Hermite-Hadamard Type Inequalities for the Functions whose Second Derivatives in Absolute Value are Convex and Concave

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1 Applied Mthemticl Sciences Vol no HIKARI Ltd Hermite-Hdmrd Type Ineulities for the Functions whose Second Derivtives in Absolute Vlue re Conve nd Concve Jekeun Prk Deprtment of Mthemtics Hnseo University Seosn Choongnm Kore Copyright c 0 Jekeun Prk. This is n open ccess rticle distributed under the Cretive Commons Attribution License which permits unrestricted use distribution nd reproduction in ny medium provided the originl work is properly cited. Abstrct In this rticle the uthor obtin some generliztion on Hermite- Hdmrd-like type ineulities which gives n new estimte between b b [ f()d nd f ( ) ] b f()f(b) for functions whose second derivtives in bsolute vlue t certin powers re respectively conve nd concve. Mthemtics Subject Clssifiction: 6D5 6A5 Keywords: Hermite-Hdmrd ineulity; Conve functions; Concve function; Hölder ineulity; Power-men ineulity; Hypergeometric function; Bett function Introduction Recll tht function f : I R R is sid to be conve on I if the ineulity f(t ( t)y) tf() ( t)f(y) () holds for ll y I nd t [0 ] nd f is sid to be concve on I if the ineulity () holds in reversed direction.

2 6 Jekeun Prk Mny ineulities hve been estblished for conve functions but the most fmous is the Hermite-Hdmrd ineulity due to its rich geometricl significnce nd pplictions which is stted s follow: Let f : I R R be conve function define on n intervl I of rel numbers nd b I with < b. Then the following double ineulities hold: ( b ) f b f(t)dt f() f(b). () Both ineulities hold in the reversed direction if f is concve. It ws first discovered by Hermite in 88 in the Journl Mthesis. This ineulity () ws no mentioned in the mthemticl literture untill 893. In [] Beckenbch leding epert on the theory of conve functions wrote tht the ineulity () ws proved by Hdmrd in 893. In 97 Mitrinovič found Hermite nd Hdmrd s note in Mthesis. Tht is why the ineulity () ws known s Hermite-Hdmrd ineulity. We note tht Hermite-Hdmrd s ineulity my be regrded s refinements of the concept of conveity nd it follows esily from Jensen s ineulity. This ineulity () hs been received renewed ttention in recent yers nd remrkble vriety of refinements nd generliztions hve been found in []-[7]. In recent pper[] Tseng et. gives refinement of (): ( b f ) f( 3b [ f l estblished the following result which ) ( f 3b ) ( b ) f()d b ] f() f(b) f() f(b) (3) f : [ b] R is conve function. In[9] Ltif estblished some new Hdmrd-type ineulities for whose derivtives in bsolute vlues re conve: Theorem.. Let f : I R R be differentible function define on the interior I 0 of n intervl I in R such tht f L([ b]) b I 0 with < b. If f is conve on [ b] for some fied then the following

3 Hermite-Hdmrd type ineulities 7 ineulity holds: (b )f(b) ( )f() f() b b ( ( ) ( 5 f )[ () f () b 6 ( f () 5 f () ) } 6 (b ) ( 5 f () f (b) ) b 6 ( f () 5 f (b) ) }] 6 ) f(u)du () for ll [ b]. Here if we choose = b in () we hve some Hermite-Hdmrd ineulities which gives n estimte between b f()d nd f ( ) ] [ b b f()f(b) for functions whose derivtives in bsolute vlue re conve. Here we recll the definitions of the Gmm function nd the Bett function Note tht Γ() = β( y) = 0 0 e t dt t ( t) y dt. β( y) = Γ()Γ(y) Γ( y). The integrl form of the hypergeometric function is defined by F [ y c z] = for z < c > y > 0. β(y c y) 0 t y ( t) c y ( zt) dt In this rticle new generl identity for continuously twice differentible functions is estblished. By mking use of this eulity uthor hs obtined new estimtes on generliztion of Hermite-Hdmrd-like type ineulities for functions whose second derivtives in bsolute vlue t certin powers re conve nd concve.

4 8 Jekeun Prk Min results In this section for the simplicity of the nottion let for ny [ b]. I f () b f(u)du ( b f() ) f () f() f(b) } In order to prove our min results we need the following lemm: Lemm. Let f : I R R be twice differentible function on the interior I 0 of n intervl I such tht f L [ b] b I with < b. Then for ny [ b] we hve the following identity: I f () = [ (t ) ( t b ) ( f t ) dt (b ) (t b) ( t b ) ( f t ) dt] Proof. By integrtion by prts we cn stte (i)i By similr wy we get (ii)i (t ) ( t b ) ( f t ) dt = ( )( b ) f () b f() ( 3 b ) f() f(u)du. (5) (t b) ( t b ) ( f t ) dt = ( b )( b ) f () b f(b) ( 3b ) b f() f(u)du. (6) By the eulities (5) nd (6) we get the desired result.

5 Hermite-Hdmrd type ineulities 9 Theorem.. Let f : I [0 ) R be twice differentible function on the interior I 0 of n intervl I such tht f L [ b] b I 0 with < b. If f is conve on [ b] then for ny [ b] the following ineulities hold: () For b we hve: I f () λ = 8( ) (b ) λ ( ) = (b ) 3 [ λ ( f ) ( λf b ) 9(b ) λ 3 f ( ) λ f ( b ) ] (7) λ 3 = 8( ) ( b 3) ( b )(7b 6) λ = ( b ) (3b ) (b ) 3. (b) For > b we hve: I f () λ 5 = (b ) 3 λ 6 = 8( )(b ) [ λ ( 5f ) ( λ6f b ) 9(b ) λ 7 f ( ) λ8 f ( b ) ] (8) λ 7 = 8(b ) (3 b) ( b) (6 7 b) λ 8 = ( b ) ( 3 b) (b ) 3. Proof. From Lemm we hve I f ( b; ) = (b ) (b ) Since f is conve on [ b] we hve: () For b we hve: [ ( b) ( (t ) t f t ) dt ( b) ( (t b) t f t ) ] dt } I I (Sy) (9)

6 0 Jekeun Prk (i) I = (t ) ( t b ) f ( t ) dt = (t ) ( b t ) ( f t ) dt = (t ) ( b t ) ( f t t ) dt (t ) ( b t ) t f ( ) t f ( ) }dt ( ) = (b ) f ( ) f ( ( b 3) ) }. (0) By similr wy we get Here note tht (ii) I = (t b) ( t b ) f ( t ) dt = b (b t) ( b t ) ( f t ) dt (b t) ( t b ) ( f t ) dt. () b nd b = (b t) ( b t ) ( f t ) dt b (b t) ( b t ) ( f t b ( b ) b t ) dt b ( b ) 96 (3b ) ( f b ) ( ) ( 7b 6 f ) } () b = (b t) ( t b ) ( f t ) dt b (b t) ( t b ( f b (b )3 96 ) f ( t b b b b b t b b ( b) ) dt ) f ( b ) }. (3)

7 Hermite-Hdmrd type ineulities By substituting () nd (3) in () we hve: [ ( I b ) (7b 6) f ( ) (b ) 3 f ( b ) 96 ( b ) (3b ) (b ) 3} ( f b )]. () By substituting (0) nd () in (9) we hve the desired result (8). (b) For > b we hve: (i) I = (t ) ( t b ) f ( t ) dt = Note tht b (t ) ( b = b b (t ) ( b b (t ) ( b (t ) ( t b t ) f ( t ) dt t ) ( f t b t ) f ( t ) dt ( b ) f ( t ) dt. (5) ) b t ) dt b nd b = (b )3 96 (t ) ( t b ) ( f t ) dt b f ( ) f ( b) } (t ) ( t b ) f ( t b b t b ( b) 96 (6 ) ( 7 b f ) ( ) ( 3 b f b ( b) ) dt By substituting (6) nd (7) in (5) we get [ (b ) 3f I ( ) ( ) ( ) ( b 6 7 b f ) 96 ( b ) 3 ( b ) ( 3 b )} f ( b (6) ) }. (7) ) ]. (8)

8 Jekeun Prk By similr wy we get (ii) I = (t b) ( t b ) f ( t ) dt = (b t) ( t b ) f ( t b b b t ) dt b (b ) ( ) ( f b ) ( ) ( 3 b f ) }. (9) By substituting (8) nd (9) in (9) we hve the desired result (8). Theorem.. Let f : I [0 ) R be twice differentible function on the interior I 0 of n intervl I such tht f L [ b] b I 0 with < b. If f is conve on [ b] for some fied > with = then p for ny [ b] the following ineulities hold: () For b we hve: [ I f () µ p ( f ( ) f ( ) )} (b ) µ p b ( f ( b) f ( ) ) b ( f ( b) f ( b ) )} ] (0) µ = (b )p ( ) p ( ) F p [ p p p ( p) b ] (b )p ( p)(b ) p µ = β( p ( p) 3p p)( ( )p ) ( b) p (b ) } F [ p p p b ]. (b) For > b we hve: [ I f () µ p b ( f ( 3 ) f ( b) ) (b ) b ( f ( ) f ( b) )} µ p b ( f ( ) f ( b ) )} ] ()

9 Hermite-Hdmrd type ineulities 3 (b )p (b ) p µ 3 = β( p p p) (b )p µ = ( b)p p (b ) p p β( p p) ( b)p p F [ p p p b ] b F [ p p p b ] b } }. Proof. From Lemm nd using the well-known Hölder integrl ineulity we hve I f () [ (t ) ( t b ) p } p dt f ( t ) } dt (b ) (t b) ( t b ) p } p dt f ( t ) } ] dt. () Since f is conve on [ b] we hve: () For b we hve: (i) (ii) (iii) (iv) (t ) ( t b ) p dt = µ (t b) ( t b ) p dt = µ f ( t ) f ( dt ) f ( ) } f ( t ) b f ( dt b) f ( ) } b f ( b ) f ( b ) }. (3) By substituting the bove eulities ():(i)-(ii) nd the bove ineulities ():(iii)-(iv) in () we hve the desired result (0).

10 Jekeun Prk (b) For > b we hve: (i) (ii) (iii) (iv) (t ) ( t b ) p dt = µ3 (t b) ( t b ) p dt = µ f ( t ) b f ( dt b) f ( ) } b f ( ) f ( b f ( t ) b dt ) } f ( ) f ( b ) }. () By substituting the bove eulities (b):(i)-(ii) nd the bove ineulities (b):(iii)-(iv) in () we hve the desired result (). Theorem.3. Let f : I [0 ) R be twice differentible function on the interior I 0 of n intervl I such tht f L [ b] b I 0 with < b. If f is conve on [ b] for some fied > with = then p for ny [ b] the following ineulities hold: () For b we hve: I f () [ µ p f 3 (λ ( 3 ) f ( λ3 ) )} (b ) µ p ( f 3 λ ( 33 ) f ( b) ) λ3 b ( f λ ( b) f ( 35 λ36 b ) )} ] (5) b

11 Hermite-Hdmrd type ineulities 5 µ 3 = (b )p ( b ) p p ( p) µ 3 = (b )p ( b ) p p ( p) ( ) λ 3 = ( )( ) λ 3 = ( ) λ 33 = (b ) (b ) ( ) b ( ) } ( )( ) λ 3 = (b ) (b ) ( ) (3 )b ( ) } ( )( ) (b ) λ 35 = ( ) (b ) λ 36 = ( )( ). (b) For > b we hve: I f ( b; ) [ µ p ( f λ ( ) f ( b) ) λ b ( f λ ( b) f ( 3 λ ) )} b f (λ ( 5 ) f ( λ6 b ) )} ] (6) b (b ) µ p

12 6 Jekeun Prk µ = (b )p ( b) p p ( p) µ = (b )p ( b) p p ( p) (b ) λ = ( )( ) λ = (b ) ( ) λ 3 = ( ) (b ) (3 ) ( )b ( ) } ( )( ) λ = (b ) ( ) ( )b ( ) } ( )( ) (b ) λ 5 = (b ) λ 6 = ( )( ). Proof. From Lemm nd using the well-known Hölder integrl ineulity we hve I f ( b; ) [ ( t b ) p } p dt ( t ) ( f t ) } dt (b ) ( t b ) p } p dt ( t b ) ( f t ) } ] dt. (7) Since f is conve on [ b] we hve: () For b we hve: (i) (ii) (iii) t b p dt = µ 3 t b p dt = µ 3 ( t ) ( f t ) dt λ 3 f ( ) λ3 f ( ) }

13 Hermite-Hdmrd type ineulities 7 (iv) ( t b ) ( f t ) dt f λ ( 33 ) f ( b) } λ3 b f λ ( b) f ( 35 λ36 b ) }. b By substituting the bove eulities ():(i)-(ii) nd the bove ineulities ():(iii)-(iv) in (7) we hve the desired result (5). (b) For > b we hve: (i) t b p dt = µ (ii) t b p dt = µ (iii) t ( f t ) dt (iv) b b t b ( f t ) dt f λ ( ) f ( b) } λ f λ ( b 3 ) λ f ( ) } ( ) f = ( b t t b b b t b ) dt f λ ( 5 ) f ( λ6 b ) }. b By substituting the bove eulities (b):(i)-(ii) nd the bove ineulities (b):(iii)-(iv) in (7) we hve the desired result (6). Theorem.. Let f : I [0 ) R be twice differentible function on the interior I 0 of n intervl I such tht f L [ b] b I 0 with < b. If f is conve on [ b] for some fied then for ny [ b] the following ineulities hold: () For b we hve: [ I f () µ f 5 λ ( 5 ) f ( λ5 ) )} (b ) µ f 5 λ ( 53 ) f ( b) ) λ5 ( f λ ( b) f ( 55 b ) )} ] (8)

14 8 Jekeun Prk µ 5 = ( ) ( 3b ) µ 5 = (b )3 ( b ) (5b ) 8 λ 5 = ( ) (b ) λ 5 = ( ) ( b 3) λ 53 = ( b ) (7b 6) 96 λ 5 = ( b ) (3b ) λ 55 = (b )3. 96 (b) For > b we hve: I f () [ (b ) 96 ( µ f 6 λ ( 6 ) f ( b) ) f λ ( 6 ) f ( b) )} λ63 f λ ( 6 ) f ( λ65 b ) )} ] (9) µ 6 µ 6 = (b )3 ( 5 b)( b ) 8 µ 6 = (b ) ( 3 b) λ 6 = (b )3 96 λ 6 = ( b) (6 7 b) 96 λ 63 = ( b) ( 3 b) 96 λ 6 = (b ) (3 b) λ 65 = (b ) ( ).

15 Hermite-Hdmrd type ineulities 9 Proof. Suppose tht. From Lemm nd using the well-known power-men integrl ineulity we hve [ I f () ( t )( t b ) } dt (b ) ( t )( t b f ( t ) } dt ( t b )( t b ) } dt ( t b )( t b ) f ( t ) } ] dt. (30) Since f is conve on [ b] we hve: () For b we hve: (i) (ii) (iii) (iv) (t )(t b dt ) = µ5 (t b)(t b dt ) = µ5 (t )( ( t b ) f ( t ) dt λ 5 () f ( ) λ5 () f ( ) ( t b ) ( ( t b ) f ( t ) dt λ 53 () f ( ) λ5 () f ( b) ( f λ 55 () ( b) f ( b ) ). By substituting the bove eulities ():(i)-(ii) nd the bove ineulities ():(iii)-(iv) in (30) we hve the desired result (8). (b) For > b we hve: (i) (ii) (t )(t b dt ) = µ6 (t b)(t b dt ) = µ6

16 30 Jekeun Prk (iii) (iv) (t )(t b f ) ( t ) dt f λ ( b) f ( 6 ) } λ 6 () f ( ) λ63 () f ( b) } (t b)(t b ) ( f t ) dt λ 6 f ( ) λ65 f ( b ) }. By substituting the bove eulities (b):(i)-(ii) nd the bove ineulities (b):(iii)-(iv) in (30) we hve the desired result (9). Theorem.5. Let f : I [0 ) R be twice differentible function on the interior I 0 of n intervl I such tht f L [ b] b I 0 with < b. If f is concve on [ b] for some fied > with = then p for ny [ b] the following ineulities hold: () For b we hve: I f () [ µ p 7 (b ) µ p ( b 7 ( ) f ( ) ( ( f b 6 ) } µ 7 = (b )p ( ) p F p [ p p p ( p) (b )p (b ) p µ 7 = ( ( ) p )β( p 3p p) ( b)p F [ p p p p (b) For > b we hve: I f () [ µ p 7 (b ) f ( b (b ) µ p ( b ) ( ( 73 f 3 b) f ( b b ) f ( 3b) } ] (3) ( ) b ] (b ) b ] }. (3) ) } ) } ] (33)

17 Hermite-Hdmrd type ineulities 3 (b )p (b ) p µ 73 = β( p p p) ( b)p F [ p p p b } ] p b (b )p (b ) p µ 7 = β( p p p) ( b)p F [ p p p b } ]. (3) p b Proof. From Lemm nd using the well-known Hölder ineulity fot > with = we hve p I f ( b; ) [ ( t )( t b ) p } p dt f ( t ) } dt (b ) ( t b )( t b ) p } p dt f ( t ) } ] dt. (35) Since f is concve on [ b] we hve: () For b we hve: (i) (ii) (iii) (iv) (t )(t b ) p dt = µ7 (t b)(t b ) d t = µ7 f ( t ) f ( dt ( ) ) f ( t ) (b ) ( dt f b 6 ) f ( 3b) }. By substituting the bove eulities ():(i)-(ii) nd the bove ineulities ():(iii)-(iv) in (35) we hve the desired result (3).

18 3 Jekeun Prk (b) For > b we hve: (i) (t )(t b dt ) = µ73 (ii) (t b)(t b dt ) = µ7 (iii) f ( t ) dt (iv) ( b ) ( f 3 b) ( ) ( f b ) b f ( t ) f ( dt (b ) b). By substituting the bove eulities (b):(i)-(ii) nd the bove ineulities (b):(iii)-(iv) in (35) we hve the desired result (33). Theorem.6. Let f : I [0 ) R be twice differentible function on the interior I 0 of n intervl I such tht f L [ b] b I 0 with < b. If f is concve on [ b] for some fied then for ny [ b] the following ineulity holds: () For b we hve: I f ( b) [ µ f 8 λ ( ( 3b )b 3(b ) ) } 8 (b ) 3b µ f 8 λ ( ( 3b)( b) 8(b ) ) 8 ( 5b) f λ ( 3b) } ] 83 (36) µ 8 = ( ) ( 3b ) µ 8 = ( b ) (5b ) 8 λ 8 = ( ) ( 3b ) λ 8 = ( b ) (5b ) λ 83 = (b )3. 8 8

19 Hermite-Hdmrd type ineulities 33 (b) For > b we hve: I f ( b) [ µ 9 () λ 9 () f ( 3 b) (b ) λ 9 () f ( 3 b )( b ) 8( b ) (5 b ) µ 9 () λ 93 () f ( ( b) ( )( ) 3b b µ 9 () = (b )3 ( b ) ( 5 b) 8 µ 9 () = (b ) ( 3 b) λ 9 () = (b )3 8 λ 9 () = ( b ) ( 5 b) 8 λ 93 () = (b ) ( 3 b). ) } ) } ] (37) Proof. From Lemm nd using the well-known power-men integrl ineulity fot we hve I f ( b; ) [ ( t )( t b (b ) ( t )( t b ( t b )( t b ) } p dt ( t b )( t b ) dt } p ) f ( t ) dt } ) f ( t ) dt } ]. (38) Since f is concve on [ b] we hve:

20 3 Jekeun Prk () For b we hve: (i) (ii) (iii) (iv) (t )(t b dt ) = µ8 ( p) (t b)(t b dt ) = µ8 ( p) (t )(t b f ) ( t ) dt λ 8 () f ( ( 3b )b 3(b ) ) 3b (t b)(t b f ) ( t ) dt λ 8 () f ( ( 3b)( b) 8(b ) ) ( 5b) λ 83 () f ( 3b). By substituting the bove eulities ():(i)-(ii) nd the bove ineulities ():(iii)-(iv) in (38) we hve the desired result (36). (b) For > b we hve: (i) (ii) (iii) (iv) (t )(t b dt ) = µ9 () (t b)(t b dt ) = µ9 () (t )(t b f ) ( t ) dt λ 9 () f ( (3 b )( b ) 8( b ) ) (5 b ) λ 9 () f ( 3 b) (t b)(t b f ) ( t ) dt λ 93 () f ( ( b) ( )( ) ). 3 b By substituting the bove eulities (b):(i)-(ii) nd the bove ineulities (b):(iii)-(iv) in (38) we hve the desired result (37).

21 Hermite-Hdmrd type ineulities 35 References [] E. F. Beckenbch Conve functions Bull. Amer. Mth. Soc. 5 (98) [] S. S. Drgomir C. E. M. Perce Selected topics on Hermite-Hdmrd integrl ineulities nd pplictions Melbourne nd Adelide December (000). [3] S. S. Drgomir S. Fitzptrick The Hdmrd s ieulity for s-conve functions in the second sense Demonstrtio Mth. 3() (999) [] S. S. Drgomir R. P. Agrwl Two ineulities for differentible mppings nd pplictions to specil mens of rel numbers nd to trpezoidl formul Applied Mth. Lett. (5) (998) [5] Imdt Işcn New estimtes on generliztion of some integrl ineulities for ds-conve functions nd their pplictions Int. J. Pure Appl. Mth. 86() (03) [6] Imdt Işcn On generliztion of different type integrl ineulities for s-conve functions vi frctionl integrls presented [7] H. Kvurmci M. Avci M. E. Özdemir New ineulities of Hermite- Hdmrd s type for conve functions with pplictions Journ. of Ineul. nd Appl. 0:86 (0) [8] U. S. Kirmci K. Klrričić M. E. Özdemir J. Pečrić Hdmrd-type ineulities for s-conve functions Appl. Mth. Comput. 93() (007) [9] M. A. Ltif Ineulities of Hermite-Hdmrd type for functions whose derivtives in bsolute vlue re conve with pplictions Arb J. Mth. Sci. (0) Article in press. /0.06/j.jmsc [0] V. G. Miheşn A generliztion of the conveity Seminr on Functionl Eutions Appro. nd Conve Cluj-Npoc Romni (993). [] M. E. Özdemir M. Avic H. Kvurmci Hermite-Hdmrd type ineulities for s-conve nd s-concve functions vi frctionl integrls rxiv: v[mth.CA].

22 36 Jekeun Prk [] Jekeun Prk Generliztion of some Simpson-like type ineulities vi differentible s-conve mppings in the second sense Inter. J. of Mth. nd Mth. Sci. 0 Art No: pges. doi:0.55/0/9353. [3] M. Z. Sriky E. Set H. Yildiz N. Bşk Hermite-Hdmrd s ineulities for frctionl integrls nd relted frctionl ineulities Mth. nd Comput. Model. 0 (0). doi:0.06/j.mcm [] K. L. Tseng S. R. Hwng S. S. Drgomir Fejér-type ineulities(i) J. Ineul. Appl. 00 (00) Art ID: pges. [5] Gh. Toder On generliztion of the conveity Mthemtic 30(53) (988) [6] M. Tunç On some new ineulities for conve functions Turk. J. Mth. 35 (0) -7. [7] M. Tunç New integrl ineulities for s-conve functions RGMIA Reserch Report Collection 3() (00) RGMIA/ v3n.php. Received: November 5 0; Published: December 0