✣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✱ ❝ị♥❣ ♠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 ✷✾ ●✐↔ sû < s ≤ 1✳ ◆➳✉ f, g ∈ Ks1 ✈➔ ♥➳✉ F : R2 → R ❧➔ ❤➔♠ ❦❤æ♥❣ ❣✐↔♠ ỗ t tứ t h : R+ R ỵ h(u) = F f (u), g(u) ❧➔ ❤➔♠ t❤✉ë❝ Ks1✳ ✣➦❝ ❜✐➺t ♥➳✉ f, g ∈ Ks1 t❤➻ f + g✱ max{f, g} ∈ Ks1 ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ u, v ∈ R+ t❤➻ ✈ỵ✐ ♠å✐ α, β ≥ ✈ỵ✐ αs + β s = t❛ ❝â h(αu + βv) = F f (αu + βv), g(αu + βv) ≤ F αs f (u) + β s f (v), αs g(u) + β s g(v) ≤ αs F f (u), g(u) + β s F f (v), g(v) = αs h(u) + β s h(v) ❱➻ F (u, v) = u + v ✈➔ F (u, v) = max{u, v} ỗ ổ tr R2 t❛ s✉② r❛ ❝→❝ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❝õ❛ ✤à♥❤ ỵ ỵ f Ks1 t❤➻ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✶✸✮ t❤ä❛ ♠➣♥ ✈ỵ✐ ♠å✐ u, v ∈ R+ ✈➔ ✈ỵ✐ ♠å✐ α, β ≥ s❛♦ ❝❤♦ αs + β s ≤ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f (0) ≤ 0✳ (ii) ◆➳✉ f ∈ Ks2 t❤➻ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✶✸✮ t❤ä❛ ♠➣♥ ✈ỵ✐ ♠å✐ u, v ∈ R+ ✈➔ α, β ≥ ✈ỵ✐ α + β ≤ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f (0) = 0✳ (i) ❈❤ù♥❣ ♠✐♥❤✳ (i) ✣✐➲✉ ❦✐➺♥ ❝➛♥✿ ❍✐➸♥ ♥❤✐➯♥ ✤ó♥❣ ❦❤✐ u = v = ✈➔ α = β = ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ t❤✐➳t r➡♥❣ u, v ∈ R+ α, β ≥ ✈➔ < γ = αs + β s < 1✳ ❱ỵ✐ a = αγ −1/s vb = βγ −1/s ✳ ❑❤✐ ✤â as + bs = αs /γ + β s /γ = ✈➔ t❛ ❝â f αu + βv = f aγ 1/s u + bγ 1/s v ≤ as f (γ 1/s u) + bs f (γ 1/s v) = as f γ 1/s u + (1 − γ)1/s + bs f γ 1/s v + (1 − γ)1/s ≤ as γf (u) + (1 − γ)f (0) + bs γf (v) + (1 − γ)f (0) = as γf (u) + bs γf (v) + (1 − γ)f (0) ≤ αs f (u) + β s f (v) ✸✵ (ii) ✣✐➲✉ ❦✐➺♥ ❝➛♥✿ ✣➦t u = v = α = β = t❛ ♥❤➟♥ ✤÷đ❝ f (0) ≤ ✈➔ sû ❞ư♥❣ ✣à♥❤ ỵ s r f (0) õ f (0) = 0✳ ✣✐➲✉ ❦✐➺♥ ✤õ✿ ❇➙② ❣✐í ❧➜② u, v ∈ R+ ✈➔ α, β ≤ ✈ỵ✐ < γ = αs + β s < 1✳ ✣➦t a = α/γ ✈➔ b = β/γ t❤➻ a + b = α/γ + β/γ = ✈➔ f αu + βv = f αγu + βγv ≤ as f (γu) + bs f (γv) = as f γu − (1 − γ)0 + bs f γv + (1 − γ)0 ≤ as γ s f (u) + (1 − γ)s f (0) + bs γ s f (v) + (1 − γ)s f (0) = as γ s f (u) + bs γ s f (v) + (1 − γ)s f (0) = αs f (u) + β s f (v) ỵ sỷ < s ≤ 1✳ ◆➳✉ f ∈ Ks2 ✈➔ f (0) = t❤➻ f ∈ Ks1✳ (ii) ●✐↔ sû < s1 ≤ s2 ≤ 1✳ ◆➳✉ f ∈ Ks2 ✈➔ f (0) = t❤➻ f ∈ Ks1 ✳ (iii) ●✐↔ sû < s1 ≤ s2 ≤ 1✳ ◆➳✉ f ∈ Ks1 ✈➔ f (0) ≤ t❤➻ f ∈ Ks1 ✳ (i) 2 ❈❤ù♥❣ ♠✐♥❤✳ (i) ●✐↔ sû f ∈ Ks2 ✈➔ f (0) = 0✳ ❱ỵ✐ u, v ∈ R+ ✈➔ α, β ≥ t❤ä❛ ♠➣♥ αs + β s = t❤➻ α + β ≤ αs + β s = tứ ỵ t ✤÷đ❝ f αu + βv ≤ αs f (u) + β s f (v), ♥❣❤➽❛ ❧➔ f ∈ Ks1 ✳ (ii) ●✐↔ sû f ∈ Ks22 ✈➔ u, v ≥ 0✱ α, β ≥ ✈ỵ✐ α + β = ❑❤✐ ✤â✱ f αu + βv ≤ αs2 f (u) + β s2 f (v) ≤ αs1 f (u) + β s1 f (v) ◆❤÷ ✈➟② f ∈ Ks11 ✳ (iii) ●✐↔ sû f ∈ Ks12 ✈➔ u, v ≥ 0✱ α, β ≥ ✈ỵ✐ αs1 + β s1 = ❑❤✐ ✤â αs2 + β s2 ≤ s1 + s1 = t ỵ ✷✳✷✳✻ f αu + βv ≤ αs2 f (u) + β s2 f (v) ≤ αs1 f (u) + β s1 f (v), ✸✶ ♥❣❤➽❛ ❧➔ f ∈ Ks11 ✳ f : [a, b] R sỗ ✈➔ a ≤ y1 ≤ x1 ≤ x2 ≤ y2 ≤ b ✈ỵ✐ x1 + x2 = y1 + y2 ✳ ❑❤✐ ✤â ❇ê ✤➲ ✷✳✷✳✽ ✭①❡♠ ❬✷❪✮ ✭✷✳✶✻✮ f (x1 ) + f (x2 ) ≤ f (y1 ) + f (y2 ) ỵ sû ❤➔♠ f : [a, b] → R ❧➔ ❤➔♠ sỗ tr [a, b] F : [0, 1] → R ①→❝ ✤à♥❤ ❜ð✐ F (t) = b × a (s + 1)(b − a) 1+t f a+ 1−t x +f 1+t b+ 1−t x dx õ (i) F sỗ tr [0, 1]✳ (ii) ❍➔♠ F ✤ì♥ ✤✐➺✉ t➠♥❣ tr➯♥ [0, 1]✳ ❈❤ù♥❣ ♠✐♥❤✳ (i) ❱ỵ✐ ♠å✐ α, β ≥ 0✱ α + β = ✈➔ t1 , t2 ∈ [0, 1] t❛ ❝â b 1 + (αt1 + βt2 ) − (αt1 + βt2 ) f a+ x dx b−a a 2 b + (αt1 + βt2 ) 1 + (αt1 − βt2 ) + f b+ dx b−a a 2 b (1 + t1 )a + (1 − t1 )x (1 + t2 )a + (1 − t2 )x) f α = +β x dx b−a a 2 b (1 + t1 )b + (1 − t1 )x (1 + t2 )b + (1 − t2 )x) f α +β dx + b−a a 2 b αs (1 − t1 ) (1 + t1 ) (1 − t1 ) (1 + t1 ) ≤ a+ x +f b+ x dx f b−a a 2 2 b βs (1 + t2 ) (1 − t2 ) (1 + t2 ) (1 − t2 ) + f a+ x +f b+ x dx b−a a 2 2 F (αt1 + βt2 ) = = αs F (t1 ) + β s F (t2 ) ✸✷ (ii) ●✐↔ sû ≤ t1 ≤ t2 ≤ 1✱ a ≤ x ≤ b✳ ❚ø b (1 + t1 ) (1 − t1 ) b+ x dx 2 b (1 + t1 ) (1 − t1 ) = f b+ (b + a − x) dx 2 a f a t❛ s✉② r❛ b (1 + t1 ) (1 − t1 ) a+ x 2 a (1 + t1 ) (1 − t1 ) +f b+ (b + a − x) 2 F (t1 ) = b−a f dx ❱➔ tø + t2 − t2 (1 + t1 ) (1 − t1 ) a+ x≤ a+ x 2 2 (1 − t1 ) (1 + t1 ) b+ (b + a − x) ≤ 2 (1 + t2 ) (1 − t2 ) ≤ b+ (b + a − x) 2 s✉② r❛ + t1 − t1 + t1 − t1 a+ x + b+ (b + a − x) 2 2 + t2 + t2 − t2 − t2 = a+ x + b+ (b + a − x) 2 2 ❱➻ f ❧➔ sỗ tr [a, b] sỷ t❛ ❝â b (1 + t2 ) (1 − t2 ) a+ x 2 a (1 + t2 ) (1 − t2 ) +f b+ (b + a − x) 2 b (1 + t2 ) (1 − t2 ) ≤ f a+ x b−a a 2 (1 + t2 ) (1 − t2 ) +f b+ x dx 2 F (t1 ) ≤ b−a = F (t2 ) f dx ✸✸ ✷✳✸ ❇➜t tự rtr sỗ t tự ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ❘➜t ♥❤✐➲✉ ❜➜t ✤➥♥❣ t❤ù❝ q✉❛♥ trå♥❣ ✤÷đ❝ t❤✐➳t tứ ợ ỗ ởt tr ỳ t ✤➥♥❣ t❤ù❝ ♥ê✐ t✐➳♥❣ ♥❤➜t ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕ ❍❛❞❛♠❛r❞ ✭❝á♥ ❣å✐ ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ❍❛❞❛♠❛r❞✮ ✭✶✳✶✮✳ ✣➸ t✐➺♥ t❤❡♦ ❞ã✐✱ t❛ ✈✐➳t ❧↕✐ ❜➜t ✤➥♥❣ t❤ù❝ ♥➔②✿ a+b f ≤ b−a b f (x)dx ≤ a f (a) + f (b) ✭✷✳✶✼✮ ❑➳t q rở sỗ ữ s sỷ ❤➔♠ f : [0, +∞) → [0, +∞) ❧➔ ♠ët sỗ t tự ợ s (0, 1)✳ ●✐↔ sû a, b ∈ [0 + ∞)✱ a < b✳ ❑❤✐ ✤â ♥➳✉ f ∈ L1[a, b] t❤➻ t tự s tọ ỵ ✭①❡♠ ❬✻❪✮ 2s−1 f ✷✳✸✳✷ b a+b ≤ b−a f (x)dx ≤ a f (a) + f (b) s+1 ởt số t tự ợ ữủ t❤✐➳t ❧➟♣ tø ❜➜t ✤➥♥❣ t❤ù❝ ❍❡r♠✐t❡✕❍❛❞❛♠❛r❞ ❙❛✉ ✤➙② ❧➔ ởt số t tự ợ ữủ tt tứ t tự rtr rữợ t t õ s❛✉✳ ❈❤♦ f : C ⊂ R → R ❧➔ ❤➔♠ ❦❤↔ ✈✐ tr➯♥ C o✱ ♠✐➲♥ tr♦♥❣ ❝õ❛ C ✱ a, b ∈ C ✈ỵ✐ a < b✳ ◆➳✉ f ∈ L1[a, b] t❤➻ t❛ ❝â ✤➥♥❣ t❤ù❝✿ ❇ê ✤➲ ✷✳✸✳✷ ✭①❡♠ ❬✻❪✮ b−a b a+b (b − a)2 = 16 f (x)dx − f a t2 f (t − 1)2 f + a+b + (1 − t)a dt a+b tb + (1 − t) dt t ✭✷✳✶✾✮ ✸✹ ❈❤ù♥❣ ♠✐♥❤✳ ❙û ❞ö♥❣ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ t❛ ❝â a+b + (1 − t)a dt 1 a+b a+b tf t = t2 f t + (1 − t)a + (1 − t)a dt − b−a b−a 2 a+b a+b = f − t f t + (1 − t)a b−a b−a b−a a+b f t + (1 − t)a dt − b−a a+b a+b f − f = b−a (b − a)2 a+b t + (1 − t)a ✭✷✳✷✵✮ + (b − a) I1 = t2 f t ❙û ❞ö♥❣ ♣❤➨♣ ✤ê✐ ❜✐➳♥ x = t a+b + (1 − t)a ✈ỵ✐ t ∈ [0, 1] ✈➔ ♥❤➙♥ ❝↔ ❤❛✐ (b − a)2 ✈➳ ❝õ❛ ✭✷✳✷✵✮ ✈ỵ✐ t❛ ♥❤➟♥ ✤÷đ❝ 16 (b − a)2 16 t2 f t b−a = f a+b + (1 − t)a dt a+b a+b − f − 2 b−a a+b f (x)dx ✭✷✳✷✶✮ f (x)dx ✭✷✳✷✷✮ a ❚÷ì♥❣ tü t❛ t➼♥❤ ✤÷đ❝ (b − a)2 a+b (t − 1)2 f tb + (1 − t) dt 16 a+b b−a a+b =− f − f + 2 b−a b a+b ❑➳t ❤ñ♣ ✭✷✳✷✶✮ ✈➔ ✭✷✳✷✷✮ t❛ ♥❤➟♥ ✤÷đ❝ ✭✷✳✶✾✮✳ ●✐↔ sû f : C ⊂ [0, +∞) → R ❧➔ ❤➔♠ ❦❤↔ ✈✐ tr➯♥ ∈ L1 [a, b]✱ ð ✤➙② a, b ∈ C ✈ỵ✐ a < b |f | ỵ Co s f sỗ tr [a, b] ✈ỵ✐ s ∈ (0, 1] t❤➻ ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ✤➙② t❤ä❛ ♠➣♥ b a+b f − f (x)dx b−a a (b − a)2 |f (a)| + (s + 1)(s + 2) f ≤ 8(s + 1)(s + 2)(s + 3) a+b + |f (b)| ✭✷✳✷✸✮ [1 + (s + 2)21−s ](b − a)2 |f (a)| + |f (b)| ≤ 8(s + 1)(s + 2)(s + 3) ✭✷✳✷✹✮ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø ❇ê ✤➲ ✷✳✸✳✷ t❛ ❝â b a+b f − f (x)dx b−a a a+b (b − a)2 + (1 − t)a dt ≤ t2 f t 16 a+b + (t − 1)2 f tb + (1 − t) dt a+b (b − a)2 s t t f ( ) + (1 − t)s |f (a)| dt ≤ 16 2 (b − a) a+b (t − 1)2 ts |f (b)| + (t − 1)s f ( + ) dt 16 (b − a)2 a+b ) + |f (a)| = f ( 16 s+3 (s1 + 1)(s2 + 2) + (s3 + 3) (b − a)2 a+b + |f (b)| + f ( ) 16 (s1 + 1)(s2 + 2) + (s3 + 3) s+3 (b − a)2 a+b = |f (a)| + (s + 1)(s + 2) f + |f (b)| , 8(s + 1)(s + 2)(s + 3) ✭✷✳✷✺✮ ð ✤➙② t❛ sû ❞ö♥❣ ❝→❝ ❦➳t q✉↔ 1 t2 (1 − t)s dt = (s1 + 1)(s2 + 2) + (s3 + 3) 1 s+2 t dt = (t − 1)s+2 dt = (s3 + 3) 0 (t − 1)2 ts dt = ◆❤÷ ✈➟② t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✷✸✮✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✷✹✮✱ tø ❤➔♠ |f | sỗ tr [a, b] ợ s (0, 1)✱ tø ✭✷✳✶✽✮ ✸✻ t❛ ❝â 2s−1 f a+b f (a) + f (b) ≤ 2 ✭✷✳✷✻✮ ❑➳t ❤đ♣ ✭✷✳✷✺✮ ✈➔ ✭✷✳✷✻✮ t❛ ♥❤➟♥ ✤÷đ❝ b a+b f (x)dx f − b−a a (b − a)2 a+b |f (a)| + (s + 1)(s + 2) f + |f (b)| ≤ 8(s + 1)(s + 2)(s + 3) f (a) + f (b) (b − a)2 ≤ |f (a)| + (s + 1)(s + 2)21−s + |f (b)| 8(s + 1)(s + 2)(s + 3) s+1 [1 + (s + 2)21−s ](b − a)2 = |f (a)| + |f (b)| , 8(s + 1)(s + 2)(s + 3) ✤➙② ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✷✹✮✳ ❍➺ q✉↔ r ỵ t s = t❤➻ b a+b f (x)dx f − b−a a (b − a)2 a+b ≤ |f (a)| + f 192 2 (b − a) |f (a)| + |f (b)| ≤ 48 + |f (b)| ✭✷✳✷✼✮ ●✐↔ sû f : C ⊂ [0, +∞) → R ❧➔ ❤➔♠ ❦❤↔ ✈✐ tr➯♥ C o s❛♦ ❝❤♦ f ∈ L1 [a, b]✱ ð ✤➙② a, b ∈ C ✈ỵ✐ a < b✳ ◆➳✉ ❤➔♠ |f |q 1 sỗ tr [a, b] ợ s ∈ (0, 1] ✈➔ q > ✈ỵ✐ + = t❤➻ ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ p q ✤➙② t❤ä❛ ỵ a+b f b−a (b − a)2 ≤ 16 × 2p + b f (x)dx a p |f (a)|q + f s+1 a+b q q q + f a+b q + |f (a)|q ✭✷✳✷✽✮ ✸✼ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû q > 1✳ ❚ø ❇ê ✤➲ ✷✳✸✳✷ ✈➔ sû ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❍☎♦❞❡r t❛ ❝â b a+b f (x)dx − b−a a (b − a)2 a+b ≤ + (1 − t)a dt t2 f t 16 a+b dt + (t − 1)2 f tb + (1 − t) f (b − a)2 ≤ 16 1 p t2p dt a+b + (1 − t)a t f 0 (b − a)2 + 16 (t − 1)2p dt 1 p dt a+b tb + (1 − t) f q q q q dt ❱➻ |f |q sỗ a+b + (1 t)a f t f tb + (1 − t) dt ≤ s+1 |f (a)|q + f dt ≤ s+1 f q a+b q ✈➔ a+b q a+b q + |f (a)|q ❙û ❞ö♥❣ t2p dt = 2p + ✈➔ 1 (t − 1)2p dt = (1 − t)2p dt = 0 2p + t❛ ♥❤➟♥ ✤÷đ❝ a+b f − b−a (b − a)2 ≤ 16 × 2p + b f (x)dx a p |f (a)|q + f s+1 a+b q q q + f a+b q + |f (a)|q ✸✽ ●✐↔ sû f : C ⊂ [0, +∞) → R ❧➔ ❤➔♠ ❦❤↔ ✈✐ tr➯♥ C o s❛♦ ❝❤♦ f ∈ L[a, b]✱ ð ✤➙② a, b ∈ C ✈ỵ✐ a < b✳ ◆➳✉ ❤➔♠ |f |q ✱ q ≥ 1✱ ❧➔ sỗ tr [a, b] ợ s (0, 1] t t tự s tọ ỵ ✷✳✸✳✻ ✭①❡♠ ❬✻❪✮ b a+b f − b−a (b − a)2 ≤ 16 p a+b |f (a)|q + f (s + 1)(s + 2)(s + 3) s+3 q a+b |f (b)|q + (s + 1)(s + 2)(s + 3) f s+3 + f (x)dx a q q ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû q ≥ 1✳ ❚ø ❇ê ✤➲ ✷✳✸✳✷ ✈➔ sû ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❣✐→ trà tr✉♥❣ ❜➻♥❤ t❛ ♥❤➟♥ ✤÷đ❝ b a+b f − b−a f (x)dx a (b − a)2 p ≤ |f (a)|q + f 16 (s + 1)(s + 2)(s + 3) s+3 a+b q + f + |f (a)|q s+3 (s + 1)(s + 2)(s + 3) (b − a)2 ≤ 16 p 1 t dt t f (b − a)2 + 16 a+b t + (1 − t)a 2 p 1 2 (t − 1) f (t − 2) dt 0 q dt a+b q q q a+b tb + (1 − t) q dt ❱➻ |f |q sỗ t f a+b t + (1 − t)a q a+b q dt ≤ |f (a)|q (s + 1)(s + 2)(s + 3) a+b q + f s+3 ✈➔ (t − 1)2 f tb + (1 − t) dt ≤ f s+3 q a+b q 2 + |f (a)|q (s + 1)(s + 2)(s + 3) q ✸✾ ❉♦ ✤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 ❝ị♥❣ ♠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✳ ... (a) + f (b) ≤ b−a b−a p+q ✭✶✳✷✼✮ ✭✶✳✷✽✮ ữỡ ỗ s rở ự ữỡ tr J ỗ ởt số t t J ỗ tr sỗ t tự rtr sỗ ự ởt số tr tr✉♥❣ ❜➻♥❤ ✤➦❝ ❜✐➺t✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ tê♥❣ ❤ñ♣ tø ❝→❝ t➔✐... tù❝ ❧➔ u, Qx u ≥ i, j = 1, , n ✈ỵ✐ ♠å✐ u Rn ự số f ỗ tr C ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ✈ỵ✐ ♠é✐ a ∈ C ✈➔ u ∈ Rn t❤➻ ❤➔♠ sè ϕa,u (t) = f (a + tu) ỗ tr số tỹ ♠ð ∂f ✳ ❉♦ ✤â✱ {t | a + tu ∈ X}✳ ❱ỵ✐ x = a +... ❤❛✐✱ ❦❤↔ ✈✐ ❝➜♣ n r J ỗ J ỗ s rở ởt số t t ũ ♠ët ù♥❣ ❞ö♥❣ ❝❤♦ ❜➔✐ t♦→♥ ❝ü❝ trà✳ ✭✸✮ ❚r➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠✱ ✈➼ ❞ö ✈➔ ♠ët sè t➼♥❤ ❝❤➜t sỗ r ự ởt số t tự ợ rtr sỗ r ❜➔② ♠ët sè ù♥❣ ❞ö♥❣