < nd f L,. If f is convex function on,, then the following inequlities for frctionl integrls holds (3) f ( ) Γ (α 1) ( ) α J α f() J α f() with α > 0. In 5 Set et. l. otined the following lemm. f() f(), Lemm 1.1. Let f :, R e differentile mpping on (, ) with < nd let g :, R. If f, g L,, then the following identity for frctionl integrls holds: (4) where f ( ) J α g() J α ( ) ( g() ) J α (gf) () J α (fg) () ( ) ( ) k(t) = (s ) α1 g(s)ds, t ( s) α1 g(s)ds, t = 1 k(t)df(t),,,,. We give some necessry definitions nd mthemticl preliminries of frctionl clculus theory which re used throughout this pper. Definition 1.1. Let h (x) e n incresing nd positive monotone function on 0, ), lso derivtive h (x) is continuous on 0, ) nd h (0) = 0.

3 Adullh Akkurt nd Hüsey ın Yıldırım 107 The spce X p h (0, ) (1 p < ) of those rel-vlued Leesque mesurle functions f on 0, ) for which 1 (5) f X p = h 0 f(t) p h (x) dt p <, 1 p nd for the cse p = (6) f X = ess sup f(t)h (x). h 1t< Definition 1.. (6). In prticulr, when h (x) = x (1 p < ) the spce X p h (0, ) coincides with the L p0, )spce ( f X = f h ) nd lso if we tke h (x) = xk1 k 1 (k 0) the spce Xp h (0, ) coincides with the L p,k 0, )spce. Definition 1.3. (1). Let (, ) e finite intervl of the rel line R nd α > 0. Also let h (x) e n incresing nd positive monotone function on (,, hving continuous derivtive h (x) on (, ). The left- nd rightsided frctionl integrls of function f with respect to nother function h on, re defined y ) (7) (J α,h f (x) := 1 x h (x) h (t) α1 h (t) f (t) dt, x nd (8) (J α,h f ) (x) := 1 x h (t) h (x) α1 h (t) f (t) dt, x. Definition 1.4. If we tke h(x) = x, then the equlities (7) nd (8) will e (9) (J α f) (x) = 1 Γ(α) nd (10) (J α f) (x) = 1 Γ(α) x x (x t) α1 f(t)dt, x > (t x) α1 f(t)dt, > x. These integrls re clled left-sided Riemnn-Liouville frctionl integrl nd right-sided Riemnn-Liouville frctionl integrl respectively 1-3, 6, 17. In this pper, we hve estlished some generlized frctionl integrl inequlities. The results presented here would provide generliztions of those given in erlier works.

4 108 On Hermite-Hdmrd-Fejér type inequlities for convex.... Min Results Lemm.1. Let f :, R e differentile mpping on (, ) with < nd let g :, R. If f, g X p h,, then the following identity for frctionl integrls holds: f ( h ( )) J α g(h()) J α g(h()) ( ) ( ) J α (g (f h)) () J α (g (f h)) () (11) ( ) ( ) = 1 k(t)df(h(t)) where k(t) = Proof. It suffices to note tht I = = (h(s) h()) α1 g(s)h (s)ds, (h() h(s)) α1 g(s)h (s)ds, k(t)df(h(t)) = I 1 I. By integrtion y prts, we get I 1 = = t t,,,. (h(s) h()) α1 g(s)h (s)ds df(h(t)) (h() h(s)) α1 g(s)h (s)ds df(h(t)) (h(s) h()) α1 g(s)h (s)ds df(h(t)) (h(s) h()) α1 g(s)h (s)ds f(h(t))

5 Adullh Akkurt nd Hüsey ın Yıldırım 109 = f ( h ( )) = nd similrly I = = (h(t) h()) α1 g(t)f(h(t))h (t)dt (h(s) h()) α1 g(s)h (s)ds (h(t) h()) α1 g(t)f(h(t))h (t)dt f ( h ( )) J α g(h()) J α (g (f h)) () ( ) ( ) (h() h(s)) α1 g(s)h (s)ds df(h(t)) (h(s) h()) α1 g(s)h (s)ds f(h(t)) = f ( h ( )) = Thus, cn write I = I 1 I = (h() h(t)) α1 g(t)f(h(t))h (t)dt (h() h(s)) α1 g(s)h (s)ds (h() h(t)) α1 g(t)f(h(t))h (t)dt f ( h ( f ( h ( )) J α g(h()) J α (g (f h)) (). ( ) ( ) )) J α ( ) g(h()) J α ( g(h()) ) J α ( ) (g (f h)) () J α ( ) (g (f h)) ().

6 110 On Hermite-Hdmrd-Fejér type inequlities for convex... Multiplying the oth sides () 1, we otin (11) which compltes the proof. Remrk.1. If we choose h(x) = x in Lemm.1, then the inequlity (11) reduces to (1.4). Remrk.. If we choose h(x) = x, g(x) = 1 nd α = 1 in Lemm.1, we otin Lemm.1 in 1. Theorem.1. Let f : I R e differentile mpping on I nd f X p h, with < nd g :, R is continuous. If f is convex on,, then the following inequlity for frctionl integrls holds: (1) f ( h ( with α > 0. )) J α ( ) g(h()) J α ( ) g(h()) J α (g (f h)) () J α (g (f h)) () ( ) ( ) g X h,, (h()h())γ(α1) f (h( (h()) )h())α1 α1 (h() h()) f (h()) f (h()) f (h()) f (h()) (h( )h())α α (h( )h())α α h,, (h()h())γ(α1) (h()h( ))α1 α1 (h() h()) (h()h( ))α α Proof. If f is convex on,, we know tht for t, f (h(t)) ( = h() h(t) f h() h() h() h(t) h() h() h() h(t) h() h() h() h()) f (h()) h(t) h() h() h() From Lemm.1, we hve f ( h ( )) J α g(h()) J α g(h()) ( ) ( ) (h()h( ))α α f (h()).

7 Adullh Akkurt nd Hüsey ın Yıldırım 111 J α (g (f h)) () J α (g (f h)) () ( ) ( ) 1 (h(s) h()) α1 g(s)h (s)ds f (h(t)) h (t)dt h,, (h() h()) (h() h(s)) α1 g(s)h (s)ds f (h(t)) h (t)dt (h(s) h()) α1 h (s)ds ( h() h(t) f (h()) ) h (t)dt (h(s) h()) α1 h ( (s)ds h(t) h() f (h()) ) h (t)dt (h() h(s)) α1 h (s)ds h,, (h() h()) ( h() h(t) f (h()) ) h (t)dt (h() h(s)) α1 h ( (s)ds h(t) h() f (h()) ) h (t)dt. h,, (h() h()) Γ (α 1) (h(t) h()) α ( h() h(t) f (h()) ) h (t)dt (h(t) h()) α1 f (h()) h (t)dt h,, (h() h(t)) α1 f (h()) h (t)dt (h() h()) Γ (α 1)

8 11 On Hermite-Hdmrd-Fejér type inequlities for convex... (h() h(t)) α1 ( h(t) h() f (h()) ) h (t)dt h,, f (h()) (h() h()) Γ (α 1) (h( ) h())α f (h()) α h,, (h() h()) Γ (α 1) (h() h( ))α α f (h()) f (h()) (h( ) h())α1 α 1 (h( ) h())α α (h() h()) (h() h( ))α1 α 1 (h() h( ))α α (h() h()). This completes the proof. Remrk.3. If we choose h(x) = x in Theorem.1, we otin Theorem 6 in 5. Remrk.4. If we choose h(x) = x, g(x) = 1 nd α = 1 in Theorem.1, we otin Theorem. in 1. Theorem.. Let f : I R e differentile mpping on I nd f X p h, with < nd g :, R is continuous. If f q is convex on,, q 1 then the following inequlity for frctionl integrls holds: (13) f ( h ( )) J α ( J α ( ) (g (f h)) () J α ( h,, (h() h()) 1 q ) g(h()) J α ( ) g(h()) ( h ( (g (f h)) () ) ) ) 1 1 h() q f (h()) q α (α 1) (h( ) h())α1 (h() h()) α 1 f (h()) q (h( ) 1 h())α q α (h( ) h())α α ( h,, h () h( ) ) 1 1 q f (h()) q (h() h()) 1 q α (α 1)

9 with α > 0. Adullh Akkurt nd Hüsey ın Yıldırım 113 (h() h( ))α1 (h() h()) (h() h( ))α α 1 α f (h()) 1 q (h() h( ))α q. α Proof. If f q is convex on,, we know tht for t, f (h(t)) ( ) q = h() h(t) f h(t) h() q h() h() h() h() h() h() h() h(t) f (h()) q h(t) h() f (h()) q. h() h() h() h() From Lemm.1, power men inequlity nd the convexity of f q, it follows tht f ( h ( )) J α g(h()) J α g(h()) ( ) ( ) J α (g (f h)) () J α (g (f h)) () ( ) ( ) 1 t 11/q (h(s) h()) α1 g(s)h (s)ds h (t)dt 1 h,, 1/q (h(s) h()) α1 g(s)h (s)ds f (h(t)) q h (t)dt (h() h(s)) α1 g(s)h (s)ds h (t)dt 11/q 1/q (h() h(s)) α1 g(s)h (s)ds f (h(t)) q h (t)dt (h(s) h()) α1 h (s)ds h (t)dt 11/q

10 114 On Hermite-Hdmrd-Fejér type inequlities for convex... h,, 1/q (h(s) h()) α1 h (s)ds f (h(t)) q h (t)dt h,, (h() h()) 1 q (h() h(s)) α1 h (s)ds h (t)dt 11/q 1/q (h() h(s)) α1 h (s)ds f (h(t)) q h (t)dt. (h(s) h()) α1 h (s)ds ( h() h(t) f (h()) q ) h (t)dt (h(s) h()) α1 h ( (s)ds h(t) h() f (h()) q ) h (t)dt h,, t (h() h()) 1 (h() h(s)) α1 h (s)ds q ( h() h(t) f (h()) q ) h (t)dt (h() h(s)) α1 h ( (s)ds h(t) h() f (h()) q ) h (t)dt h,, (h() h()) 1 q ( h ( ) h() α (α 1) ) 11/q f (h()) q (h( ) h())α1 (h() h()) α 1 f (h()) q (h( ) h())α α f (h()) q (h( ) 1 h())α q α 1 q 1 q

11 h,, (h() h()) 1 q Adullh Akkurt nd Hüsey ın Yıldırım 115 ( h () h( ) α (α 1) ) 11/q f (h()) q (h() h( ))α1 (h() h()) α 1 (h() h( ))α α f (h()) q (h() h( ))α α 1 q. Remrk.5. If we choose h(x) = x in Theorem., we otin Theorem 7 in 5. Lemm.. Let f :, R e differentile mpping on (, ) with < nd let g :, R. If f, g X p h,, then the following identity for frctionl integrls holds: f ( h ( )) J α g(h()) J α g(h()) (14) where J α ( ) g(h()) J α ( ) g(h()) 1 (h() h()) α1 g(s)h (s)ds J α (g (f h)) () J α (g (f h)) () J α (g (f h)) () J α (g (f h)) () ( ) ( ) 1 = 1 k(t)df(h(t)) k(t) = (h() h()) α1 g(t)f(h(t))h (t)dt (h() h()) α1 (h() h(s)) α1 (h(s) h()) α1 g(s)h (s)ds t,, (h() h(s)) α1 (h(s) h()) α1 (h() h()) α1 g(s)h (s)ds t,.

12 116 On Hermite-Hdmrd-Fejér type inequlities for convex... Proof. It suffices to note tht I = = k(t)df(h(t)) t t = I 1 I. By integrtion y prts, we get (h() h()) α1 (h() h(s)) α1 (h(s) h()) α1 g(s)h (s)ds df(h(t)) (h() h(s)) α1 (h(s) h()) α1 (h() h()) α1 g(s)h (s)ds df(h(t)) I 1 = t (h() h()) α1 (h() h(s)) α1 (h(s) h()) α1 g(s)h (s)ds df(h(t)) = t (h() h()) α1 (h() h(s)) α1 (h(s) h()) α1 g(s)h (s)ds f(h(t)) (h() h()) α1 (h() h(t)) α1 (h(t) h()) α1 g(t)f(h(t))h (t)dt = f ( h ( )) (h() h()) α1 (h() h(s)) α1. (h(s) h()) α1 g(s)h (s)ds

13 Adullh Akkurt nd Hüsey ın Yıldırım 117 nd similrly I = = t (h() h()) α1 (h() h(t)) α1 (h(t) h()) α1 g(t)f(h(t))h (t)dt = f ( h ( )) J α g(h()) J α ( ) g(h()) t 1 (h() h()) α1 g(s)h (s)ds J α (g (f h)) () J α ( ) (g (f h)) () 1 (h() h()) α1 g(t)f(h(t))h (t)dt (h() h(s)) α1 (h(s) h()) α1 (h() h()) α1 g(s)h (s)ds df(h(t)) (h() h(s)) α1 (h(s) h()) α1 (h() h()) α1 g(s)h (s)ds f(h(t)) (h() h(t)) α1 (h(t) h()) α1 (h() h()) α1 g(t)f(h(t))h (t)dt = f ( h ( )) (h() h(s)) α1 (h(s) h()) α1 (h() h()) α1 g(s)h (s)ds

14 118 On Hermite-Hdmrd-Fejér type inequlities for convex... = (h() h(t)) α1 (h(t) h()) α1 Thus, cn write (h() h()) α1 g(t)f(h(t))h (t)dt f ( h ( J α ( I 1 I = )) J α ( ) g(h()) J α g(h()) 1 (h() h()) α1 g(s)h (s)ds ) (g (f h)) () J α (g (f h)) () 1 f ( h ( (h() h()) α1 g(t)f(h(t))h (t)dt. )) J α g(h()) J α g(h()) J α ( ) g(h()) J α ( ) g(h()) 1 1 (h() h()) α1 g(s)h (s)ds J α (g (f h)) () J α (g (f h)) () (h() h()) α1 g(s)h (s)ds J α ( ) (g (f h)) () J α ( ) (g (f h)) () 1 1 (h() h()) α1 g(t)f(h(t))h (t)dt (h() h()) α1 g(t)f(h(t))h (t)dt

15 Adullh Akkurt nd Hüsey ın Yıldırım 119 Multiplying the oth sides () 1, we otin (14) which completes the proof. Remrk.6. If we choose h(x) = x in Lemm., we hve f ( ) J α g() J α g() J α g() J α g() ( ) ( ) ( )α1 g(s)ds J α (gf) () J α (gf) () where J α (gf) () J α ( )α1 (gf) () ( ) ( ) = 1 k(t)df(t) k(t) = ( )) α1 ( s) α1 (s ) α1 g(s)ds, t ( s) α1 (s ) α1 ( ) α1 g(s)ds, t g(t)f(t)dt,,,. Remrk.7. If we choose h(x) = x nd g(x) = 1 in Lemm., we hve f ( ) ( ( ) α 1 1 ( )) 1 α α J α f() J α f() J α f() J α ( )α1 f() f(t)dt ( ) ( ) = 1 k(t)df(t) where k(t) = ( ) α1 ( s) α1 (s ) α1 ds, t,, ( s) α1 (s ) α1 ( ) α1 ds, t,.

16 10 On Hermite-Hdmrd-Fejér type inequlities for convex... Remrk.8. If we choose h(x) = x, α = 1 nd g(x) = 1 in Lemm., we otin Lemm.1 in 1 1 ( 1 = ( ) f(t)dt f ( ) 0 1 tf (t (1 t))dt 1 ) (t 1) f (t (1 t))dt. Theorem.3. Let f : I R e differentile mpping on I nd f X p h, with < nd g :, R is continuous. If f is convex on,, then the following inequlity for frctionl integrls holds: f ( g,, ( ) ) J α g() J α g() J α ( ) g() J α ( )α1 g() g(s)ds ( ) J α (gf) () J α (gf) () J α (gf) () ( ) J α ( )α1 (gf) () ( ) g,, ( ) with α > 0. t g(t)f(t)dt ( ) α1 ( s) α1 (s ) α1 ds ( ( t) f () (t ) f () ) dt ( s) α1 (s ) α1 ( ) α1 g(s)ds ( ( t) f () (t ) f () ) dt Proof. If f is convex on,, we know tht for t, f (t) ( = t f t ) t f () t f ().

17 Adullh Akkurt nd Hüsey ın Yıldırım 11 From Remrk.6, we hve f ( 1 ) J α g() J α g() J α ( ) g() J α ( ) g() ( )α1 g(s)ds J α (gf) () J α (gf) () J α (gf) () J α ( )α1 (gf) () ( ) ( ) 1 ( ) g(t)f(t)dt ( ) α1 ( s) α1 (s ) α1 g(s)ds f (t) dt ( s) α1 (s ) α1 ( ) α1 ds g(s) f (t) dt t g,, ( ) ( ) α1 ( s) α1 (s ) α1 g(s)ds ( ( t) f () (t ) f () ) dt t t g,, ( ) ( s) α1 (s ) α1 ( ) α1 g(s)ds ( ( t) f () (t ) f () ) dt ( ) α1 ( s) α1 (s ) α1 ds ( ( t) f () (t ) f () ) dt ( s) α1 (s ) α1 ( ) α1 g(s)ds ( ( t) f () (t ) f () ) dt

18 1 On Hermite-Hdmrd-Fejér type inequlities for convex... g,, ( ) g,, ( ) ( ) α1 ( s) α1 (s ) α1 ds ( ( t) f () (t ) f () ) dt t ( s) α1 (s ) α1 ( ) α1 g(s)ds ( ( t) f () (t ) f () ) dt. This completes the proof. References 1 A.A. Kils, H.M. Srivstv, J.J. Trujillo, Theory nd Applictions of Frctionl Differntil Equtions, Elsevier B.V., Amsterdm, 006. A. Akkurt, H. Yıldırım, On Feng Qi Type Integrl Inequlities For Generlized Frctionl Integrls, IAAOJ, Scientific Science, 1(), (013), A. Akkurt, Z. Kırty, H. Yıldırım, Generlized Frctionl Integrls Inequlities for Continuous Rndom Vriles, Journl of Proility nd Sttistics, Volume 015 (015), Article ID , 7 pges, 4 E. F. Beckench, Convex functions, Bull. Amer. Mth. Soc., 54 (1948) E. Set, İ. İşcn, M. E. Özdemir nd M. Z. Srıky, On new Hermite-Hdmrd-Fejer type inequlities for convex functions vi frctionl integrls, Applied Mthemtics nd Computtion, 59(015), H. Yıldırım, Z. Kırty, Ostrowski Inequlity for Generlized Frctionl Integrl nd Relted Inequlities, Mly Journl of Mtemtik: Volume, Issue 3, 014, pp İ. İşcn, Hermite Hdmrd Fejér type inequlities for convex functions vi frctionl integrls, Stud. Univ. Beş-Bolyi Mth. 60 (015), No. 3, J. Prk, Inequlities of Hermite-Hdmrd-Fejér Type for Convex Functions vi Frctionl Integrls,Interntionl Journl of Mthemticl Anlysis,Vol. 8, 014, no. 59, pp , 9 K. L. Tseng, S. R. Hwng, S. S. Drgomir, Fejér-type inequlities (I), J. Inequl. Appl., 010 (010), Art ID: , 7 pges L. Fejér, Uerdie Fourierreihen, II, Mth. Nturwise. Anz Ungr. Akd., Wiss 4 (1906) (in Hungrin).

19 Adullh Akkurt nd Hüsey ın Yıldırım M.Z. Sriky, E. Set, H. Yldız, N. Bşk, Hermite Hdmrd s inequlities for frctionl integrls nd relted frctionl inequlities, Mth. Comput.Modell. 57 (9) (013) M. E. Özdemir, M. Avic, H. Kvurmci, Hermite-Hdmrd type inequlities for s- convex nd s-concve functions vi frctionl integrls, Turkish Journl of Science 1 (016): M. Z. Sriky, On new Hermite Hdmrd Fejér type integrl inequlities, Stud. Univ. Bes-Bolyi Mth. 57(3) (01) M. Tunç, On some new inequlities for convex functions, Turk. J. Mth., 35 (011), M. Z. Sriky, S. Erden, On the Hermite-Hdmrd-Fejér type integrl inequlity for convex function, Turk. J. Anl. Numer Theory (3) (014) M. Z. Sriky, A. Sğlm nd H. Yıldırım, On some Hdmrd-type inequlities for h-convex functions, J. Mth. Ineq., (3) (008), S.G. Smko, A.A. Kils, nd O.I. Mrichev, Frctionl Integrls nd Derivtives - Theory nd Applictions, Gordon nd Brech, Linghorne, S. Belri, Z. Dhmni, On some new frctionl integrl inequlities, J. Ineq. Pure Appl. Mth. 10 (3) (009). Art S. Kılınç, H. Yıldırım, Generlized Frctionl Integrl Inequlities Involving Hipergeometric Opertors, IJPAM., vol.101 no.1 (015) pp Z. Dhmni, On Minkowski nd Hermite-Hdmrd integrl inequlities vi frctionl integrtion, Ann. Funct. Anl. 1(1) (010) U.S. Kırmcı, Inequlities for differentile mppings nd pplictions to specil mens of rel numers nd to midpoint formul, Appl. Mth. Comput. 147 (1) (004) Adullh Akkurt Deprtment of Mthemtics University of Khrmnmrş Sütçü İmm Khrmnmrş Turkey E-mil ddress: dullhmt@gmil.com Hüsey ın Yıldırım Deprtment of Mthemtics University of Khrmnmrş Sütçü İmm Khrmnmrş Turkey E-mil ddress: hyildir@ksu.edu.tr