Một số bất đẳng thức về hàm lồi và ứng dụng

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕ ◆●❯❨➍◆ ❚❍➚ ❍➬◆● ❍❖❆ ▼❐❚ ❙➮ ❇❻❚ ✣➃◆● ❚❍Ù❈ ❱➋ ❍⑨▼ ▲➬■ ❱⑨ Ù◆● ❉Ö◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆✱ ✶✵✴✷✵✶✽ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕ ◆●❯❨➍◆ ❚❍➚ ❍➬◆● ❍❖❆ ▼❐❚ ❙➮ ❇❻❚ ✣➃◆● ❚❍Ù❈ ❱➋ ❍⑨▼ ▲➬■ ❱⑨ Ù◆● ❉Ö◆● ❈❤✉②➯♥ ♥❣➔♥❤✿ P❤÷ì♥❣ ♣❤→♣ t♦→♥ ❝➜♣ ▼➣ sè✿ ✽✹✻✵✶✶✸ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ●■⑩❖ ❱■➊◆ ❍×❰◆● ❉❼◆ P●❙✳❚❙✳ ◆●❯❨➍◆ ❚❍➚ ❚❍❯ ❚❍Õ❨ ❚❍⑩■ ◆●❯❨➊◆✱ ✶✵✴✷✵✶✽ ✐✐✐ ▼ö❝ ❧ö❝ ỵ ữỡ ỗ t tự rtr ỗ ởt t tự rtr t tự rtr ỗ ✳ ✳ ✳ ✹ ✶✳✶✳✷ ❇➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ❝❤♦ ỗ t t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✷✳✶ Ù♥❣ ❞ö♥❣ tr♦♥❣ ✤→♥❤ ❣✐→ ❝→❝ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ✳ ✳ ✳ ✶✹ ✶✳✷✳✷ Ù♥❣ ❞ö♥❣ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ♣❤ê t❤ỉ♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ữỡ ỗ s rë♥❣ ✈➔ ù♥❣ ❞ö♥❣ ✷✳✶ ✷✳✷ ✷✳✸ ✷✶ ❍➔♠ J ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ỗ tr Rn ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✶✳✷ ❍➔♠ J ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ sỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✷✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❱➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ t sỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ❇➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ sỗ ✷✳✸✳✶ ❇➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✸✳✷ ▼ët sè ❜➜t tự ợ ữủ tt tứ t tự ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✸✳✸ ▼ët sè ù♥❣ ❞ö♥❣ ❝❤♦ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ✤➦❝ ❜✐➺t ✳ ✳ ✳ ✹✵ ✐✈ ❑➳t ❧✉➟♥ ✹✶ ❚➔✐ ❧✐➺✉ t ỵ R t số t❤ü❝ Lp [a, b] ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔ t➼❝❤ ❜➟❝ p tr➯♥ ✤♦↕♥ [a, b] Co ♣❤➛♥ tr♦♥❣ ❝õ❛ t➟♣ C A tr✉♥❣ ❜➻♥❤ ❝ë♥❣ G tr✉♥❣ ❜➻♥❤ ♥❤➙♥ H tr✉♥❣ ❜➻♥❤ ✤✐➲✉ ❤á❛ L tr✉♥❣ ❜➻♥❤ ❧æ❣❛r✐t Lp tr pổrt ỗ t ỗ ữủ ự tứ r s ▼✐♥❦♦✇s❦✐✳ ✣➦❝ ❜✐➺t ✈ỵ✐ ♥❤ú♥❣ ❝ỉ♥❣ tr➻♥❤ ❝õ❛ ❋❡♥❝❤❡❧✱ ▼♦r❡❛✉✱ ❘♦❝❦✲ ❛❢❡❧❧❛r ✈➔♦ ❝→❝ t❤➟♣ ♥✐➯♥ ✶✾✻✵ ✈➔ ✶✾✼✵ ✤➣ ữ t ỗ tr t ởt tr ỳ ✈ü❝ ♣❤→t tr✐➸♥ ♥❤➜t ❝õ❛ t♦→♥ ❤å❝✳ ❇➯♥ ❝↕♥❤ ✤â✱ ởt số ổ ỗ t ữ ❝ô♥❣ ❝❤✐❛ s➫ ♠ët ✈➔✐ t➼♥❤ ❝❤➜t ♥➔♦ ✤â ❝õ❛ ỗ ú ữủ ỗ s rë♥❣ ✭❣❡♥❡r❛❧✐③❡❞ ❝♦♥✈❡① ❢✉♥❝t✐♦♥✮✳ ✳ ✳ ▼ö❝ t✐➯✉ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ❧➔ tr➻♥❤ ❜➔② ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì t ỗ ỗ ởt ỗ J ỗ sỗ t tự rtr ỗ ỗ sỗ ✈➔ ù♥❣ ❞ö♥❣ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ t♦→♥ ♣❤ê t❤æ♥❣✱ ✤→♥❤ ❣✐→ ❝→❝ ❣✐→ trà tr✉♥❣ ❜➻♥❤✳ ▲✉➟♥ ✈➠♥ ❝ô♥❣ tr➻♥❤ ❜➔② ♠ët sè ❜➜t ✤➥♥❣ t❤ù❝ s✉② rë♥❣ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ❝❤♦ n J ỗ sỗ s✲❧ã♠ tr♦♥❣ ❝→❝ ❝æ♥❣ tr➻♥❤ ❬✼❪✱ ❬✽❪ ❝æ♥❣ ❜è ♥➠♠ ✷✵✶✷ ✈➔ ✷✵✶✼✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ✈➔ ❝❤ù♥❣ t tự rtr ỗ ởt ỗ t n ✈➔ ù♥❣ ❞ö♥❣ ✤→♥❤ ❣✐→ ♠ët sè ❣✐→ trà tr✉♥❣ ❜➻♥❤ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❜➔✐ t➟♣ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ♣❤ê t❤ỉ♥❣✳ ❈❤÷ì♥❣ ✷ tr J ỗ ởt số t t ợ J ỗ sỗ t t sỗ sỗ r t tự rtr sỗ tr tt ự ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ♥➔②✱ ❝ò♥❣ ♠ët sè ù♥❣ ❞ư♥❣ ❝❤♦ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ✤➦❝ ❜✐➺t✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ❚r♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ tèt ♥❤➜t ✤➸ t→❝ ❣✐↔ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉✳ ❚→❝ ❣✐↔ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❝→❝ t❤➛②✱ ❝ỉ tr♦♥❣ ❑❤♦❛ ❚♦→♥ ✲ ❚✐♥✱ tr♦♥❣ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ✣➦❝ ❜✐➺t✱ t→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ P●❙✳❚❙✳ ◆❣✉②➵♥ ❚❤à ữớ t t ữợ t ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❳✐♥ ❝↔♠ ì♥ ♥❤ú♥❣ ♥❣÷í✐ t❤➙♥ tr♦♥❣ ❣✐❛ ✤➻♥❤ ✤➣ ❤➳t sù❝ t❤ỉ♥❣ ❝↔♠✱ ❝❤✐❛ s➫ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ tèt ♥❤➜t ❝❤♦ tæ✐ ✤➸ tæ✐ ❝â t❤➸ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ♥❤ú♥❣ ❝ỉ♥❣ ✈✐➺❝ ❝õ❛ ♠➻♥❤✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❣û✐ ♥❤ú♥❣ ❧í✐ ❝↔♠ ì♥ ✤➦❝ ❜✐➺t ♥❤➜t tỵ✐ t➜t ❝↔ ♥❤ú♥❣ ♥❣÷í✐ ❜↕♥ t❤➙♥ ②➯✉✱ ♥❤ú♥❣ ♥❣÷í✐ ✤➣ ②➯✉ ♠➳♥✱ ❝❤✐❛ s➫ ✈ỵ✐ tỉ✐ ♥❤ú♥❣ ❦❤â ❦❤➠♥ tr♦♥❣ ❦❤✐ tỉ✐ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✶✵ ♥➠♠ ✷✵✶✽ ỗ ữỡ ỗ t tự rtr ữỡ ợ t ỗ tr ởt số t tự rtr ỗ ỗ ự ởt số ❣✐→ trà tr✉♥❣ ❜➻♥❤ ✤➦❝ ❜✐➺t ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❜➔✐ t➟♣ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ♣❤ê t❤ỉ♥❣✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ tê♥❣ ❤đ♣ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✸❪✱ ❬✹❪✱ ❬✼❪✱ ❬✽❪ ✈➔ ❬✶✵❪✳ ỗ ởt t tự rtr t tự rtr ỗ ✶✳✶✳✶ ❍➔♠ f : [a, b] ⊂ R → R ữủ ỗ ợ x, y ∈ [a, b] ✈➔ λ ∈ [0, 1] t❤➻ f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y) ❍➔♠ f ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ❧ã♠ (f ) ỗ q ❍➺ q✉↔ ✷✳✶❪✮ ❍➔♠ f (x) ❦❤↔ ✈✐ ❤❛✐ ❧➛♥ tr ỗ ✤↕♦ ❤➔♠ ❝➜♣ ❤❛✐ ❝õ❛ ♥â ❦❤æ♥❣ ➙♠ tr➯♥ t♦➔♥ ❦❤♦↔♥❣ (a, b)✳ (a, b) ⊆ R ❘➜t ♥❤✐➲✉ ❜➜t tự q trồ ữủ tt tứ ợ ỗ ởt tr ỳ t tự t ♥❤➜t ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕ ✺ ❍❛❞❛♠❛r❞ ✭❝á♥ ❣å✐ ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ❍❛❞❛♠❛r❞✮✳ ❇➜t ✤➥♥❣ t❤ù❝ ❦➨♣ ♥➔② ữủ t tr ỵ s ỵ rtr tr qt f ởt ỗ tr➯♥ [a, b] ⊂ R✱ a = b✳ ❑❤✐ ✤â a+b f ≤ b−a b f (x)dx ≤ a f (a) + f (b) ❧➔ ✭✶✳✶✮ t tự õ t t ữợ b (b − a)f a+b ≤ f (x)dx ≤ (b − a) f (a) + f (b) a ự f ỗ tr [a, b]✱ ♥➯♥ ✈ỵ✐ ♠å✐ λ ∈ [0, 1] t❛ ❝â f λa + (1 − λ)b ≤ λf (a) + (1 − λ)f (b) ▲➜② t➼❝❤ ♣❤➙♥ ❤❛✐ ✈➳ t❤❡♦ λ tr➯♥ ✤♦↕♥ [0, 1]✱ t❛ ♥❤➟♥ ✤÷đ❝ 1 f λa + (1 − λ)b dλ ≤ f (a) ❱➻ 1 (1 − λ)dλ = λdλ = (1 − λ)dλ λdλ + f (b) ✈➔ ❜➡♥❣ ♣❤➨♣ ✤ê✐ ❜✐➳♥ x = λa + (1 − λ)b, s✉② r❛ b f λa + (1 − λ)b dλ = b−a f (x)dx a ❑➳t ❤đ♣ ✈ỵ✐ ✭✶✳✸✮ t❛ ♥❤➟♥ ✤÷đ❝ ❜➜t ✤➥♥❣ t❤ù❝ t❤ù ❤❛✐ ❝õ❛ ✭✶✳✶✮✳ ❈ơ♥❣ t ỗ f f (a + (1 − λ)b) + f ((1 − λ)a + λb) λa + (1 − λ)b + (1 − λ)a + λb ≥f a+b =f ✭✶✳✸✮ ✻ ❚➼❝❤ ♣❤➙♥ ❤❛✐ ✈➲ ❜➜t ✤➥♥❣ t❤ù❝ ♥➔② t❤❡♦ λ tr➯♥ ✤♦↕♥ [0, 1] t❛ ♥❤➟♥ ✤÷đ❝   1 a+b ≤  f (λa + (1 − λ)b)dλ + f ((1 − λ)a + λb)dλ f 2 0 b = b−a f (x)dx a ❇➜t ✤➥♥❣ t❤ù❝ t❤ù ♥❤➜t ❝õ❛ ✭✶✳✶✮ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ◆➳✉ g : [a, b] → R ❧➔ ❤➔♠ ❦❤↔ ✈✐ ❤❛✐ ❧➛♥ tr➯♥ (t) ≤ M ✈ỵ✐ ♠å✐ x ∈ [a, b]✱ m✱ M ❧➔ ❤➡♥❣ sè ①→❝ ✤à♥❤✱ ❍➺ q✉↔ ✶✳✶✳✹ ✭①❡♠ ❬✸❪✮ [a, b] ⊆ R t❤➻ ✈➔ m ≤ g b m (b − a)2 ≤ 24 b−a g(x)dx − g a+b ≤ M (b − a)2 24 ✭✶✳✹✮ a m x ✈ỵ✐ ♠å✐ x ∈ [a, b]✳ ❑❤✐ ✤â✱ f (x) = g (x) − m ≥ 0, ∀x ∈ (a, b) ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t f (x) = g(x) ự tọ f ỗ tr (a, b)✳ ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ❝❤♦ ❤➔♠ f t❛ ♥❤➟♥ ✤÷đ❝ g a+b m − a+b 2 =f a+b b = b−a g(x) − m x dx a b = b−a g(x)dx − m b3 − a3 3(b − a) g(x)dx − m a2 + ab + b2 a b = b−a a ❉♦ ✤â✱ m a2 + ab + b2 m − a+b 2 b ≤ b−a g(x)dx − g a a+b ✸✾ ❉♦ ✤â t❛ ❝â a+b − f b−a b f (x)dx a (b − a)2 p a+b ≤ |f (a)|q + f 16 (s + 1)(s + 2)(s + 3) s+3 2 a+b q + |f (b)|q f + s+3 (s + 1)(s + 2)(s + 3) ❍➺ q✉↔ ✷✳✸✳✼ ✭①❡♠ ❬✻❪✮ + f q r ỵ s = t❤➻ a+b − f b−a (b − a)2 ≤ 48 q p a+b b f (x)dx a |f (a)|q a+b + f q |f (a)|q 1q + q q ✭✷✳✷✾✮ ❚❛ ❝â ❦➳t q✉↔ ❝❤♦ ❤➔♠ s✲❧ã♠ ♥❤÷ s❛✉✳ ●✐↔ sû f : C ⊂ [0, +∞) → R ❧➔ ❤➔♠ ❦❤↔ ✈✐ tr➯♥ C o s❛♦ ❝❤♦ f ∈ L1 [a, b]✱ ð ✤➙② a, b ∈ C ✈ỵ✐ a < b✳ ◆➳✉ ❤➔♠ |f |q ✱ q ≥ 1✱ ❧➔ ❤➔♠ s✲❧ã♠ ❧♦↕✐ ✷ tr➯♥ [a, b] ✈ỵ✐ s ∈ (0, 1] t❤➻ ❜➜t tự s tọ ỵ ❬✻❪✮ a+b − f b−a b f (x)dx a s−1 (b − a)2 q ≤ 16 (2p + 1) p1 ❍➺ q✉↔ ✷✳✸✳✾ ✭①❡♠ ❬✻❪✮ 1/p 2p+1 ✱ p > 1✱ t❤➻ f 3a + b + f a + 3b r ỵ ✷✳✸✳✽ ♥➳✉ ❝❤å♥ b a+b f − f (x)dx b−a a (b − a)2 3a + b ≤ f + f 48 a + 3b s = ✈➔ < ✭✷✳✸✵✮ ✹✵ ✷✳✸✳✸ ▼ët sè ù♥❣ ❞ö♥❣ ❝❤♦ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ✤➦❝ ❜✐➺t ❚❛ ①➨t ♠ët sè ù♥❣ ❞ö♥❣ ❝õ❛ ❜➜t tự rtr sỗ ởt sè ❣✐→ trà tr✉♥❣ ❜➻♥❤ ✤➦❝ ❜✐➺t✿ tr✉♥❣ ❜➻♥❤ ❝ë♥❣ ✭✶✳✶✼✮✱ tr✉♥❣ ❜➻♥❤ ❧æ❣❛r✐t ✭✶✳✶✽✮ ✈➔ tr✉♥❣ ❜➻♥❤ p✲❧æ❣❛r✐t ✭✶✳✶✾✮✳ ▼➺♥❤ ✤➲ ✷✳✸✳✶✵ ✭①❡♠ ❬✻❪✮ s A (a, b) − Lss (a, b) ●✐↔ sû < a < b ✈➔ s ∈ (0, 1)✳ ❑❤✐ ✤â✱ |s(s − 1)|(b − a)2 s−2 a+b ≤ a +6 192 s−2 + bs−2 ✭✷✳✸✶✮ ❈❤ù♥❣ ♠✐♥❤✳ ❇➜t ✤➥♥❣ t❤ù❝ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ s✉② r❛ tø ✭✷✳✷✼✮ ✈➔ →♣ ❞ư♥❣ sỗ f : [0, 1] [0, 1] ▼➺♥❤ ✤➲ ✷✳✸✳✶✶ ✭①❡♠ ❬✻❪✮ ①→❝ ✤à♥❤ ❜ð✐ f (x) = xs ●✐↔ sû < a < b ✈➔ s ∈ (0, 1)✳ ❑❤✐ ✤â✱ q s A (a, b) − Lss (a, b) |s(s − 1)|(b − a)2 ≤ 48 a+b bq(s−2) + + aq(s−2) + q a+b q(s−2) ✭✷✳✸✷✮ ❈❤ù♥❣ ♠✐♥❤✳ ❇➜t ✤➥♥❣ t❤ù❝ ✭✷✳✸✷✮ ✤÷đ❝ s r tứ sỗ ✷ f : [0, 1] → [0, 1] ▼➺♥❤ ✤➲ ✷✳✸✳✶✷ ✭①❡♠ ❬✻❪✮ ❍➺ q✉↔ ✷✳✸✳✾✳ ❑❤✐ ✤â✱ ①→❝ ✤à♥❤ ❜ð✐ f (x) = xs ●✐↔ sû < a < b ✈➔ p > ♥❤÷ ❣✐↔ t❤✐➳t ❝õ❛ As (a, b) − Lss (a, b) ≤ (b − a)2 1 + (3a + b) (a + 3b)2 ✭✷✳✸✸✮ ❈❤ù♥❣ ♠✐♥❤✳ ❈ỉ♥❣ t❤ù❝ ✭✷✳✸✸✮ ✤÷đ❝ s✉② r❛ tø ✭✷✳✸✵✮ →♣ ❞ö♥❣ ❝❤♦ ❤➔♠ ❧ã♠ ❧♦↕✐ ✷ f : [a, b] → R ①→❝ ✤à♥❤ ❜ð✐ f (x) = ln x q ✹✶ ❑➳t ❧✉➟♥ ✣➲ t➔✐ ❧✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② ♠ët sè ❜➜t ✤➥♥❣ tự rt r ỗ ỗ s rë♥❣ ✈➔ ♠ët sè ù♥❣ ❞ö♥❣ ✤→♥❤ ❣✐→ ❝→❝ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ✤➦❝ ❜✐➺t✱ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ♣❤ê t❤ỉ♥❣✳ ❈ö t❤➸✿ ✭✶✮ ❚r➻♥❤ ❜➔② ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❜➜t tự rtr ỗ ởt ỗ ởt ởt ❤❛✐✱ ❦❤↔ ✈✐ ❝➜♣ n✳ ✭✷✮ ❚r➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ J ỗ J ỗ s rở ởt sè t➼♥❤ ❝❤➜t ❝ò♥❣ ♠ët ù♥❣ ❞ư♥❣ ❝❤♦ ❜➔✐ t♦→♥ ❝ü❝ trà✳ ✭✸✮ ❚r➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠✱ ✈➼ ❞ö ✈➔ ởt số t t sỗ r ự ♠✐♥❤ ♠ët sè ❜➜t ✤➥♥❣ t❤ù❝ ♠ỵ✐ ❞↕♥❣ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ❝❤♦ sỗ r ởt số ự t tự rtr ỗ ỗ sỗ só tr ởt sè ❣✐→ trà tr✉♥❣ ❜➻♥❤ ✤➦❝ ❜✐➺t ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ♣❤ê t❤æ♥❣✳ ✹✷ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ❚r➛♥ ❱ô ❚❤✐➺✉✱ ◆❣✉②➵♥ ❚❤à ❚❤✉ ❚❤õ② ✭✷✵✶✶✮✱ t✉②➳♥✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✱ ●✐→♦ tr➻♥❤ ❚è✐ ÷✉ ♣❤✐ ❚✐➳♥❣ ❆♥❤ ❬✷❪ ▼✳ ❆❧♦♠❛r✐✱ ▼✳ ❉❛r✉s ✭✷✵✵✽✮✱ ✧❚❤❡ ❍❛❞❛♠❛r❞✬s ✐♥❡q✉❛❧✐t② ❢♦r s✲ ❝♦♥✈❡① ❢✉♥❝t✐♦♥✧✱ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ♦❢ ▼❛t❤❡♠❛t✐❝s ❆♥❛❧②s✐s✱ ✶✸✭✷✮✱ ✻✸✾✕✻✹✻✳ ❬✸❪ P✳ ❈❡r♦♥❡✱ ❙✳❙✳ ❉r❛❣♦♠✐r ✭✷✵✶✶✮✱ ▼❛t❤❡♠❛t✐❝❛❧ ■♥❡q✉❛❧✐t✐❡s✿ ❆ ♣❡r✲ s♣❡❝t✐✈❡✱ ❈❘❙ Pr❡ss✱ ❚❛②❧♦r ❛♥❞ ❋r❛♥❝✐s ●r♦✉♣✱ ▲▲❈✱ ❯❙❆✳ ❬✹❪ ❙✳❙✳ ❉r❛❣♦♠✐r✱ ❊✳▼✳P✳ ❈❤❛r❧❡s ✭✷✵✵✵✮✱ ❙❡❧❡❝t❡❞ ❚♦♣✐❝s ♦♥ ❍❡r♠✐t❡✲ ❍❛❞❛♠❛r❞ ■♥❡q✉❛❧✐t✐❡s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❘●▼■❆ ▼♦♥♦❣r❛♣❤s✱ ❱✐❝t♦✲ r✐❛ ❯♥✐✈❡rs✐t②✳ ❬✺❪ ❍✳ ❍✉❞③✐❦✱ ▲✳ ▼❛❧✐❣r❛♥❞❛ ✭✶✾✾✹✮✱ ✧❙♦♠❡ r❡♠❛r❦s ♦♥ s✲❝♦♥✈❡① ❢✉♥❝✲ t✐♦♥s✧✱ ❆❡q✉❛t✐♦♥❡s ▼❛t❤❡♠❛t✐❝❛❡✱ ❯♥✐✈❡rs✐t② ♦❢ ❲❛t❡r❧♦♦✱ ✹✽✱ ✶✵✵✕ ✶✶✶✳ ☎ ❬✻❪ ▼✳❊✳ ❖③❞❡♠✐r✱ ❈✳ ❨✙❧❞✙③✱ ❆✳❖✳ ❆❦❞❡♠✐r ❛♥❞ ❊✳ ❙❡t ✭✷✵✶✸✮✱ ✧❖♥ s♦♠❡ ✐♥❡q✉❛❧✐t✐❡s ❢♦r s✲❝♦♥✈❡① ❢✉♥❝t✐♦♥s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✧✱ ❡q✉❛❧✐t✐❡s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✷✵✶✸✿✸✸✸✳ ❏♦✉r♥❛❧ ♦❢ ■♥✲ ☎ ❬✼❪ ▼✳❊✳ ❖③❞❡♠✐r✱ ❈✳ ❨✙❧❞✙③ ✭✷✵✶✼✮✱ ✧❖♥ ❣❡♥❡r❛❧✐③❡❞ ✐♥❡q✉❛❧✐t✐❡s ♦❢ ❍❛r♠✐t❡✕❍❛❞❛♠❛r❞ t②♣❡ ❢♦r ❝♦♥✈❡① ❢✉♥❝t✐♦♥s✧✱ ♥❛❧ ♦❢ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✶✹✭✶✮✱ ✺✷✲✻✸✳ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r✲ ✹✸ ❬✽❪ ▼✳❘✳ ❚❛s❦♦✈✐✁❝ ✭✷✵✶✷✮✱ ✧■♥❡q✉❛❧✐t✐❡s ♦❢ ❣❡♥❡r❛❧ ❝♦♥✈❡① ❢✉♥❝t✐♦♥s ❛♥❞ ▼❛t❤❡♠❛t✐❝❛ ▼♦r❛✈✐❝❛✱ ✶✻✭✶✮✱ ✸✼✕✶✶✻✳ ❬✾❪ ❏✳❊✳ P❡❝❛r✐❝✱ ❋✳ Pr♦s❝❤❛♥✱ ❛♥❞ ❨✳▲✳ ❚♦♥❣ ✭✶✾✾✶✮✱ ❈♦♥✈❡① ❋✉♥❝t✐♦♥s✱ P❛rt✐❛❧ ❖r❞❡r✐♥❣s ❛♥❞ ❙t❛t✐st✐❝❛❧ ❆♣♣❧✐❝❛t✐♦♥s✱ ❆❝❛❞❡♠✐❝ Pr❡ss✱ ■♥❝✳✱ ❛♣♣❧✐❝❛t✐♦♥s✧✱ ❇♦st♦♥✱ ❙❛♥ ❉✐❡❣♦✱ ◆❡✇ ❨♦r❦✳ ❬✶✵❪ ❑✳ ❚s❡♥❣❛✱ ❙✳ ❍✇❛♥❣❜✱ ❑✳ ❍s✉ ✭✷✵✶✷✮✱ ✧❍❛❞❛♠❛r❞✲t②♣❡ ❛♥❞ ❇✉❧❧❡♥✲ t②♣❡ ✐♥❡q✉❛❧✐t✐❡s ❢♦r ▲✐♣s❝❤✐t③✐❛♥ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥s✧✱ ❈♦♠♣✉t❡rs ✫ ▼❛t❤❡♠❛t✐❝s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✱ ✻✹✭✹✮✱ ✻✺✶✕✻✻✵✳ ❬✶✶❪ ❍✳ ❚✉② ✭✶✾✾✽✮✱ ❈♦♥✈❡① ❆♥❛❧②s✐s ❛♥❞ ●❧♦❜❛❧ ❖♣t✐♠✐③❛t✐♦♥✱ ■♥ ❙❡r✐❡ ◆♦♥❝♦♥✈❡① ❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s✱ ❑❧✉✇❡r ❆❝❛❞❡♠✐❝ P✉❜✲ ❧✐s❤❡rs✱ ❉♦r❞r❡❝❤t✱ ❚❤❡ ◆❡t❤❡r❧❛♥❞s✳ ... ✳ ✳ t tự rtr sỗ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✸✳✶ ❇➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ởt số t tự ợ ữủ t❤✐➳t ❧➟♣ tø ❜➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✸✳✸... ù♥❣ ❞ö♥❣ ❝❤♦ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ✤➦❝ ❜✐➺t ✳ ✳ ✳ ✹✵ ✐✈ ❑➳t ❧✉➟♥ ✹✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✷ ✶ ỵ R t số tỹ Lp [a, b] ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔ t➼❝❤ ❜➟❝ p tr➯♥ ✤♦↕♥ [a, b] Co ♣❤➛♥ tr♦♥❣ ❝õ❛ t➟♣ C A tr✉♥❣... ❜➔✐ t➟♣ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ♣❤ê t❤ỉ♥❣✳ ❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ J ỗ ởt số t t ợ J ỗ sỗ t t sỗ sỗ r t tự rtr sỗ tr ✸ ❝❤✐ t✐➳t ❝→❝ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❜➜t ✤➥♥❣