HERMITE-HADAMARD TYPE INEQUALITIES FOR FUNCTIONS WHOSE DERIVATIVES ARE (α -CONVEX İMDAT İŞCAN Dertent of Mthetics Fculty of Science nd Arts Giresun University 8 Giresun Turkey idtiscn@giresunedutr Abstrct: In this er severl ineulities of the right-hnd side of Herite- Hdrd ineulity re obtined for the clss of functions whose derivtives in bsolutely vlue t certin owers re (α -convex Soe lictions to secil ens of ositive rel nubers re lso given Keywords: (α -convex functionsherite-hdrd s ineulity Hölder s integrl ineulity INTRODUCTION Let f : I R R be convex function defined on the intervl I of rel nubers nd b I with < b then ( + b ( f b f( + f(b This doubly ineulity is known in the literture s Herite-Hdrd integrl ineulity for convex functions In 4 Miheşn introduced the clss of (α -convex functions s the following: The function f : b R b > is sid to be (α -convex (α if we hve f (tx + ( ty t α f(x + ( t α f(y for ll x y b nd t It cn be esily tht for (α {( (α ( ( ( (α }one obtins the following clsses of functions: incresing α-strshed strshed -convex convex α-convex Denote by K α (b the set of ll (α -convex functions on b for which f( For recent results nd generliztions concerning -convex nd (α -convex functions (see 8 6 5 7 9 In 3 Drgoir nd Agrwl estblished the following result connected with the right-hnd side of ( Theore Let f : I R R be differentible ing on I b I with < bif f is convex on b then the following ineulity holds: ( f( + f(b b b f ( + f (b 8
HERMITE-HADAMARD TYPE INEQUALITIES In the following ineulity of Herite-Hdrd tye for (α -convex functions holds: Theore Let f : R be n (α -convex function with (α ( If < b < nd f L b then one hs the ineulity: (3 b { ( f( + αf b in f(b + αf ( } α + α + In the following Herite-Hdrd tye ineulities for nd (α - convex functions were obtined Theore 3 Let I be n oen rel intervl such tht I Let f : I R be differentible ing on I such tht f L b < b < If f is -convex on b for soe fixed ( nd ( then (4 f( + f(b µ = in µ = in b { f ( + ( f +b { f (b + f ( +b b 4 b 4 ( ( µ + µ ( µ + µ f ( +b + ( f f ( +b + f ( b } } Theore 4 Let I be n oen rel intervl such tht I Let f : I R be differentible ing on I such tht f L b < b < If f is (α -convex on b for soe fixed α ( nd then (5 nd f( + f(b b b { ( in ν f ( + ν f ( b ν = ν = α + (α + (α + ( } (ν f (b f + ν ( ( α α + α + (α + (α + ( α The in i of this er is to estblish new ineulities of Herite-Hdrd tye for the clss of functions whose derivtives in bsolutely vlue t certin owers re (α -convex
HERMITE-HADAMARD TYPE INEQUALITIES 3 INEQUALITIES FOR FUNCTIONS WHOSE DERIVATIVES ARE (α CONVEX In order to rove our in resuls we need the following le: Le Let f : I R R be differentible ing on I b I with < b If f L b nd µ > then the following eulity holds: f( + µf(b b Proof integrtion by rts we hve I = = b (t f (tb + ( tdt = (t = f( + µf(b b f(tb + ( t b b (t f (tb + ( tdt b f(tb + ( tdt Setting x = tb + ( t nd dx = (b dt gives f(tb + ( tdt Therefore I = which coletes the roof f( + µf(b b (b ( b f( + µf(b I = b The next theore gives new refineent of the uer Herite-Hdrd ineulity for (α -convex functions Theore 5 Let f : I R be differentible ing on I such tht f L b b I with < b If f is (α -convex on b for soe fixed (α ( µ with > nd then the following ineulity holds: ( f( + µf(b in b b ( + µ ( { ( γ f (b f + γ ( (γ 3 f ( + γ 4 f ( b }
4 HERMITE-HADAMARD TYPE INEQUALITIES nd γ = γ 3 = α+ (α + (α + α+ + (α + µ ( µ α+ (α + (α + α+ + (α + µ ( γ = ( γ γ 4 = ( γ 3 Proof Suose tht = Fro Le nd using the (α -convexity of f we hve f( + µf(b b ( b b = b We hve nd (3 = = (t f (tb + ( t dt (t t α f (b + ( t α f ( dt (t t α f (b + ( t α (t f ( dt (t t α dt (t t α dt + (α + (α + hence f( + µf(b b Since α+ (t t α dt α+ + (α + µ ( (t ( t α dt = b (t f (tb + ( t dt = ( γ = γ = γ (γ f (b + γ f ( (t µ f (t + ( tb dt
HERMITE-HADAMARD TYPE INEQUALITIES 5 Anlogously we obtin f( + µf(b b γ 3 = (α + (α + b α+ α+ + (α + µ ( (γ 3 f ( + γ 4 f ( b nd γ 4 = ( γ 3 which coletes the roof for this cse Suose now tht ( Fro Le nd using the Hölder s integrl ineulity we hve (4 (t f (tb + ( t dt b (t dt (t f (tb + ( t dt Since f is (α -convex on b we know tht for every t (5 f (tb + ( t t α f (b + ( t α f ( Fro ( (3 (4 nd (5 we hve f( + µf(b b b ( + µ (γ ( f (b f + γ ( nd nlogously f( + µf(b b which coletes the roof b ( + µ ( γ ( 3 f ( + γ 4 f ( b Corollry Suose tht ll the ssutions of Theore 5 re stisfied ( In the ineulity ( If we choose = µ we obtin the ineulity in (5 ( In the ineulity ( If we choose = µ = = nd α = we obtin the ineulity in ( (3 In the ineulity ( If we choose = α = we hve f( + µf(b (6 b b ( + µ ( { (γ in f (b + γ f ( ( γ 3 f ( + γ 4 f (b }
6 HERMITE-HADAMARD TYPE INEQUALITIES nd γ = 3 6 ( + µ γ = ( γ γ 3 = µ 3 6 ( + µ γ 4 = ( γ 3 Theore 6 Let f : I R be differentible ing on I such tht f L b b I with < b If f is (α -convex on b for soe fixed (α ( µ with > nd > then the following ineulity holds: (7 f( + µf(b b ( M + + µ M ( f ( + α f b+µ ( M = in α + ( f (b + α f b+µ ( M = in α + nd + = b ( f ( b+µ + α f ( α + f ( b+µ + α f ( b α + Proof Fro Le nd using the Hölder ineulity we hve f( + µf(b b b + b b ( (t dt (t dt + ( + ( = b ( + M ( ( M + + µ f (tb + ( t dt M f (tb + ( t dt µ + ( + ( ( µ M
HERMITE-HADAMARD TYPE INEQUALITIES 7 we use the fct tht (t dt = nd by Theore we get (t = + ( + ( µ + ( + ( = = f (tb + ( t dt b+µ f (x dx (b ( f ( + α f b+µ ( in α + µ µ (b f (tb + ( t dt b+µ f (x dx ( f (b + α f b+µ ( in α + f ( b+µ + α f ( α + f ( b+µ + α f ( b α + which coletes the roof Corollry Suose tht ll the ssutions of Theore 6 re stisfied in this cse: (8 ( In the ineulity (7 if we choose = µ nd α = we obtin the ineulity in (4 ( In the ineulity (7 if we choose = α = we hve f( + µf(b b b ( ( M + + µ M ( f ( + f b+µ ( f (b + f b+µ M = M =
8 HERMITE-HADAMARD TYPE INEQUALITIES Theore 7 Let f : I R be differentible ing on I such tht f L b b I with < b If f is (α -convex on b for soe fixed (α ( µ with > nd > then the following ineulity holds: f( + µf(b (9 b nd + = b ( + + µ + ( { } in K ( + ( α + K K = f (b + α f ( K = f ( + α f ( b Proof Fro Le nd using the Hölder s integrl ineulity we hve f( + µf(b b (t f (tb + ( t dt b (t dt b ( + + µ + ( + ( f (tb + ( t dt t α f (b + ( t α f ( dt = b ( + + µ + ( ( f (b + α f ( ( + ( α + Anlogously we obtin f( + µf(b b b ( + + µ + ( ( + ( α + ( f ( + α f ( b Corollry 3 Suose tht ll the ssutions of Theore 7 re stisfied in this cse:
HERMITE-HADAMARD TYPE INEQUALITIES 9 ( In the ineulity (9 if we choose = µ then the following ineulity holds: f( + f(b b b ( ( { } in K + α + K ( In the ineulity (9 if we choose = α = we hve f( + µf(b ( b b ( + + µ + ( + ( ( f ( + f (b 3 S Let us recll the following secil ens of two nonnegtive nuber b with b nd α : ( The weighted rithetic en ( The unweighted rithetic en A α ( b := α + ( αb b A ( b := + b b (3 The weighted hronic en ( α H α ( b := + α b > b (4 The unweighted hronic en (5 The Logrithic en L ( b := H ( b := (6 The -Logrithic en ( b + + L ( b := (+(b b { b ln b ln b b > + b if b b if = b if b if = b b > b > Z\ { } Proosition Let b R with < < b nd n Z n Then we hve the following ineulity: A ( n b n L n n ( b b ( + µ ( { ( n in γ b (n + γ (n ( γ 3 (n + γ 4 b (n }
HERMITE-HADAMARD TYPE INEQUALITIES γ = 3 6 ( + µ γ = ( γ γ 3 = µ 3 6 ( + µ γ 4 = ( γ 3 µ with > nd Proof The ssertion follows fro ineulity (6 in Corollry for f : ( R f(x = x n Proosition Let b R with < < b nd n Z n Then we hve the following ineulity: A ( n b n L n n ( b b ( n ( M + + µ M ( M = A ( M = A (n A (n (b b (n A (n (b µ with > nd > with + = Proof The ssertion follows fro ineulity (8 in Corollry for f : ( R f(x = x n Proosition 3 Let b R with < < b nd n Z n Then we hve the following ineulity: ( n b n L n n ( b A b ( + + µ + ( n A ( (n b (n + µ with > nd > with + = Proof The ssertion follows fro ineulity ( in Corollry 3 for f : ( R f(x = x Proosition 4 Let b R with < < b Then we hve the following ineulity: ( b L ( b H b ( + µ ( { ( in γ b + γ (γ 3 + γ 4 b } γ = 3 6 ( + µ γ = ( γ
HERMITE-HADAMARD TYPE INEQUALITIES γ 3 = µ 3 6 ( + µ γ 4 = ( γ 3 µ with > nd Proof The ssertion follows fro ineulity (6 in Corollry for f : ( R f(x = x n Proosition 5 Let b R with < < bthen we hve the following ineulity: H ( b L ( b b ( ( M + + µ M M = H ( A (b M = H (b A (b µ with > nd > with + = Proof The ssertion follows fro ineulity (8 in Corollry for f : ( R f(x = x Proosition 6 Let b R with < < b Then we hve the following ineulity: H ( b L ( b b ( ( A ( µ ( H b + µ with > nd > with + = Proof The ssertion follows fro ineulity ( in Corollry 3 for f : ( R f(x = x R Bkul MK Ozdeir ME nd Pecric J Hdrd tye ineulities for -convex nd (α -convex functions J Ineul Pure Al Mth 9 Article 96 8 Online: htt://jivueduu Bkul MK Pecric J nd Ribiˇcić M Conion ineulities to Jensen s ineulity for -convex nd (α -convex functions J Ineul Pure Al Mth 7 Article 94 6 Online: htt://jivueduu 3 Drgoir SS nd Agrwl RP Two ineulities for diff erentible ings nd lictions to secil ens of rel nubers nd to trezoidl forul Al Mth Lett 9-95 998 4 Miheşn VG A generliztion of the convexity Seinr on Functionl Eutions Arox nd Convex Cluj-Noc Roni 993 5 Ozdeir ME Kvurcı H nd Set E Ostrowski s tye ineulities for (α -convex functions Kyungook Mth J 5 37-378 6 Ozdeir ME Avcı M SetE On soe ineulities of Herite-Hdrd tye vi - convexity Alied Mthetics Letters 3 65-7 7 Ozdeir ME Set E nd Srıky MZ Soe new Hdrd s tye ineulities for coordinted -convex nd (α -convex functions Hcettee Journl of Mthetics nd Sttistics volue 4 ( 9-9 8 Ozdeir ME Avcı M nd Kvurcı H Herite-Hdrd-tye ineulities vi (α - convexity Couters nd Mthetics with Alictions 6 64-6
HERMITE-HADAMARD TYPE INEQUALITIES 9 Srıky MZ Ozdeir ME nd Set E Ineulities of Herite-Hdrd s tye for functions whose derivtive s bsolute vlues re -convex RGMIA Res Re Coll 3 Suleent Article 5 SetE Srdri MOzdeir ME nd Rooin J On generliztions of the Hdrd ineulity for (α -convex functions RGMIA Res Re Coll (4 Article 4 9