Bất đẳng thức dạng Hermite - Hadamard cho hàm tựa lồi

Header Page of 258 B GIO DC V O TO TRNG I HC S PHM H NI DNG BCH HNG BT NG THC DNG HERMITE-HADAMARD CHO HM TA LI LUN VN THC S TON HC H Ni - 2016 Footer Page of 258 Header Page of 258 B GIO DC V O TO TRNG I HC S PHM H NI DNG BCH HNG BT NG THC DNG HERMITE-HADAMARD CHO HM TA LI LUN VN THC S TON HC Chuyờn ngnh: Toỏn gii tớch Mó s: 60 46 01 02 NGI HNG DN KHOA HC PGS.TS T DUY PHNG H Ni - 2016 Footer Page of 258 Header Page of 258 LI CM N Sau mt thi gian c ti liu v dt nghiờn cu khoa hc, lun ca tụi ó c hon thnh Tụi xin by t lũng bit n sõu sc n thy giỏo PGS TS T Duy Phng ó tn tỡnh ch bo, hng dn, to iu kin cho tụi thi gian lm lun Tụi xin chõn thnh cm n s giỳp quý bỏu ca cỏc thy cụ giỏo b mụn Toỏn Gii tớch núi riờng v khoa Toỏn, trng i hc S phm H Ni núi chung Tụi xin cm n s ng viờn, giỳp ca gia ỡnh v bn bố ó dnh cho tụi quỏ trỡnh nghiờn cu v hon thnh lun H Ni, ngy 05 thỏng 07 nm 2016 Tỏc gi lun Dng Bớch Hng Footer Page of 258 Header Page of 258 LI CAM OAN Tụi xin cam oan lun ny l kt qu nghiờn cu ca riờng tụi di s hng dn ca PGS TS T Duy Phng Trong quỏ trỡnh nghiờn cu, tụi ó k tha thnh qu khoa hc ca cỏc nh khoa hc vi s trõn trng v bit n Cỏc kt qu trớch dn lun ny ó c ch rừ ngun gc H Ni, ngy 05 thỏng 07 nm 2016 Tỏc gi lun Dng Bớch Hng Footer Page of 258 Header Page of 258 i Mc lc Li M u Chng Bt ng thc Hermite-Hadamard cho lp hm ta li 1.1 1.2 1.3 Hm li v mt s c trng c bn ca hm li kh vi Bt ng thc Hermite-Hadamard Bt ng thc Hermite-Hadamard cho hm ta li 1.3.1 Hm ta li 1.3.2 15 15 Bt ng thc Hermite-Hadamard cho hm ta li 22 Chng Mt s m rng bt ng thc Hermite-Hadamard cho hm ta li 25 2.1 Bt ng thc Hermite-Hadamard cho lp hm cú o 2.2 2.3 hm l ta li Bt ng thc dng Hermite-Hadamard cho o hm bc hai l ta li Bt ng thc dng Hermite-Hadamard cho cú cú 26 o hm bc ba l ta li 58 Ti liu tham kho Footer Page of 258 lp lp hm hm 49 64 Header Page of 258 Li M u Lý chn ti Gii tớch li ó v ang úng mt v trớ quan trng toỏn hc Gii tớch li liờn quan n rt nhiu ngnh ca toỏn hc nh gii tớch, gii tớch hm, gii tớch s, hỡnh hc, toỏn kinh t, ti u phi tuyn, Mt kt qu kinh in cho hm li l Bt ng thc Hermite-Hadamard (H-H Inequality), c phỏt biu nh lớ di õy nh lớ (Hermite, 1883, [14]; Hadamard, 1893, [13]) Nu f : R R l hm li trờn on [a; b] thỡ ta cú b f a+b ba f (t)dt f (a) + f (b) (1) a Bt ng thc trờn cú th vit di dng b (b a)f a+b f (t)dt (b a) f (a) + f (b) a í ngha hỡnh hc ca bt ng thc ny l: Nu f : R R l hm li trờn on [a; b] thỡ din tớch hỡnh thang cong chn bi trc honh v th hm s y = f (x) (cựng vi hai ng thng x = a v x = b) luụn ln hn din tớch hỡnh ch nht cú cnh l b a v f a+b , v luụn nh hn hỡnh thang vuụng chiu cao l b a, hai ỏy l f (a) v f (b) Footer Page of 258 Header Page of 258 Tc l din tớch hỡnh thang cong khụng ln hn din tớch hỡnh thang vuụng ABCD v khụng nh hn din tớch hỡnh ch nht ABMN T õy ta cng suy din tớch hỡnh tam giỏc cong NDP bao gi cng nh hn din tớch tam giỏc cong MCP Hỡnh 1: í ngha hỡnh hc ca bt ng thc Hermite-Hadamard Trong [15], Fejer ó m rng bt ng thc (1) thnh bt ng thc (2), m sau ny c gi l bt ng thc Fejer nh lớ Nu f : R R l li trờn [a, b] v g : [a, b] R l mt hm a+b khụng õm, kh tớch v i xng qua im x = thỡ a+b f b b g(t)dt a a f (a) + f (b) f (t)g(t)dt b g(t)dt (2) a Khi g(x) thỡ Bt ng thc Fejer tr thnh Bt ng thc HermiteHadamard Sau ú, nhiu tỏc gi ó m rng cỏc bt ng thc Hermite-Hadamrd v s dng chỳng c trng v nghiờn cu cỏc tớnh cht ca hm li Xem thớ d cun sỏch chuyờn kho [6], [7] v cỏc Ti liu tham kho khỏc Nhiu bi toỏn thc t mụ t bi cỏc hm khụng nht thit l li Vỡ vy, Footer Page of 258 Header Page of 258 cn phi m rng khỏi nim hm li v nghiờn cu cỏc tớnh cht ca hm li suy rng, nhm ỏp dng vo cỏc bi toỏn ti u ny sinh thc t Mt bi toỏn hin nhiờn c t l phỏt biu v chng minh cỏc bt ng thc dng Hermite-Hadamard cho cỏc lp hm li suy rng Vn ny ó c nhiu nh toỏn hc nghiờn cu v phỏt trin Thớ d, bt ng thc Hermite-Hadamard ó c m rng cho cỏc lp hm ta li, lp hm log-li, lp hm r - li, Mt nhng cỏch xõy dng v nghiờn cu lp hm li suy rng, l gi li mt (mt s) tớnh cht c trng ca hm li Thớ d, ta ó bit, hm li cú mc di l li v hm li liờn tc trờn compact t giỏ tr ln nht ti biờn Hai tớnh cht ny cũn ỳng cho lp hm ta li Do ý ngha toỏn hc v ý ngha thc t, cú th núi, s cỏc lp hm li suy rng, lp hm ta li c nghiờn cu y hn c Mc ớch chớnh ca Lun ny l trỡnh by tng quan v Bt ng thc Hermite-Hadamard cho cỏc lp hm ta li Mc ớch nghiờn cu Nghiờn cu bt ng thc dng Hermite-Hadamard cho cỏc lp hm ta li Nhim v nghiờn cu Tỡm hiu chng minh cỏc bt ng thc dng Hermite-Hadamard cho cỏc lp hm ta li v mt s liờn quan Footer Page of 258 Header Page of 258 4 i tng v phm vi nghiờn cu i tng nghiờn cu: Cỏc bt ng thc dng Hermite-Hadamard cho cỏc lp hm ta li Phm vi nghiờn cu: Cỏc ti liu, cỏc sỏch bỏo liờn quan n bt ng thc Hermite-Hadamard cho cỏc lp hm ta li Phng phỏp nghiờn cu Thu thp ti liu, cỏc sỏch bỏo v cỏc bt ng thc dng HermiteHadamard cho cỏc lp hm ta li Tng hp, phõn tớch, h thng cỏc kin thc v bt ng thc HermiteHadamard cho cỏc lp hm ta li D kin úng gúp ca lun vn: C gng xõy dng lun thnh mt bn tng quan tt v Bt ng thc Hermite-Hadamard cho hm ta li H Ni, thỏng nm 2016 Tỏc gi Dng Bớch Hng Footer Page of 258 Header Page 10 of 258 Chng Bt ng thc Hermite-Hadamard cho lp hm ta li Trong chng ny, chỳng tụi trỡnh by mt s c trng c bn ca hm li, chng minh cỏc bt ng thc dng Hermite-Hadamard cho hm li mt bin v mt s m rng ca Bt ng thc Hermite-Hadamard, Bt ng thc Hermite-Hadamar cho hm ta li Ni dung Chng ch yu theo Ti liu [11], [6], [7] v tham kho thờm mt s ti liu khỏc 1.1 Hm li v mt s c trng c bn ca hm li kh vi nh ngha 1.1 Tp X Rn c gi l li nu vi mi [0; 1] v x1 X , x2 X ta cú x := x1 + (1 )x2 X Ngha l, li X cha mi on thng ni hai im bt k ca nú nh ngha 1.2 Hm f : X Rn R c gi l hm li nu X l li v vi mi [0; 1], x1 X , x2 X ta cú f (x ) f (x1 ) + (1 )f (x2 ) Footer Page 10 of 258 Header Page 57 of 258 52 Vỡ |f | l ta li trờn [a, b] nờn ta cú |f (ta + (1 t)b)| max{|f (a)|, |f (b)| v |f (tb + (1 t)a)| max{|f (b)|, |f (a)| Suy b a+b f (x)dx f ba a 1/2 (b a)2 t2 max{|f (a)|, |f (b)|}dt (1 t)2 max{|f (b)|, |f (a)|}dt + 1/2 = (b a) max{|f (a)|, |f (b)|} 24 Ta cú iu phi chng minh nh lý 2.14 ([10], Theorem 7) Gi s a, b I R ,v a < b, hm f : [a, b] R kh vi trờn (a, b) Nu f L[a, b] v |f |q vi q l ta li trờn [a, b] thỡ: b a+b f (x)dx f ba a 2 (b a) (max{|f (a)|q , |f (b)|q })1/q , 1/p 8(2p + 1) ú q = p p1 Chng minh T B 2.4 ta cú b a+b I= f (x)dx f ba a (b a)2 = m(t)[f (ta + (1 t)b) + f (tb + (1 t)a)]dt Footer Page 57 of 258 Header Page 58 of 258 53 (b a)2 |m(t)|[|f (ta + (1 t)b)| + |f (tb + (1 t)a)|]dt p dng Bt ng thc Hoălder ta c 1/p (b a)2 I 1/q p q |m(t)| dt |f (ta + (1 t)b| dt 0 1/q |f (tb + (1 t)a)|q dt + Vi 1/2 p |m(t)| dt = 2p (1 t)2p dt = t dt + 1/2 4p (2p + 1) Suy 1/q (b a)2 I 16(2p + 1)1/p q q max{|f (a)| , |f (b)| }dt 1/q q q max{|f (b)| , |f (a)| dt + = (b a)2 (max{|f (a)|q , |f (b)|q })1/q 1/p 8(2p + 1) Ta cú iu phi chng minh nh lý 2.15 ([10], Theorem 8) Gi s a, b I R ,v a < b, hm f : [a, b] R kh vi trờn (a, b) Nu f L[a, b] v |f |q vi q l ta li trờn [a, b] thỡ: ba b a a+b f (x)dx f (b a)2 (max{|f (a)|q , |f (b)|q })1/q 24 Chng minh T B 2.4 v tớnh cht b b f (x)dx a ta cú: I ba Footer Page 58 of 258 b f (x)dx f a a+b |f (x)|dx a Header Page 59 of 258 54 (b a)2 = (b a)2 (b a)2 = m(t)[f (ta + (1 t)b) + f (tb + (1 t)a)]dt |m(t)|[|f (ta + (1 t)b)| + |f (tb + (1 t)a)|]dt |m(t)|1/p |m(t)|1/q [|f (ta + (1 t)b)| + |f (tb + (1 t)a)|]dt D dng tớnh c 1/2 t2 dt + |m(t)|dt = (1 t)2 dt = 1/2 12 p dng Bt ng thc Hoălder ta c 1/p (b a)2 I 1/q q |m(t)|dt |m(t)||f (ta + (1 t)b)| dt 0 1/q q |m(t)||f (tb + (1 t)a)| dt + 1/2 (b a)2 (12)1/p t2 max{|f (a)|q , |f (b)|q }dt 1/q q q (1 t) max{|f (a)| , |f (b)| }dt + 1/2 1/2 t2 max{|f (b)|q , |f (a)|q }dt + 1/q q q (1 t) max{|f (b)| , |f (a)| }dt + 1/2 (b a)2 = (max{|f (a)|q , |f (b)|q })1/q 1/p 1/q (12) (12) Ta cú iu phi chng minh Trong [3] cng ó chng minh mt s bt ng thc tng t dng Hermite-Hadamard cho lp hm cú o hm bc hai l hm ta li Ta cú: B 2.5 ([3], Lemma 1) Gi s a, b I R, a < b, hm f : [a, b] R kh vi trờn (a, b), f L[a, b] y f (a) + f (b) ba Footer Page 59 of 258 b f (x)dx = a (b a)2 t(1 t)f (ta + (1 t)b)dt Header Page 60 of 258 55 nh lý 2.16 ([3], Theorem 3) Gi s a, b I R, a < b, hm f : [a, b] R kh vi trờn (a, b) Nu f L[a, b] v |f | l ta li trờn [a, b] thỡ: f (a) + f (b) ba b a (b a)2 max{|f (a)|, |f (b)|} f (x)dx 12 Chng minh T B 2.5 ta cú b f (a) + f (b) I= f (x)dx ba a (b a)2 t(1 t)f (ta + (1 t)b)dt = (b a)2 t(1 t)|f (ta + (1 t)b)|dt Vỡ |f | l ta li nờn |f (ta + (1 t)b)| max{|f (a)|, |f (b)|} (b a)2 I t(1 t) max{|f (a)|, |f (b)|}dt (b a)2 t(1 t)dt ) max{|f (a)|, |f (b)|} (b a)2 = ) max{|f (a)|, |f (b)|} Ta cú iu phi chng minh nh lý 2.17 ([3], Theorem 5) Gi s a, b I R ,v a < b, hm f : [a, b] R kh vi trờn (a, b) Nu f L[a, b] v |f |q vi q l ta li trờn [a, b] thỡ: f (a) + f (b) ba Footer Page 60 of 258 b a (b a)2 f (x)dx max{|f (a)|q , |f (b)|q } 12 1/q Header Page 61 of 258 56 Chng minh T B 2.5 ta cú b f (a) + f (b) I= f (x)dx ba a (b a)2 t(1 t)f (ta + (1 t)b)dt = (b a)2 t(1 t)|f (ta + (1 t)b)|dt (b a)2 = (t t2 )1/p (t t2 )1/q |f (ta + (1 t)b)|dt p dng Bt ng thc Hoălder ta cú (b a)2 I 11/q 1/q 2 (t t )dt q (t t )|f (ta + (1 t)b)| dt 0 Vỡ |f |q l ta li nờn |f (ta + (1 t)b)|q max{|f (a)|q , |f (b)|q } 11/q 1 (b a)2 ã ã max{|f (a)|q , |f (b)q } I 6 (b a)2 1/q = max{|f (a)|q , |f (b)q } 1/q Ta cú iu phi chng minh nh lý 2.18 ([3], Theorem 5) Gi s a, b I R ,v a < b, hm p f : [a, b] R kh vi trờn (a, b) Nu f L[a, b] v |f | p vi p > l ta li trờn [a, b] thỡ: f (a) + f (b) ba 1/p (b a)2 ú q = Footer Page 61 of 258 p p1 b f (x)dx a (1 + p ( + p) 1/p (max{|f (a)|q , |f (b)|q })1/q , Header Page 62 of 258 57 Chng minh T B 2.5 ta cú b f (a) + f (b) I= f (x)dx ba a (b a)2 t(1 t)f (ta + (1 t)b)dt = (b a)2 t(1 t)|f (ta + (1 t)b)|dt p dng Bt ng thc Hoălder ta cú (b a)2 I 1/p 1/q p q (t t ) dt |f (ta + (1 t)b)| dt Hm Beta v hm Gamma c xỏc nh nh sau tx1 (1 t)y1 dt, B(x, y) = x, y > 0 v (x) = et tx1 dt, x > 0 Ta cú 1 p (1 t)p dt = (p + 1, p + 1) (t t ) dt = 0 S dng tớnh cht ca hm Beta B(x, x) = 212x B( , x) v B(x, y) = (x)(y) , (x + y) ta cú ( )(p + 1) 12(p+1) 2p1 B(p + 1, p + 1) = B ,p + = ( + p) Footer Page 62 of 258 Header Page 63 of 258 58 Vỡ ( ) = nờn ta cú (p + 1) B(p + 1, p + 1) = 212(p+1) B , p + = 22p1 ( + p) Vỡ |f |q l ta li nờn |f (ta + (1 t)b)|q max{|f (a)|q , |f (b)|q } Suy (b a)2 212p (1 + p) I ( + p) 1/p (b a) (1 + p = ( + p) 2.3 1/p ã (max{|f (a)q |, |f (b)|q })1/q 1/p (max{|f (a)|q , |f (b)|q })1/q Bt ng thc dng Hermite-Hadamard cho lp hm cú o hm bc ba l ta li Trong [4] ó chng minh mt s bt ng thc dng Hermite-Hadamard cho lp hm cú o hm bc ba l ta li Ta cú: nh lý 2.19 ([4], Lemma 1) Gi s a, b I R v a < b, hm f : [a, b] R l liờn tc tuyt i trờn (a, b) Nu f L[a, b] l ta li trờn [a, b] thỡ: b f (x)dx a a+b (b a) f (a) + 4f = (b a) p(t)f (ta + (1 t)b)dt, Footer Page 63 of 258 + f (b) Header Page 64 of 258 59 ú 1 t2 (t ), p(t) = 1 (t 1)2 (t ), t [0, ]; t [ , 1] Chng minh Ta cú: I= 1/2 f (ta + (1 t)b)dt p(t)f (ta + (1 t)b)dt = t t 1 (t 1)2 t f (ta + (1 t)b)dt + 1/2 Tớch phõn tng phn ta c 1 f (ta + (1 t)b) I = t2 t ab + t f (ta + (1 t)b) (a b)3 1/2 1/2 f (ta + (1 t)b) t(3t 1) (a b)2 1/2 0 1 f (ta + (1 t)b) + (t 1)2 t (a b) f (ta + (1 t)b) (3t 2)(t 1) (a b)2 1/2 f (ta + (1 t)b) dt (a b)3 1/2 1/2 f (ta + (1 t)b) f (ta + (1 t)b) dt (a b)3 (a b)3 1/2 1/2 a+b a+b f f 1/2 f (b) f (ta + (1 t)b) 2 = + + dt 24 (a b)2 (a b)3 (a b)3 (a b)3 a+b a+b f f 1 f (a) f (ta + (1 t)b) 2 + + + dt 24 (a b)2 (a b)3 (a b)3 (a b)3 1/2 + t Thay x = ta + (1 t)b, v dx = (a b)dt,ta c b (b a)4 I = f (x)dx a Ta cú iu phi chng minh Footer Page 64 of 258 (b a) a+b f (a) + 4f + f (b) Header Page 65 of 258 60 nh lý 2.20 ([4], Theorem 2) Gi s a, b I R v a < b, hm f : [a, b] R l liờn tc tuyt i trờn (a, b) Nu f L[a, b] l ta li trờn [a, b] thỡ: b (b a) a+b f (a) + 4f a (b a) a+b max |f (a)|, f 1152 a+b + max f , |f (b)| f (x)dx + f (b) Chng minh T nh lý 2.19 ta cú: b f (x)dx I= a a+b (b a) f (a) + 4f + f (b) = (b a) p(t)f (ta + (1 t)b)dt (b a)4 |p(t)||f (ta + (1 t)b)|dt (b a)4 = (b a)4 + 1/2 t2 t |f (ta + (1 t)b)|dt (t 1)2 t 1/2 |f (ta + (1 t)b)|dt Vỡ |f | l ta li nờn 1/2 |f (ta + (1 t)b)|dt max f a+b , |f (b)|q , |f (ta + (1 t)b)|dt max f a+b , |f (b)|q v 1/2 Suy (b a)4 (b a)4 + 1/2 t2 t I Footer Page 65 of 258 ã max |f (b)|, f (1 t)2 t 1/2 ã max f a+b dt a+b , |f (a)| dt Header Page 66 of 258 61 (b a)4 a+b max |f (a)|, f = 1152 a+b + max f , |f (b)| Ta cú iu phi chng minh nh lý 2.21 ([4], Theorem 3) Gi s a, b I R v a < b, hm f : [a, b] R l liờn tc tuyt i trờn (a, b) Nu f L[a, b] v |f |q vi q l ta li trờn [a, b] thỡ b f (x)dx a (b a) a+b f (a) + 4f 21/p (b a)4 (p + 1)(2p + 1) 48 (3p + 2) + max f 1/p max q , |f (b)| , |f (a)| max q a+b f 1/q q q 21/p (b a)4 = (B(p + 1, 2p + 1))1/p 48 + max f a+b 1/q q a+b + f (b) f a+b 1/q q , |f (b)| 1/q q , |f (a)| Chng minh T nh lý 2.19 ta cú b f (x)dx I= a a+b (b a) f (a) + 4f + f (b) = (b a) p(t)f (ta + (1 t)b)dt (b a)4 |p(t)f (ta + (1 t)b)|dt (b a)4 = (b a)4 + Footer Page 66 of 258 1/2 t2 t |f (ta + (1 t)b)|dt (t 1)2 t 1/2 q |f (ta + (1 t)b)|dt Header Page 67 of 258 62 p dng Bt ng thc Hoălder ta cú 1/2 (b a)4 I t p t 1/p (b a) (t 1)2 t 1/2 q |f (ta + (1 t)b)| dt 1/p p dt 1/q q ì 1/q 1/2 dt + |f (ta + (1 t)b)| dt 1/2 Khi |f |q l ta li, theo Bt ng thc Hermite-Hadamard ta cú: 1/2 q |f (ta + (1 t)b)| dt max f a+b f a+b q , |f (b)|q , v q |f (ta + (1 t)b)| dt max 1/2 q , |f (b)|q Hm Beta v hm Gamma c xỏc nh nh sau tx1 (1 t)y1 dt, B(x, y) = x, y > 0 v (x) = et tx1 dt, x>0 Kt hp cỏc bt ng thc trờn ta thu c: b f (x)dx a 1/p (b a) a+b f (a) + 4f (b a)4 (p + 1)(2p + 1) 48 (3p + 2) + max f a+b q 1/p max 1/q q , |f (a)| Ta cú iu phi chng minh Footer Page 67 of 258 + f (b) f a+b q 1/q q , |f (b)| Header Page 68 of 258 63 Kt lun Lun trỡnh by c mt s kt qu sau: - Mt s c trng c bn ca hm li kh vi v hm ta li - Chng minh cỏc Bt ng thc dng Hermite-Hadamard cho hm li mt bin - Chng minh cỏc Bt ng thc dng Hermite-Hadamard cho hm ta li mt bin - Mt s m rng Bt ng thc dng Hermite-Hadamard cho hm cú o hm cp mt, o hm cp hai, o hm cp ba l ta li Footer Page 68 of 258 Header Page 69 of 258 64 Ti liu tham kho Ting Anh [1] M Alomari, M Darus, Some Ostrowski type inequalities for quasiconvex functions with applications to special means, RGMIA13 (2) (2010), article No Preprint [2] M Alomari, M Darus and U S Kirmaci, Refinements of Hadamardtype inequalities for quasi-convex functions with applications to trapezoidal formula and to special means, Comput Math Appl., 59, (2010), 225-232 [3] M Alomari, M Darus and S S Dragomir, New inequalities of Hermite Hadamard for functions whose second derivatives absolute values are quasi-convex, Tamkang Journal of Mathematics, Volume 41, No 4, 2010, 353-359 [4] M Alomari, S Hussain, Two Inequalities of Simpson type for Quasiconvex and Applications, Applied Mathematics E-Notes, 11, (2011), 110-117 [5] Peter Bullen, Dictionary of Inequalities, Second Edition, CRS Press, Taylor and Francis Group, LLC, USA Footer Page 69 of 258 Header Page 70 of 258 65 [6] P Cerone, Sever S Dragomir, Mathematical Inequalities: A perspective, (2011), CRS Press, Taylor and Francis Group, LLC, USA [7] S S Dragomir, Charles E M Pearce, Selected Topics on HermiteHadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000 [8] D A Ion, Some estimates on the Hermite Hamdamard inequalities through quasi convex functions, Annals of University of Craiova, Math Comp Sci Ser., Volume 34, 2007, 82-87 scan, On new general integral inequalities for quasi-convex [9] Imdat Iá functions and their applications, Palestine Journal of Mathematics, Vol 4(1), 2015, 21-29 [10] M Z Sarikaya, A Saglam, and H Yildirim, New inequalities of Hermite-Hadamard type for functions whose second derivatives absolute values are quasi-convex [11] Hoang Tuy, Convex Analysis and Global Optimization, In Serie Nonconvex Optimization and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998 Ting Phỏp [12] J L W V Jensen, Sur les fonctions convexes et les inộgalitộs entre les valeurs moyennes, Acta Math, 30 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Chương Bất đẳng thức Hermite- Hadamard cho lớp hàm tựa lồi 1.1 1.2 1.3 Hàm lồi số đặc trưng hàm lồi khả vi Bất đẳng thức Hermite- Hadamard Bất đẳng thức Hermite- Hadamard cho hàm tựa lồi. .. 1.3.1 Hàm tựa lồi 1.3.2 15 15 Bất đẳng thức Hermite- Hadamard cho hàm tựa lồi 22 Chương Một số mở rộng bất đẳng thức Hermite- Hadamard cho hàm tựa lồi 25 2.1 Bất đẳng thức Hermite- Hadamard. .. Hermite- Hadamard cho lớp hàm có đạo 2.2 2.3 hàm tựa lồi Bất đẳng thức dạng Hermite- Hadamard cho đạo hàm bậc hai tựa lồi Bất đẳng thức dạng Hermite- Hadamard cho có