Desigualdades entre as médias geométrica e aritmética e de CauchySchwarz

❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ ❈❊❆❘➪ ❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙

❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆

P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲

▲❯■❩ ❊❉❯❆❘❉❖ ▲❆◆❉■▼ ❙■▲❱❆

❉❊❙■●❯❆▲❉❆❉❊❙ ❊◆❚❘❊ ❆❙ ▼➱❉■❆❙ ●❊❖▼➱❚❘■❈❆ ❊ ❆❘■❚▼➱❚■❈❆ ❊ ❉❊ ❈❆❯❈❍❨✲❙❈❍❲❆❘❩

▲❯■❩ ❊❉❯❆❘❉❖ ▲❆◆❉■▼ ❙■▲❱❆

❉❊❙■●❯❆▲❉❆❉❊❙ ❊◆❚❘❊ ❆❙ ▼➱❉■❆❙ ●❊❖▼➱❚❘■❈❆ ❊ ❆❘■❚▼➱❚■❈❆ ❊ ❉❊ ❈❆❯❈❍❨✲❙❈❍❲❆❘❩

❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠á✲ t✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧✱ ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡✲ ♠át✐❝❛✳ ➪r❡❛ ❞❡ ❝♦♥❝❡♥tr❛çã♦✿ ❊♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛

❖r✐❡♥t❛❞♦r✿

Pr♦❢✳ ❉r✳ ▼❛r❝♦s ❋❡rr❡✐r❛ ❞❡ ▼❡❧♦✳

▲❯■❩ ❊❉❯❆❘❉❖ ▲❆◆❉■▼ ❙■▲❱❆

❉❊❙■●❯❆▲❉❆❉❊❙ ❊◆❚❘❊ ❆❙ ▼➱❉■❆❙ ●❊❖▼➱❚❘■❈❆ ❊ ❆❘■❚▼➱❚■❈❆ ❊ ❉❊ ❈❆❯❈❍❨✲❙❈❍❲❆❘❩

❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠á✲ t✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧✱ ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡✲ ♠át✐❝❛✳ ➪r❡❛ ❞❡ ❝♦♥❝❡♥tr❛çã♦✿ ❊♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛

❆♣r♦✈❛❞❛ ❡♠ ✷✸✴✵✸✴✷✵✶✸

❇❆◆❈❆ ❊❳❆▼■◆❆❉❖❘❆

Pr♦❢✳ ❉r✳ ▼❛r❝♦s ❋❡rr❡✐r❛ ❞❡ ▼❡❧♦ ✭❖r✐❡♥t❛❞♦r✮ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá ✭❯❋❈✮

Pr♦❢✳ ❉r✳ ▼❛r❝❡❧♦ ❋❡rr❡✐r❛ ❞❡ ▼❡❧♦ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá ✭❯❋❈✮

❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ❛ ❉❡✉s✱ ♣♦r t❡r ♠❡ ❞❛❞♦ ❢♦rç❛✱ s❛ú❞❡✱ ❝♦r❛❣❡♠ ❡ ❞❡t❡r♠✐♥❛çã♦ ❞✐❛♥t❡ ❞❡ t❛♥t❛s ❞✐✜❝✉❧❞❛❞❡s q✉❡ ❛ ✈✐❞❛ ♥♦s ♦❢❡r❡❝❡✳

❆♦s ♠❡✉s ♣❛✐s ♣❡❧♦ ❛♠♦r✱ ❡①❡♠♣❧♦ ❡ ✐♥❝❡♥t✐✈♦ ❞❛❞♦✱ ❞❡s❞❡ ❝❡❞♦✱ ♣❛r❛ q✉❡ ♠❡ ❞❡❞✐❝❛ss❡ ❛♦s ❡st✉❞♦s✳

❆♦s ♠❡✉s ✜❧❤♦s q✉❡ ♠❡ ♠♦t✐✈❛♠ ❛ ♠❡❧❤♦r❛r ❝♦♠♦ ♣❡ss♦❛ ❡ ♣r♦✜ss✐♦♥❛❧✳

❆♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✭P❘❖❋▼❆❚✮ ♣♦r ♣♦ss✐❜✐✲ ❧✐t❛r ❛ ♠✐♥❤❛ ♣❛rt✐❝✐♣❛çã♦ ♥✉♠❛ ♣ós✲❣r❛❞✉❛çã♦ str✐❝t♦✲s❡♥s✉✱ ❡st❛♥❞♦ ❡♠ ♣❧❡♥♦ ❡①❡r❝í❝✐♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳

❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ♣r♦❢❡ss♦r ▼❛r❝♦s ▼❡❧♦✱ ♣❡❧❛ s✉❛ ♦r✐❡♥t❛çã♦ ❡ ❛♣♦✐♦ à r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

❆♦s ♣r♦❢❡ss♦r❡s ❞❛ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá q✉❡ ❛❝r❡❞✐t❛r❛♠ ❡ ♣❛rt✐❝✐♣❛r❛♠ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ✭P❘❖❋▼❆❚✮✳ ❆♦ ■♥st✐t✉t♦ ❋❡❞❡r❛❧ ❞❡ ❊❞✉❝❛çã♦✱ ❈✐ê♥❝✐❛ ❡ ❚❡❝♥♦❧♦❣✐❛ ❞♦ ❈❡❛rá q✉❡ ♠❡ ❛❥✉❞♦✉ ❝✉s✲ t❡❛♥❞♦ ♠✉✐t❛s ❞❛s ♠✐♥❤❛s ♣❛ss❛❣❡♥s ❡ ❝♦♥❝❡♥tr♦✉ ❡♠ ❛♣❡♥❛s três ❞✐❛s ❞❛ s❡♠❛♥❛ ❛ ♠✐♥❤❛ ❝❛r❣❛ ❤♦rár✐❛ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳

➚ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡ P❡ss♦❛❧ ❞❡ ❊♥s✐♥♦ ❙✉♣❡r✐♦r ✭❈❆P❊❙✮ q✉❡ ♠❡ ❝♦♥❝❡❞❡✉ ✉♠❛ ❜♦❧s❛ ❞❡ ❡st✉❞♦✳

❘❡s✉♠♦

❊st❡ tr❛❜❛❧❤♦ tr❛t❛ ❞❡ ❞✉❛s ❞❛s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞❡s✐❣✉❛❧❞❛❞❡s ❞❛ ▼❛t❡♠át✐❝❛✿ ❛ ❞❡s✐✲ ❣✉❛❧❞❛❞❡ ❡♥tr❡ ❛s ♠é❞✐❛s ❣❡♦♠étr✐❝❛ ❡ ❛r✐t♠ét✐❝❛ ❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✳ ❆♣r❡s❡♥t❛♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ ❞✐✈❡rs❛s ❞❡♠♦♥str❛çõ❡s ♣❛r❛ ♦ ❝❛s♦ n = 2✱ ❛♣ós ❛s q✉❛✐s s❡✲

❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✼

✶✳✶ ❏✉st✐✜❝❛t✐✈❛ ❡ ♦❜❥❡t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷ ▼❡t♦❞♦❧♦❣✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸ ❆♣r❡s❡♥t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✷ ❉❡s✐❣✉❛❧❞❛❞❡ ▼● ✲ ▼❆ ✾

✷✳✶ ❉❡s✐❣✉❛❧❞❛❞❡ ▼● ✲ ▼❆✿ ❝❛s♦ n= 2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾

✷✳✷ ❉❡s✐❣✉❛❧❞❛❞❡ ▼● ✲ ▼❆✿ ❝❛s♦ ❣❡r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✸ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ ✶✾

✸✳✶ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✿ ❝❛s♦n = 2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

✸✳✷ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✿ ❝❛s♦ ❣❡r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✹ ❆♣❧✐❝❛çõ❡s ✷✼

✺ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✹✺

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

✶✳✶ ❏✉st✐✜❝❛t✐✈❛ ❡ ♦❜❥❡t✐✈♦s

❆♣ós ❛❧❣✉♥s ❛♥♦s ❞❡❞✐❝❛❞♦s ❛♦ ❡♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛ q✉❡r ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✱ q✉❡r ♥♦ ❡♥✲ s✐♥♦ s✉♣❡r✐♦r✱ ♣✉❞❡ ❝♦♥st❛t❛r ♦ q✉❛♥t♦ ♦s ❡st✉❞❛♥t❡s ❛♣r❡s❡♥t❛♠ ❞✐✜❝✉❧❞❛❞❡s ❡♠ tr❛❜❛❧❤❛r ❝♦♠ ❞❡s✐❣✉❛❧❞❛❞❡s✱ ❧❡✈❛♥❞♦✲♠❡ ❛ ❛❝r❡❞✐t❛r q✉❡ t❛❧ ❛ss✉♥t♦ é ♣♦✉❝♦ ❛❜♦r❞❛❞♦ ❞✉r❛♥t❡ ♦ ❡♥s✐♥♦ ♠é❞✐♦✳

❖ ❧✐✈r♦ ❊①❛♠❡ ❞❡ ❚❡①t♦s✿ ❆♥á❧✐s❡ ❞❡ ▲✐✈r♦s ❞❡ ▼❛t❡♠át✐❝❛ ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦ ♣✉❜❧✐✲ ❝❛❞♦ ♣❡❧❛ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ ✭❙❇▼✮ ❡♠ ✷✵✵✶ r❡❧❛t❛ ❞✐✈❡rs♦s ♣r♦❜❧❡♠❛s ❞♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ✉t✐❧✐③❛❞♦s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ ♥♦ ❇r❛s✐❧✳ ❆✐♥❞❛ ❤♦❥❡✱ ❛ ♠❛✐♦r ♣❛rt❡ ❞❡s✲ s❡s ❧✐✈r♦s ❧✐♠✐t❛♠✲s❡ ❛ ❛❜♦r❞❛r❡♠ ❛❧❣✉♥s t✐♣♦s ❞❡ ✐♥❡q✉❛çõ❡s✱ ❛♣r❡s❡♥t❛♥❞♦ ♠ét♦❞♦s ❞❡ r❡s♦❧✉çã♦ r❡♣❡t✐t✐✈♦s ❡ q✉❡ ♥ã♦ r❡q✉❡r❡♠ q✉❛❧q✉❡r ❡♥❣❡♥❤♦s✐❞❛❞❡✳ ❆❧✐❛❞♦ ❛ ✐ss♦ t❡♠♦s ❛ ♠á ❢♦r♠❛çã♦ ❞❡ ♣r♦❢❡ss♦r❡s✱ ♠✉✐t♦s ❞♦s q✉❛✐s t❡♠ ♥♦ ❧✐✈r♦ ❞✐❞át✐❝♦ ❛❞♦t❛❞♦ ♥❛s ❡s❝♦❧❛s ❛ s✉❛ ú♥✐❝❛ ❢♦♥t❡ ❞❡ ❡st✉❞♦✳

P♦r ♦✉tr♦ ❧❛❞♦ ❤á ♠✉✐t♦ t❡♠♣♦ ❛s ❞❡s✐❣✉❛❧❞❛❞❡s sã♦ ❜❛st❛♥t❡ tr❛❜❛❧❤❛❞❛s ❝♦♠ ♦s ❡s✲ t✉❞❛♥t❡s ❞❡ ♦❧✐♠♣í❛❞❛s ❡ ♥ã♦ sã♦ r❛r♦s ♦s ♣r♦❜❧❡♠❛s q✉❡ ❡♥✈♦❧✈❡♠ ❞❡s✐❣✉❛❧❞❛❞❡s ♥❡ss❛s ❝♦♠♣❡t✐çõ❡s✳ Pr♦❜❧❡♠❛s ❡ss❡s q✉❡ ♥ã♦ r❡q✉❡r❡♠ ❝♦♥❤❡❝✐♠❡♥t♦s ❛✈❛♥ç❛❞♦s ❞❡ ▼❛t❡♠á✲ t✐❝❛✱ ❛♣❡♥❛s ♦ ❡st✉❞♦ ❞❡ ❛❧❣✉♠❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❡ ♠✉✐t❛ ❝r✐❛t✐✈✐❞❛❞❡ ♥❛s s✉❛s ✉t✐❧✐③❛çõ❡s✳ ❊ss❛ s✐t✉❛çã♦ ♥♦s ♠♦t✐✈♦✉ ❛ ❡s❝r❡✈❡r ❡st❡ tr❛❜❛❧❤♦ s♦❜r❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❛s ♠é✲ ❞✐❛s ❛r✐t♠ét✐❝❛ ❡ ❣❡♦♠étr✐❝❛ ❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✱ ❝♦♥s✐❞❡r❛❞❛s ❞✉❛s ❞❛s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞❡s✐❣✉❛❧❞❛❞❡s✱ q✉❡r ♣❡❧❛s s✉❛s ❢r❡q✉❡♥t❡s ✉t✐❧✐③❛çõ❡s ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s✱ q✉❡r ♣❡❧♦ ♥ú♠❡r♦ ❞❡ ❞❡♠♦♥str❛çõ❡s ❡①✐st❡♥t❡s ♣❛r❛ ❡❧❛s✳

Pr❡t❡♥❞❡♠♦s ❝♦♠ ❡st❡ tr❛❜❛❧❤♦

✭❛✮ ❛♣r❡s❡♥t❛r ❞✐✈❡rs❛s ❞❡♠♦♥str❛çõ❡s ❞❡ss❛s ❞✉❛s ❞❡s✐❣✉❛❧❞❛❞❡s❀

✭❜✮ ♠♦str❛r q✉❡ ❛s ❞❡s✐❣✉❛❧❞❛❞❡s ♣♦❞❡♠ s❡r ❡♥s✐♥❛❞❛s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ❥á q✉❡ ♥ã♦ r❡q✉❡r❡♠ ❝♦♥❤❡❝✐♠❡♥t♦s ♠❛t❡♠át✐❝♦s ♠❛✐s ❛✈❛♥ç❛❞♦s❀

❈❆P❮❚❯▲❖ ✶✳ ■◆❚❘❖❉❯➬➹❖ ✽

✭❞✮ ❞✐✈✉❧❣❛r ❞✐✈❡rs❛s ❛♣❧✐❝❛çõ❡s ❞❡ss❛s ❞❡s✐❣✉❛❧❞❛❞❡s❀ ✭❡✮ ❡st✐♠✉❧❛r ♦ ❡♥s✐♥♦ ❞❡ ❞❡s✐❣✉❛❧❞❛❞❡s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✳

✶✳✷ ▼❡t♦❞♦❧♦❣✐❛

❊st❡ tr❛❜❛❧❤♦ r❡s✉❧t♦✉ ❞❡ ♣❡sq✉✐s❛ ❡♠ ❧✐✈r♦s ❡ ❛rt✐❣♦s s♦❜r❡ ❞❡s✐❣✉❛❧❞❛❞❡s✱ ❛❧❣✉♥s ❞♦s q✉❛✐s ✈♦❧t❛❞♦s ❡s♣❡❝í✜❝❛♠❡♥t❡ ♣❛r❛ ♦❧✐♠♣í❛❞❛s ❞❡ ▼❛t❡♠át✐❝❛✳ ❉✉r❛♥t❡ ❛ ❛♥á❧✐s❡ ❞❡ ❝❛❞❛ ♠❛t❡r✐❛❧✱ ✐♥t❡r❡ss❛✈❛ ❛ ❜✉s❝❛ ♣♦r ❞❡♠♦♥str❛çõ❡s ❞✐st✐♥t❛s ♣❛r❛ ❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❡s✲ ❝♦❧❤✐❞❛s✱ ❜❡♠ ❝♦♠♦ ❞✐✈❡rs❛s ❛♣❧✐❝❛çõ❡s ❞❡❧❛s✳

❆ ♦❜t❡♥çã♦ ❞❡ss❛s ❢♦♥t❡s ♥ã♦ ❢♦✐ rá♣✐❞❛ ❡ s❡ ❞❡✉ ❞❡ ❞✐✈❡rs❛s ❢♦r♠❛s✿ ❡♥❝♦♥tr❛❞♦s ❡♠ ❢♦r♠❛t♦ ❞✐❣✐t❛❧ ♥❛ ✐♥t❡r♥❡t✱ ❛❝❡ss❛❞♦s ❡♠ ❜✐❜❧✐♦t❡❝❛s✱ ❛❞q✉✐r✐❞♦s ❡♠ ❧✐✈r❛r✐❛s✱ ♦❜t✐❞♦s ❝♦♠ ♣r♦❢❡ss♦r❡s✳ ❉✉r❛♥t❡ ❛ tr✐❛❣❡♠✱ ♥♦t❛♠♦s ❛ r❡♣❡t✐çã♦ ❞❡ ❞✐✈❡rs❛s ✐♥❢♦r♠❛çõ❡s ❢❛③❡♥❞♦ ❝♦♠ q✉❡ ❞❡s❝❛rt❛ss❡♠♦s ❛❧❣✉♥s ♠❛t❡r✐❛✐s✳

✶✳✸ ❆♣r❡s❡♥t❛çã♦

❆❧é♠ ❞❡st❡ ❝❛♣ít✉❧♦ ✐♥tr♦❞✉tór✐♦ ❤á ♦✉tr♦s q✉❛tr♦ q✉❡ ❝♦♠♣õ❡♠ ♥♦ss♦ t❡①t♦✳

◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ é ❛❜♦r❞❛❞❛ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❛s ♠é❞✐❛s ❣❡♦♠étr✐❝❛ ❡ ❛r✐t♠é✲ t✐❝❛✱ ♦♥❞❡ sã♦ ❛♣r❡s❡♥t❛❞❛s ✐♥✐❝✐❛❧♠❡♥t❡ ❝✐♥❝♦ ❞❡♠♦♥str❛çõ❡s ♣❛r❛ ♦ ❝❛s♦ n = 2 ❡ s❡✐s

❞❡♠♦♥str❛çõ❡s ♣❛r❛ ♦ ❝❛s♦ ❣❡r❛❧✳

❖ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ tr❛t❛ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ ❛tr❛✈és ❞❡ três ❞❡♠♦♥s✲ tr❛çõ❡s ❞♦ ❝❛s♦n = 2 ❡ ❝✐♥❝♦ ❞❡♠♦♥str❛çõ❡s ❞♦ ❝❛s♦ ❣❡r❛❧✳

❆s ❛♣❧✐❝❛çõ❡s ❞❡st❛s ❞✉❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❡♥❝♦♥tr❛♠✲s❡ ♥♦ q✉❛rt♦ ❝❛♣ít✉❧♦✱ s❡♥❞♦ s❡♠✲ ♣r❡ q✉❡ ♣♦ssí✈❡❧ ♠♦str❛❞♦ ♠❛✐s ❞❡ ✉♠❛ ❢♦r♠❛ ❞❡ ✉t✐❧✐③❛çã♦ ❞❛s ❞❡s✐❣✉❛❧❞❛❞❡s ♥✉♠❛ ❝❡rt❛ ❛♣❧✐❝❛çã♦❀ ❛❝r❡❞✐t❛♠♦s t❛♠❜é♠ q✉❡ ❛s ❛♣❧✐❝❛çõ❡s ❡stã♦ ❞✐s♣♦st❛s ❡♠ ♦r❞❡♠ ❞❡ ❞✐✜❝✉❧❞❛❞❡ s❡♠ q✉❡ ❤❛❥❛ q✉❛❧q✉❡r s❡♣❛r❛çã♦ ♦✉ ✐♥❞✐❝❛çã♦ ❞❡ q✉❛❧ ❛ ♠❡❧❤♦r ❞❡s✐❣✉❛❧❞❛❞❡ ❛ ✉s❛r✳

❋✐♥❛❧♠❡♥t❡✱ ❛♣r❡s❡♥t❛♠♦s ♥♦ss❛s ❝♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ♥♦ ú❧t✐♠♦ ❝❛♣ít✉❧♦ ❞❡st❡ tr❛❜❛✲ ❧❤♦✳

❈❛♣ít✉❧♦ ✷

❉❡s✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❛s ♠é❞✐❛s

❣❡♦♠étr✐❝❛ ❡ ❛r✐t♠ét✐❝❛

❉❛❞❛ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❝♦♠♦ ♠é❞✐❛ ❞❡❧❡s ✉♠ ♥ú♠❡r♦ r❡❛❧ q✉❡✱ ❛♦ s✉❜st✐t✉✐r ❝❛❞❛ ✉♠ ❞♦s t❡r♠♦s ❞❛ s❡q✉ê♥❝✐❛✱ ♠❛♥t❡♥❤❛ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ♣r♦♣r✐❡❞❛❞❡✳ ❆ s❡❣✉✐r ❞❡✜♥✐♠♦s ❞✉❛s ♠é❞✐❛s✳

❉❡✜♥✐çã♦ ✷✳✶✳ ❉❛❞♦sn > 1 ♥ú♠❡r♦s r❡❛✐s x1, x2, . . . , xn ✱ ❛ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ ❞❡❧❡s é ♦

♥ú♠❡r♦ r❡❛❧ A(x1, . . . , xn) =

x1+x2+. . .+xn

n ✳

❉❡✜♥✐çã♦ ✷✳✷✳ ❉❛❞♦sn > 1 ♥ú♠❡r♦s r❡❛✐s ♣♦s✐t✐✈♦s x1, x2, . . . , xn ✱ ❛ ♠é❞✐❛ ❣❡♦♠étr✐❝❛

❞❡❧❡s é ♦ ♥ú♠❡r♦ r❡❛❧G(x1, . . . , xn) = √nx1x2. . . xn✳

❆ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ ❝♦♥s❡r✈❛ ❛ s♦♠❛ ❡ ❛ ♠é❞✐❛ ❣❡♦♠étr✐❝❛ ♦ ♣r♦❞✉t♦ ❞♦s ♥ú♠❡r♦s✳ ❆❧é♠ ❞✐ss♦✱ ❡ss❛s ♠é❞✐❛s ❡stã♦ r❡❧❛❝✐♦♥❛❞❛s ❛tr❛✈és ❞❡ ✉♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ q✉❡ ♣♦ss✉✐ ❞✐✲ ✈❡rs❛s ❛♣❧✐❝❛çõ❡s ♥❛ ▼❛t❡♠át✐❝❛✱ s❡♥❞♦ ♣♦r ✐ss♦ ❝♦♥s✐❞❡r❛❞❛ ✉♠❛ ❞❛s ♠❛✐s ✐♠♣♦rt❛♥t❡s✳ Pr♦♣♦s✐çã♦ ✷✳✶✳ P❛r❛ q✉❛✐sq✉❡rn >1 ♥ú♠❡r♦s r❡❛✐s ♣♦s✐t✐✈♦s x1, x2, . . . , xn t❡♠♦s

G(x1, x2, . . . , xn)≤A(x1, x2, . . . , xn)

♦❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱x1 =x2 =. . .=xn✳

✷✳✶ ❉❡s✐❣✉❛❧❞❛❞❡ ▼● ✲ ▼❆✿ ❝❛s♦

n

_{= 2}

■♥✐❝✐❛❧♠❡♥t❡ ♣r♦✈❛♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❛s ♠é❞✐❛s ❣❡♦♠étr✐❝❛ ❡ ❛r✐t♠ét✐❝❛ ❞❡ ❞♦✐s ♥ú♠❡r♦s r❡❛✐s ♣♦s✐t✐✈♦s✳ ❆♣r❡s❡♥t❛♠♦s ❝✐♥❝♦ ❞❡♠♦♥str❛çõ❡s✱ s❡♥❞♦ ❛s ❞✉❛s ♣r✐♠❡✐r❛s ❛❧❣é❜r✐❝❛s ❡ ❛s ❞❡♠❛✐s ❣❡♦♠étr✐❝❛s✳

❉❡♠♦♥str❛çã♦ ✶✳

❙❡♥❞♦ x1 ❡ x2 ♥ú♠❡r♦s r❡❛✐s ♣♦s✐t✐✈♦s✱ t❡♠♦s √x1 ❡ √x2 ❜❡♠ ❞❡✜♥✐❞♦s ♥♦s r❡❛✐s ❡

❈❆P❮❚❯▲❖ ✷✳ ❉❊❙■●❯❆▲❉❆❉❊ ▼● ✲ ▼❆ ✶✵

(√x2−√x1)2 ≥0 ⇐⇒x2−2√x2√x1+x1 ≥0

⇐⇒x2+x1 ≥2√x2x1

⇐⇒ x2+x1

2 ≥

√_x

2x1

⇐⇒A(x1, x2)≥G(x1, x2)

❆ ✐❣✉❛❧❞❛❞❡ ♦❝♦rr❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱

(√x2−√x1)2 = 0 ⇐⇒√x2 =√x1 ⇐⇒x2 =x1

❉❡♠♦♥str❛çã♦ ✷✳ ❙❡❥❛a= x1+x2

2 ❡d=

x2−x1

2 ✱ t❡♠♦sx1 =a−d❡x2 =a+d❡ ♣♦rt❛♥t♦x1x2 =a2−d2✳ ❖r❛

d2 _≥0 _{⇐⇒ −}d2 _≤0

⇐⇒a2₋d2 _≤a2

⇐⇒x1x2 ≤ x1+₂x2

2

⇐⇒√x1x2 ≤ x1+₂x2

❆ ✐❣✉❛❧❞❛❞❡ ♦❝♦rr❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ d= 0✱ ♦✉ s❡❥❛✱ x1 =x2

❉❡♠♦♥str❛çã♦ ✸✳

▼❛r❝❛♠♦s s♦❜r❡ ✉♠❛ r❡t❛ r ♦s s❡❣♠❡♥t♦s ❛❞❥❛❝❡♥t❡s AB = x1 ❡ BC = x2✳ ❆ s❡❣✉✐r

tr❛ç❛♠♦s ♦ ❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦AC ❡ ❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛ r ♣♦r B ❛té ✐♥st❡rs❡❝t❛r ♦ ❝ír❝✉❧♦ ❡♠ D✳

❈❆P❮❚❯▲❖ ✷✳ ❉❊❙■●❯❆▲❉❆❉❊ ▼● ✲ ▼❆ ✶✶

❖ â♥❣✉❧♦ADCb é r❡t♦ ❡ s❡♥❞♦DB =h❛ ❛❧t✉r❛ ❞♦∆ADC r❡❧❛t✐✈❛ à ❤✐♣♦t❡♥✉s❛ t❡♠♦s h=√x1x2✳

❙❡♥❞♦ O ♦ ❝❡♥tr♦ ❞♦ ❝ír❝✉❧♦✱ s❡ x1 = x2 t❡♠♦s O = B ❡ ♣♦rt❛♥t♦ BD = OD ⇐⇒

√_x

1x2 =

x1 +x2

2 ⇐⇒ G(x1, x2) = A(x1, x2)✳ ❈❛s♦ x1 6= x2✱ t❡♠♦s q✉❡ ♦ ∆OBD é

r❡tâ♥❣✉❧♦ ❡♠ B ❡ ♣♦rt❛♥t♦ t❡♠♦s BD < OD _⇐⇒ √x1x2 <

x1+x2

2 ⇐⇒ G(x1, x2) <

A(x1, x2)✳

❉❡♠♦♥str❛çã♦ ✹✳

❈♦♥str✉✐♠♦s ✉♠ q✉❛❞r❛❞♦ ABCD ❞❡ ❧❛❞♦ √x1+x2 ❡✱ s♦❜r❡ ❝❛❞❛ ❧❛❞♦ ❞❡st❡✱ ✉♠

tr✐â♥❣✉❧♦ ❞❡ ♦✉tr♦s ❧❛❞♦s √x1 ❡ √x2✱ ❝♦♥❢♦r♠❡ ❛ ✜❣✉r❛✳ ❊st❡s sã♦ ❝♦♥❣r✉❡♥t❡s ❡✱ ♣❡❧❛

r❡❝í♣r♦❝❛ ❞♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ sã♦ r❡tâ♥❣✉❧♦s✳

❋✐❣✉r❛ ✷✳✷✿ Pr♦✈❛ ♣♦r ár❡❛

❈♦♠♦ ❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦ ABCD s❡rá ♠❛✐♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛ s♦♠❛ ❞❛s ár❡❛s ❞♦s q✉❛tr♦ tr✐â♥❣✉❧♦s✱ t❡♠♦s

(√x1+x2)2 ≥4

√_x

1√x2

2 ⇐⇒ x1+x2 ≥2

√_x

1x2

⇐⇒ x1+x2

2 ≥

√_x

1x2

⇐⇒A(x1, x2)≥G(x1, x2)

❖❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦ q✉❛❞r❛❞♦ ❝❡♥tr❛❧ s❡ r❡❞✉③ ❛ ✉♠ ♣♦♥t♦ ♦ q✉❡ ♦❝♦rr❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱|√x2−√x1 |= 0✱ ♦✉ s❡❥❛✱ x1 =x2✳

❉❡♠♦♥str❛çã♦ ✺✳

❈♦♥s✐❞❡r❡ ✉♠ s✐st❡♠❛ ♦rt♦❣♦♥❛❧ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s ❡ ♦s ♣♦♥t♦s O = (0,0)✱

❈❆P❮❚❯▲❖ ✷✳ ❉❊❙■●❯❆▲❉❆❉❊ ▼● ✲ ▼❆ ✶✷

❋✐❣✉r❛ ✷✳✸✿ ❙✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s

❈♦♠♦ ❛ ár❡❛ ❞♦ r❡tâ♥❣✉❧♦ ODCE ♥ã♦ ❡①❝❡❞❡ ❛ s♦♠❛ ❞❛s ár❡❛s ❞♦s tr✐â♥❣✉❧♦s OBD ❡OAE✱ t❡♠♦s

√_x

1√x2 ≤ 1₂(√x1)2 +1₂(√x2)2 ⇐⇒√x1x2 ≤ x1+₂x2

⇐⇒G(x1, x2)≤A(x1, x2)

❖❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ár❡❛ ❞♦∆ABC ❢♦r ♥✉❧❛✱ ♦ q✉❡ ♦❝♦rr❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x1 =x2✳

✷✳✷ ❉❡s✐❣✉❛❧❞❛❞❡ ▼● ✲ ▼❆✿ ❝❛s♦ ❣❡r❛❧

◆❡ss❛ s❡çã♦ ♣r♦✈❛♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❛s ♠é❞✐❛s ❣❡♦♠étr✐❝❛ ❡ ❛r✐t♠ét✐❝❛ ❞❡ n > 1

♥ú♠❡r♦s r❡❛✐s ♣♦s✐t✐✈♦sx1, x2, . . . , xn✳

❆♥t❡s ❞❡ ✐♥✐❝✐❛r♠♦s ❛ ♣r✐♠❡✐r❛ ❞❡♠♦♥str❛çã♦✱ ♣r♦✈❛r❡♠♦s ♦ s❡❣✉✐♥t❡ ❧❡♠❛✳

▲❡♠❛ ✶✳ ❙❡ x1, x2, . . . , xn sã♦ n > 1 r❡❛✐s ♣♦s✐t✐✈♦s t❛✐s q✉❡ x1x2... xn = 1✱ ❡♥tã♦ x1+

x2+. . .+xn ≥n✱ ♦❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x1 =x2 =. . .=xn✳

❉❡♠♦♥str❛çã♦✳ ❋❛r❡♠♦s ❛ ♣r♦✈❛ ♣♦r ✐♥❞✉çã♦ s♦❜r❡ n✳

❙❡ x1x2 = 1✱ ❡♥tã♦ x1 = x2 = 1 ♦✉ x1 6= 1 ❡ x2 6= 1✳ ◆♦ ♣r✐♠❡✐r♦ ❝❛s♦✱ t❡♠♦s

x1+x2 = 2✳ ❏á ♥♦ s❡❣✉♥❞♦ ❝❛s♦✱ t❡♠♦s ✉♠ ❞❡❧❡s ♠❡♥♦r ❞♦ ✶ ❡ ♦ ♦✉tr♦ ♠❛✐♦r ❞♦ q✉❡ ✶✱

❝❛s♦ ❝♦♥trár✐♦ ♦ ♣r♦❞✉t♦ s❡r✐❛ ♠❛✐♦r ❞♦ q✉❡ ✶ ✭s❡ x1 > 1 ❡ x2 > 1✮ ♦✉ ♠❡♥♦r ❞♦ q✉❡ ✶

❈❆P❮❚❯▲❖ ✷✳ ❉❊❙■●❯❆▲❉❆❉❊ ▼● ✲ ▼❆ ✶✸

(1₋x1)(x2−1)>0 ⇐⇒ −1 +x1+x2−x1x2 >0

⇐⇒x1+x2−2>0

⇐⇒x1+x2 >2

❈♦♥❝❧✉✐♠♦s ❡♥tã♦ q✉❡ s❡ x1x2 = 1✱ ❡♥tã♦ x1 +x2 ≥ 2✱ ♦❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡

s♦♠❡♥t❡ s❡✱ x1 =x2 = 1✳

❙✉♣♦♥❤❛ ❝♦♠♦ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦ q✉❡ ♦ ❧❡♠❛ s❡❥❛ ✈á❧✐❞♦ ♣❛r❛ n = k ❝♦♠ k _≥ 2✳

▼♦str❡♠♦s ❡♥tã♦ q✉❡ ❛ ♣r♦♣♦s✐çã♦ ❝♦♥t✐♥✉❛ ✈á❧✐❞❛ ♣❛r❛n =k+ 1✳

❙❡ x1x2. . . xk+1 = 1✱ ❡♥tã♦ x1 =x2 = . . .= xk+1 = 1 ♦✉ t❡r❡♠♦s t❡r♠♦s ♠❡♥♦r❡s ❞♦

q✉❡ ✶ ❡ t❡r♠♦s ♠❛✐♦r❡s ❞♦ q✉❡ ✶✳

◆♦ ♣r✐♠❡✐r♦ ❝❛s♦✱ t❡♠♦sx1+x2+. . .+xk+1 =k+1✳ ❏á ♥♦ s❡❣✉♥❞♦ ❝❛s♦✱ ❝♦♥s✐❞❡r❡♠♦s

s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡x1 <1❡xk+1 >1✳ ❋❛③❡♥❞♦y1 =x1xk+1✱ t❡♠♦sy1x2. . . xk= 1

❡ ♣❡❧❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦y1+x2+. . .+xk≥k✳

P♦r ♦✉tr♦ ❧❛❞♦✱

x1+x2+. . .+xk+1 = (y1+x2+. . .+xk) +xk+1+x1−y1

≥k+xk+1+x1−y1

=k+ 1₋1 +xk+1+x1−y1 =k+ 1₋1 +xk+1+x1−x1xk+1 =k+ 1 +xk+1(1−x1)−(1−x1) =k+ 1 + (xk+1−1)(1−x1)

> k+ 1

▲♦❣♦ s❡x1x2. . . xn = 1✱ ❡♥tã♦x1+x2+. . .+xn≥n♣❛r❛ t♦❞♦ ✐♥t❡✐r♦n >1✱ ♦❝♦rr❡♥❞♦

❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x1 =x2 =. . .=xn = 1✳

❉❡♠♦♥str❛çã♦ ✶✳ ❈♦♥s✐❞❡r❡ g = √n_x

❈❆P❮❚❯▲❖ ✷✳ ❉❊❙■●❯❆▲❉❆❉❊ ▼● ✲ ▼❆ ✶✹

n

√_x

1x2. . . xn

g = 1 ⇐⇒

n r_x

1x2. . . xn

gn = 1

⇐⇒ n r_x

1

g x2

g . . . xn

g = 1

⇐⇒ x_g1x_g2 . . .xn g = 1 P❡❧♦ ❧❡♠❛ ✶ t❡♠♦s

x1

g x2

g . . . xn

g = 1 =⇒ x1

g + x2

g +. . .+ xn

g ≥n

⇐⇒ x1+x2 +_n. . .+xn ≥g _⇐⇒A(x1, . . . , xn)≥G(x1, . . . , xn)

❖❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x1

g = x2

g =. . . = xn

g = 1✱ ♦✉ s❡❥❛✱ x1 =x2 =

. . .=xn=g

❉❡♠♦♥str❛çã♦ ✷✳

❉✐✈✐❞✐r❡♠♦s ❛ ♥♦ss❛ ❞❡♠♦♥str❛çã♦ ❡♠ ❞✉❛s ❡t❛♣❛s✳

❊t❛♣❛ ✶✿ Pr♦✈❛♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♣❛r❛ n= 2m_{✱ ❝♦♠m ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✳}

❯t✐❧✐③❛♠♦s ✐♥❞✉çã♦ s♦❜r❡ m✱ s❡♥❞♦ q✉❡ ♣❛r❛ m = 1 ❥á ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♠❛s ♣r♦✈❛s

♥♦ ✐♥í❝✐♦ ❞♦ ❝❛♣ít✉❧♦✳ ❙✉♣♦♥❤❛ ❛❣♦r❛ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✈á❧✐❞❛ ♣❛r❛m =k ❡ ♠♦str❡♠♦s q✉❡ t❛♠❜é♠ s❡rá ✈á❧✐❞❛ ♣❛r❛m =k+ 1✳

A(x1, x2, . . . , x2k+1) =

x1+x2+. . .+x2k+1

2k+1

= x1+x2+. . .+x2k +x2k+1+x2k+2+. . .+x2k+1 2k+1

= 1 2

_x

1+x2+. . .+x2k

2k +

x₂k₊₁+x₂k₊₂+. . .+x₂k+1

2k

≥

r

x1+x2+. . .+x2k

2k ·

x₂k₊₁+x₂k₊₂+. . .+x₂k+1

2k

≥

q

2√k _x

❈❆P❮❚❯▲❖ ✷✳ ❉❊❙■●❯❆▲❉❆❉❊ ▼● ✲ ▼❆ ✶✺

= 2k+1√_x

1x2. . . x2k+1

=G(x1, x2, . . . , x2k+1)

▲♦❣♦ A(x1, x2, . . . , x2k+1)≥G(x₁, x₂, . . . , x₂k+1) ♣❛r❛ t♦❞♦ n q✉❡ s❡❥❛ ✉♠❛ ♣♦tê♥❝✐❛ ❞❡

❜❛s❡ ✷ ❝♦♠ ❡①♣♦❡♥t❡ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✳ ❆ ✐❣✉❛❧❞❛❞❡ ♦❝♦rr❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱

✶✳ x1+x2+. . .+x2k

2k =

x2k₊₁+x₂k₊₂+. . .+x₂k+1

2k

✷✳ x1 =x2 =. . .=x2k ❡ x₂k₊₁ =x₂k₊₂ =. . .=x₂k+1

❉❡ ✶ ❡ ✷ t❡♠♦s 2k·x2k

2k =

2k_·x

2k+1

2k ⇐⇒x2k =x2k+1

❧♦❣♦ x1 =x2 =. . .=x2k+1

❊t❛♣❛ ✷✿ Pr♦✈❛♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦n >1

❈♦♥s✐❞❡r❡ g = √n_x

1x2. . . xn✳ ❉❛ ♣r✐♠❡✐r❛ ❡t❛♣❛ t❡♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ s♦❜r❡ ❛s ♠é❞✐❛s

❞♦s ♥ú♠❡r♦s x1, x2, . . . , xn ❡ 2m−n ♥ú♠❡r♦s ✐❣✉❛✐s ❛g

x1+. . .+xn+ (g+. . .+g)

2m ≥

2mp

x1. . . xn·g2

m_−n

= 2mp

gn_·g2m_−n

=g ❆ss✐♠ t❡♠♦s

x1+x2+. . .+xn+ (2m−n)g ≥2mg ⇐⇒x1+x2+. . .+xn≥ng

⇐⇒ x1+x2 +_n. . .+xn ≥g

⇐⇒A(x1, . . . , xn)≥G(x1, . . . , xn)

❖❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x1 =x2 =. . .=xn =g✳

❉❡♠♦♥str❛çã♦ ✸✳

❙❡♥❞♦ ❝ô♥❝❛✈❛ ❛ ❢✉♥çã♦ ❧♦❣❛r✐t♠♦ ♥❛t✉r❛❧✱ ❡♥tã♦ ♣❛r❛ t♦❞♦ x1, x2, . . . , xn > 0 ❡

t1, t2, . . . , tn≥0 ❝♦♠ t1+t2 +. . .+tn= 1✱ t❡♠♦s

ln(t1x1+t2x2+. . .+tnxn)≥t1ln(x1) +t2ln(x2) +. . .+tnln(xn)

❈❆P❮❚❯▲❖ ✷✳ ❉❊❙■●❯❆▲❉❆❉❊ ▼● ✲ ▼❆ ✶✻

❯t✐❧✐③❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ❧♦❣❛r✐t♠♦s✱ t❡♠♦s

t1ln(x1) +t2ln(x2) +. . .+tnln(xn) = ln (x1t1) + ln (x2t2) +. . .+ ln (xntn) =

= ln (x1t1x2t2. . . xntn)

❉❡st❡ ♠♦❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✐♥✐❝✐❛❧ é ❡q✉✐✈❛❧❡♥t❡ ❛

ln(t1x1+t2x2+. . .+tnxn)≥ln x1t1x2t2. . . xntn

❈♦♠♦ ❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ♥❛ ❜❛s❡e é ❝r❡s❝❡♥t❡✱ t❡♠♦s eln(t1x1+...+tnxn)

≥eln(x1t1...xntn)

⇐⇒t1x1+. . .+tnxn≥x1t1. . . xntn

❚♦♠❛♥❞♦t1 =t2 =. . .=tn = _n1✱ t❡♠♦s

x1+x2+. . .+xn

n ≥

n

√_x

1x2. . . xn⇐⇒A(x1, x2, . . . , xn)≥G(x1, x2, . . . xn)

❆ ♦❝♦rrê♥❝✐❛ ❞❛ ✐❣✉❛❧❞❛❞❡ t❡♠ ❝♦♠♦ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ❛ ♠❡s♠❛ ❝♦♥❞✐çã♦ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✐♥✐❝✐❛❧✱ ♦✉ s❡❥❛✱x1 =x2 =. . .=xn✳

❉❡♠♦♥str❛çã♦ ✹✳

P♦❞❡♠♦s ❞❡✜♥✐r ♦ln(a)❝♦♠♦ s❡♥❞♦ ❛ ár❡❛ ❞❛ r❡❣✐ã♦ ❧✐♠✐t❛❞❛ ♣❡❧❛ ❝✉r✈❛y= _x1 ❡ ♣❡❧❛s

r❡t❛s x= 1✱ x=a ❡ y= 0✱ s❡ a_≥1❀ ♦✉ ♦ ♦♣♦st♦ ❞❡ss❛ ár❡❛✱ s❡0< a_≤1✳

❈❆P❮❚❯▲❖ ✷✳ ❉❊❙■●❯❆▲❉❆❉❊ ▼● ✲ ▼❆ ✶✼

❊st❛ r❡❣✐ã♦ ❡stá ❝♦♥t✐❞❛ ♥♦ r❡tâ♥❣✉❧♦ ❞❡ ❜❛s❡ | a₋1 _| ❡ ❛❧t✉r❛ 1✱ ❞❡ ♦♥❞❡ ❝♦♥❝❧✉í♠♦s

q✉❡ln(a)_≤a₋1✱ ✈❛❧❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ a= 1✳ ❈♦♠♦ ❛ ❢✉♥çã♦ y =ex _é

❝r❡s❝❡♥t❡ t❡♠♦sa_≤ea−1_✳

❉❡st❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ t❡♠♦s✱

exiA−1 ≥ xi A ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦1_≤i_≤n✱ ♦♥❞❡ A=A(x1, . . . , xn)✳

◆❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❛♦ ♠✉❧t✐♣❧✐❝❛r♠♦s ♠❡♠❜r♦ ❛ ♠❡♠❜r♦✱ t❡♠♦s ex1 +...A+xn−n≥ x1x2...xn

An ⇐⇒e nA

A −n≥ x1x2...xn

An

⇐⇒An

≥x1x2. . . xn

⇐⇒A_≥ √n_x

1x2. . . xn

⇐⇒A(x1, x2, . . . , xn)≥G(x1, x2, . . . , xn)

❖❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦❝♦rr❡ ❛ ✐❣✉❛❧❞❛❞❡ ❡♠ t♦❞❛s ❛sn❞❡s✐❣✉❛❧❞❛❞❡s

xi

A = 1 ♣❛r❛ t♦❞♦i

✱ ♦✉ s❡❥❛✱ x1 =x2 =. . .=xn=A✳

❉❡♠♦♥str❛çã♦ ✺✳

❖r❞❡♥❡♠♦s ♦s n ♥ú♠❡r♦s r❡❛✐s ❞❡ ♠♦❞♦ q✉❡ x1 ≤ x2 ≤ . . . ≤ xn ❡ s❡❥❛♠ a ❡ g ❛s

♠é❞✐❛s ❛r✐t♠ét✐❝❛s ❡ ❣❡♦♠étr✐❝❛s ❞❡❧❡s r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❙❡ t♦❞♦s ♦s ♥ú♠❡r♦s ❢♦r❡♠ ✐❣✉❛✐s✱ t❡♠♦sa=g✳ ❙✉♣♦♥❤❛ ❡♥tã♦ q✉❡ ♥❡♠ t♦❞♦s s❡❥❛♠ ✐❣✉❛✐s✳

❙✉❜st✐t✉✐♥❞♦ x1 ❡ xn r❡s♣❡❝t✐✈❛♠❡♥t❡ ♣♦r g ❡ x1_gxn✱ ♠❛♥t❡♥❞♦ ♦s ❞❡♠❛✐s ♥ú♠❡r♦s

✐♥❛❧t❡r❛❞♦s✱ ❛ ♠é❞✐❛ ❣❡♦♠étr✐❝❛ ❞♦s ♥♦✈♦s ♥ú♠❡r♦s ❝♦♥t✐♥✉❛g✱ ♣♦✐s g_·x1xn

g =x1xn✱ ❝♦♥✲

s❡r✈❛♥❞♦ ❛ss✐♠ ♦ ♣r♦❞✉t♦ ❞♦s ♥ú♠❡r♦s✳ ❈♦♠♦x1 ≤g ≤xn✱ s❡❣✉❡ q✉❡

x1+xn−

g+x1xn

g

=

=x1−g+xn−x1_gxn

=x1−g+xn

g₋x1

g

= (x1−g)

1₋xn

g

≥0

❉❡st❡ ♠♦❞♦ ❛ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ a1 ❞♦s ♥♦✈♦s ♥ú♠❡r♦s é ♠❡♥♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛ a✱

♦❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x1 = g ♦✉ xn = g✱ ♦ q✉❡ ❡♠ ❛♠❜♦s ♦s ❝❛s♦s é

❈❆P❮❚❯▲❖ ✷✳ ❉❊❙■●❯❆▲❉❆❉❊ ▼● ✲ ▼❆ ✶✽

❘❡♦r❞❡♥❞❡ ♦s ♥♦✈♦s ♥ú♠❡r♦s ❞❡ ♠♦❞♦ q✉❡ xk1 ≤ xk2 ≤ . . . ≤ xkn✳ ❙✉❜st✐t✉✐♥❞♦ xk1

❡ xkn ♣♦r g ❡ xk1_gxkn✱ t❡r❡♠♦s ❞❛ ♠❡s♠❛ ❢♦r♠❛ ❛ ♠é❞✐❛ ❣❡♦♠étr✐❝❛ ✐♥❛❧t❡r❛❞❛ ❡ ❛ ♠é❞✐❛

❛r✐t♠ét✐❝❛a2 ❞♦s ♥♦✈♦s ♥ú♠❡r♦s ♠❡♥♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛ a1✳

❆♣ósn ♣r♦❝❡❞✐♠❡♥t♦s ❞❡st❡s✱ t❡r❡♠♦s ♥❡❝❡ss❛r✐❛♠❡♥t❡nt❡r♠♦s ✐❣✉❛✐s ❛g✱ ♦❝♦rr❡♥❞♦ g = an ≤ an₋1 ≤ . . .≤ a✱ ♦✉ s❡❥❛✱ G(x1, . . . , xn) ≤ A(x1, . . . , xn)✱ ♦❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡

s❡✱ ❡ s♦♠❡♥t❡ s❡✱an =an−1 =. . .=a✱ ♦✉ ❞❡ ♠♦❞♦ ❡q✉✐✈❛❧❡♥t❡✱ x1 =x2 =. . .=xn=g✳

❉❡♠♦♥str❛çã♦ ✻✳

❈♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ f : A _⊂ Rn −→ R✱ ♦♥❞❡ A é ♦ ❝♦♥❥✉♥t♦ ❞❛s ♥✲✉♣❧❛s ❞❡ ♥ú✲ ♠❡r♦s r❡❛✐s ♣♦s✐t✐✈♦s✱ ❞❡✜♥✐❞❛ ♣♦r f(x1, x2, . . . , xn) = √nx1x2. . . xn✱ s✉❥❡✐t❛ à ❝♦♥❞✐çã♦

g(x1, x2, . . . , xn) = x1+x2+. . .+xn=S✳ ❯t✐❧✐③❛r❡♠♦s ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡ ♣❛r❛

❞❡t❡r♠✐♥❛r♠♦s ♦ ✈❛❧♦r ♠á①✐♠♦ ❞❡f s✉❥❡✐t❛ à ❝♦♥❞✐çã♦ ❞❛❞❛✳ P❡r❝❡❜❛ q✉❡

∂f ∂xi =

1

n ·(x1x2. . . xn)

1

n−1·(x₁x₂. . . x_i

−1xi+1. . . xn) = _n1x

1

n

1x

1

n

2 . . . x

1

n−1

i . . . x

1

n

n

♣❛r❛ t♦❞♦ ✐♥t❡✐r♦1_≤i_≤n✳

❆❧é♠ ❞✐ss♦✱ ∂g

∂xi = 1 ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ 1≤i≤n✳

❘❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦ ✈❡t♦r✐❛❧ ∇f = λ_∇g✱ t❡♠♦s (x1x2. . . xn)

1

n = nλx_i✱ ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦1_≤i_≤n❀ ♦✉ s❡❥❛✱nλx1 =nλx2 =. . .=nλxn✱ ♦✉ ❛✐♥❞❛✱x1 =x2 =. . .=xn= S_n✳

❙❛❜❡♠♦s ♣♦rt❛♥t♦ q✉❡ ❛ ❢✉♥çã♦ f ❛♣r❡s❡♥t❛ ✈❛❧♦r ♠á①✐♠♦ s♦❜ ❛s ❝♦♥❞✐çõ❡s ❞❛❞❛s q✉❛♥❞♦x1 =. . .=xn = S_n✱ ❧♦❣♦

f(x1, . . . , xn)≤

n r

S n

S n . . .

S n

⇐⇒ √n_x

1x2. . . xn ≤

S n

⇐⇒G(x1, x2, . . . , xn)≤A(x1, x2, . . . , xn)✳

❈❛♣ít✉❧♦ ✸

❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③

◗✉❛♥t♦ ❛♦ ♥ú♠❡r♦ ❞❡ ❛♣❧✐❝❛çõ❡s✱ s✉r❣❡ ❛♦ ❧❛❞♦ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❛s ♠é❞✐❛s ❛r✐t✲ ♠ét✐❝❛ ❡ ❣❡♦♠étr✐❝❛✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ q✉❡ ❛❜♦r❞❛r❡♠♦s ♥❡ss❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♥❞♦ ❛❧❣✉♠❛s ❞❡ s✉❛s ❞❡♠♦♥str❛çõ❡s✳

Pr♦♣♦s✐çã♦ ✸✳✶✳ ❉❛❞♦s2n ♥ú♠❡r♦s r❡❛✐s x1, x2, . . . , xn ❡y1, y2, . . . , yn✱ t❡♠♦s

|x1y1+x2y2+. . .+xnyn| ≤

p

x12+x22+. . .+xn2·

p

y12+y22 +. . .+yn2

♦❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st✐r ✉♠ ♥ú♠❡r♦ r❡❛❧ λ t❛❧ q✉❡ xi = λyi ♣❛r❛

t♦❞♦ ✐♥t❡✐r♦1_≤i_≤n✳

✸✳✶ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✿ ❝❛s♦

n

_{= 2}

❖❝♦rr❡♥❞♦ ♥❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♥♦ ❝❛s♦n = 1✱ ❞❡♠♦♥str❛♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡

❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡n = 2✱ ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ ♣r♦✈❛ ❛❧❣é❜r✐❝❛ ❡ ♦✉tr❛s

❞✉❛s ❣❡♦♠étr✐❝❛s✱ ♣❡r♠✐t✐♥❞♦ ✉♠❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡✳ ❉❡♠♦♥str❛çã♦ ✶✳

(x12+x22)(y12+y22) = x12y12+x22y22+x12y22+x22y12

= (x1y1)2+ 2(x1y1)(x2y2) + (x2y2)2+ (x1y2)2

−2(x1y2)(x2y1) + (x2y1)2

= (x1y1+x2y2)2+ (x1y2−x2y1)2

❈❆P❮❚❯▲❖ ✸✳ ❉❊❙■●❯❆▲❉❆❉❊ ❉❊ ❈❆❯❈❍❨✲❙❈❍❲❆❘❩ ✷✵

▲♦❣♦

(x1y1+x2y2)2 ≤(x12+x22)(y12+y22)

⇐⇒|x1y1+x2y2 | ≤√x12+x22·

p

y12+y22

❖❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱x1y2 =x2y1✱ ♦✉ ❞❡ ♠♦❞♦ ❡q✉✐✈❛❧❡♥t❡x1 =λy1

❡x2 =λy2✳

❉❡♠♦♥str❛çã♦ ✷✳

❈♦♥s✐❞❡r❡ ♦s ♣♦♥t♦s O = (0,0)✱ A = (x1, x2) ❡ B = (y1, y2)✳ ❙❡ ♦s ♣♦♥t♦s O✱ A ❡ B

sã♦ ❝♦❧✐♥❡❛r❡s✱ t❡♠♦s x1 =λy1 ❡ x2 =λy2 ❡ ❛ss✐♠

|x1y1 +x2y2 | =|λy12+λy22 | =_|λ _|(y12+y22) =_|λ _|py12+y22·

p

y12+y22

=

q

(λy1)2+ (λy2)2_·py12+y22

=√x12 +x22·

p

y12+y22

❙❡ ♦s ♣♦♥t♦s O✱ A ❡ B ♥ã♦ sã♦ ❝♦❧✐♥❡❛r❡s✱ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ❛ ❧❡✐ ❞♦s ❝♦ss❡♥♦s ♥♦

∆OAB✱ ♦❜t❡♥❞♦

AB2 =OA2+OB2₋2_·OA_·OB_·cosθ ♦♥❞❡ θ é ♦ ❛♥❣✉❧♦ AOBb ✳

❆♣❧✐❝❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s ❡ ❛❧❣✉♠❛s ♠❛♥✐♣✉❧❛çõ❡s✱ ♦❜t❡♠♦s

|cosθ _|= _√ |x1y1+x2y2 |

x12+x22·

p

y12+y22

❈♦♠♦ |cosθ _|<1✱ t❡♠♦s

|x1y1+x2y2 |<

p

x12+x22·

p

y12+y22

❉❡♠♦♥str❛çã♦ ✸✳

❊ss❛ ♣r♦✈❛ ✉t✐❧✐③❛rá ár❡❛s ❞❡ ❞✉❛s ✜❣✉r❛s ❝✉❥❛ ❝♦♥str✉çã♦ ✐♥❞✐❝❛♠♦s ❛ s❡❣✉✐r✳

❈♦♥str✉í♠♦s ♦ r❡tâ♥❣✉❧♦ABCD ❞❡ ❧❛❞♦s _|x1 |+|y2 |❡ |y1 |+|x2 |✱ ❡ ❛ ♣❛rt✐r ❞❡❧❡

❈❆P❮❚❯▲❖ ✸✳ ❉❊❙■●❯❆▲❉❆❉❊ ❉❊ ❈❆❯❈❍❨✲❙❈❍❲❆❘❩ ✷✶

❋✐❣✉r❛ ✸✳✶✿ P❛r❛❧❡❧♦❣r❛♠♦

❊♠ s❡❣✉✐❞❛ ❝♦♥str✉í♠♦s ♦ r❡tâ♥❣✉❧♦A′_B′_C′_D′ _{❞❡ ❞✐♠❡♥sõ❡s}√_x

12+x22❡

p

y12+y22✱

❡ s♦❜r❡ ❝❛❞❛ ♦s s❡✉s ❧❛❞♦s ❝♦♥str✉í♠♦s ♦s tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s ✐♥❞✐❝❛❞♦s ♥❛ ✜❣✉r❛ ❛❜❛✐①♦✳

❋✐❣✉r❛ ✸✳✷✿ ❘❡tâ♥❣✉❧♦

◆♦t❡ q✉❡ ❛ ár❡❛ ❞❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ é s❡♠♣r❡ ♠❡♥♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛ ár❡❛ ❞❡ ✉♠ r❡tâ♥❣✉❧♦ ❞❡ ♠❡s♠♦s ❧❛❞♦s✱ t❡♠♦s

S(EF GH)_≤S(A′B′C′D′)

⇔(_|x1 |+|y2 |)(|y1 |+|x2 |)≤

√

x12 +x22·

p

y12+y22+|x1x2 |+|y1y2 |

⇔|x1y1 |+|x2y2 |+|x1x2 |+|y1y2 | ≤√x12+x22·

p

y12+y22+|x1x2 |+|y1y2 |

⇔|x1y1 |+|x2y2 | ≤√x12+x22·

p

y12+y22

⇒|x1y1 +x2y2 | ≤

√

x12+x22·

p

❈❆P❮❚❯▲❖ ✸✳ ❉❊❙■●❯❆▲❉❆❉❊ ❉❊ ❈❆❯❈❍❨✲❙❈❍❲❆❘❩ ✷✷

❖❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦ ♣❛r❛❧❡❧♦❣r❛♠♦ EF GH ❢♦r ✉♠ r❡tâ♥❣✉❧♦ ❡ ✈❛❧❡r ❛ ✐❣✉❛❧❞❛❞❡ ♥❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r ✉t✐❧✐③❛❞❛ ♥❛ ✐♠♣❧✐❝❛çã♦✱ ♦ q✉❡ ♦❝♦rr❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦s tr✐â♥❣✉❧♦s AEH ❡ BEF ❢♦r❡♠ s❡♠❡❧❤❛♥t❡s✱x1y1 ≥0 ❡ x2y2 ≥0✱ ♦✉ s❡❥❛✱

x1 =λy1 ❡ x2 =λy2✳

✸✳✷ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✿ ❝❛s♦ ❣❡r❛❧

◆❡st❛ s❡çã♦ sã♦ r❡❛❧✐③❛❞❛s ❛❧❣✉♠❛s ❞❡♠♦♥str❛çõ❡s ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ ♣❛r❛ q✉❛❧q✉❡r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n✳

❉❡♠♦♥str❛çã♦ ✶✳

❈♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ f :R−→R ❞❡✜♥✐❞❛ ♣♦r

f(u) = (x1u−y1)2+ (x2u−y2)2+. . .+ (xnu−yn)2

❉❡s❡♥✈♦❧✈❡♥❞♦ ❛ ❡①♣r❡ssã♦✱ ♦❜t❡♠♦s

f(u) = (x12+x22+. . .+xn2)u2−2(x1y1+x2y2+. . .+xnyn)u+ (y12+y22+. . .+yn2)

❈♦♠♦ ❝❛❞❛ ✉♠❛ ❞❛s ♣❛r❝❡❧❛s(xiu−yi)2 é ♥ã♦ ♥❡❣❛t✐✈❛✱ t❡♠♦sf(u)≥0♣❛r❛ t♦❞♦ u r❡❛❧

♦ q✉❡ ♦❝♦rr❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱∆_≤0✱ ♦✉ s❡❥❛✱

4(x1y1+. . .+xnyn)2−4(x12+. . .+xn2)·(y12+. . .+yn2)≤0

⇐⇒(x1y1+. . .+xnyn)2 ≤(x12+. . .+xn2)·(y12+. . .+yn2)

⇐⇒|x1y1+. . .+xnyn | ≤

√

x12+. . .+xn2·

p

y12+. . .+yn2

❖❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ∆ = 0 ♦ q✉❡ s✐❣♥✐✜❝❛ q✉❡ ❛ ❢✉♥çã♦ ❛♣r❡s❡♥t❛

✉♠ ú♥✐❝♦ ③❡r♦ r❡❛❧u′✳ ❖r❛✱f(u′) = 0 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❝❛❞❛ ✉♠❛ ❞❛s ♣❛r❝❡❧❛s(xiu′−yi)2

❢♦r ♥✉❧❛✱ ♦✉ ❞❡ ♠♦❞♦ ❡q✉✐✈❛❧❡♥t❡xi = _u1′yi ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ 1≤i≤n✳

❉❡♠♦♥str❛çã♦ ✷✳

❈♦♥s✐❞❡r❡ ♦s ♥ú♠❡r♦s A = √x12+. . .+xn2✱ B =

p

y12+. . .+yn2 ❡ ❛ ♣❛rt✐r ❞❡❧❡s

xi = x_Ai ❡yi = yi

B ❝♦♠ ♦ ✐♥t❡✐r♦i ✈❛r✐❛♥❞♦ ❞❡ 1 ❛ n✳

P❡r❝❡❜❛ q✉❡ x12+x22+. . .+xn2 = 1 ❡ y12+y22+. . .+yn2 = 1

❆❧é♠ ❞✐ss♦✱ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❛s ♠é❞✐❛s ❣❡♦♠étr✐❝❛ ❡ ❛r✐t♠ét✐❝❛✱ t❡♠♦s

|xiyi | ≤

xi2+yi2

❈❆P❮❚❯▲❖ ✸✳ ❉❊❙■●❯❆▲❉❆❉❊ ❉❊ ❈❆❯❈❍❨✲❙❈❍❲❆❘❩ ✷✸

❙♦♠❛♥❞♦ ♠❡♠❜r♦ ❛ ♠❡♠❜r♦ ❛sn ❞❡s✐❣✉❛❧❞❛❞❡s ♦❜t✐❞❛s q✉❛♥❞♦ ❢❛③❡♠♦s i✈❛r✐❛r ❞❡ 1

❛n✱ t❡♠♦s

|x1y1 |+. . .+|xnyn| ≤

x12+. . .+xn2+y12+. . .+yn2

2

⇐⇒ |x_AB1y1 | +. . .+ |xnyn|

AB ≤1

⇐⇒|x1y1 |+. . .+|xnyn | ≤AB

⇐⇒|x1y1 |+. . .+|xnyn | ≤

p

x12+. . .+xn2·

p

y12+. . .+yn2

❖r❛ |x1y1+. . .+xnyn| ≤ |x1y1 |+. . .+|xnyn |✱ ❧♦❣♦

|x1y1+x2y2+. . .+xnyn| ≤

p

x12+x22+. . .+xn2·

p

y12+y22 +. . .+yn2

❖❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x1 =y1 ✱ . . .✱xn =yn✱ ♦✉ s❡❥❛✱ x1 = _BAy1✱. . .✱

xn= _BAyn✳

❉❡♠♦♥str❛çã♦ ✸✳

❉❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❛s ♠é❞✐❛s ❣❡♦♠étr✐❝❛ ❡ ❛r✐t♠ét✐❝❛✱ t❡♠♦s ♣❛r❛ t♦❞♦ λ r❡❛❧ ♥ã♦ ♥✉❧♦

G(xi 2

λ , λyi

2₎

≤A(xi 2

λ , λyi

2₎

⇐⇒

r xi2

λ ·λyi

2 _≤ 1 2

xi2

λ +λyi

2

⇐⇒|xiyi |≤

1 2

_x

i2

λ +λyi

2

❚♦♠❛♥❞♦ λ =

s _n X

i=1

xi2

s_Xn i=1

yi2

❡ s♦♠❛♥❞♦ ♠❡♠❜r♦ ❛ ♠❡♠❜r♦ ❛s n ❞❡s✐❣✉❛❧❞❛❞❡s ♦❜t✐❞❛s

❈❆P❮❚❯▲❖ ✸✳ ❉❊❙■●❯❆▲❉❆❉❊ ❉❊ ❈❆❯❈❍❨✲❙❈❍❲❆❘❩ ✷✹

n

X

i=1

|xiyi |≤

1 2      

s_Xn i=1

xi2

s _n X

i=1

yi2

·

n

X

i=1

yi2+

s_Xn i=1

yi2

s _n X

i=1

xi2

·

n

X

i=1

xi2

      ⇐⇒ n X i=1

|xiyi |≤

1 2   v u u t n X i=1

xi2·

v u u t n X i=1

yi2+

v u u t n X i=1

yi2·

v u u t n X i=1

xi2

  ⇐⇒ n X i=1

|xiyi |≤

v u u t n X i=1

xi2·

v u u t n X i=1

yi2

=_⇒|

n

X

i=1

xiyi |≤

v u u t n X i=1

xi2·

v u u t n X i=1

yi2

❖❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦❝♦rr❡r ❛ ✐❣✉❛❧❞❛❞❡ ♥❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❛s ♠é❞✐❛s ❡ ♥❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✱ ♦✉ ❞❡ ♠♦❞♦ ❡q✉✐✈❛❧❡♥t❡✱ xi2

λ =λyi

2 _❡x

iyi ≥0✱ ❧♦❣♦xi =λyi

♣❛r❛ t♦❞♦ ✐♥t❡✐r♦1_≤i_≤n✳

❉❡♠♦♥str❛çã♦ ✹✳

❈♦♥s✐❞❡r❡ ♦s ✈❡t♦r❡s u= (x1, x2, . . . , xn) ❡ v = (y1, y2, . . . , yn)✳

P♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ ❝♦♠♦|u_·v _{|≤ k}u_{k · k}v_k✳ ❙❡u= 0 ♦✉v = 0✱ t❡♠♦su_·v = 0 ❡_ku_{k · k}v_k= 0✱ s❡♥❞♦ ♣♦rt❛♥t♦ ✈á❧✐❞❛ ❛ ✐❣✉❛❧❞❛❞❡✳

❙✉♣♦♥❤❛ ❡♥tã♦ q✉❡ u❡ v s❡❥❛♠ ✈❡t♦r❡s ♥ã♦ ♥✉❧♦s✳ ❙❡ u ❡v sã♦ ✉♥✐tár✐♦s✱ t❡♠♦s

0_{≤ k}u_±v_k2 = (u_±v)_·(u_±v) =u_·v _±2u_·v+v_·v

= 1_±2u_·v+ 1 = 2(1_±u_·v)

❧♦❣♦ 1_±u_·v _≥0_{⇐⇒ ∓}u_·v _≤1_⇐⇒|u_·v _{| ≤}1✱ ♦❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡

s❡✱u_±v = 0✳

❈❆P❮❚❯▲❖ ✸✳ ❉❊❙■●❯❆▲❉❆❉❊ ❉❊ ❈❆❯❈❍❨✲❙❈❍❲❆❘❩ ✷✺

| _ku1_ku·

1

kv_kv | ≤1

⇐⇒ 1

ku_k·kv_k |u·v | ≤1

⇐⇒|u_·v _{|≤ k}u_{k · k}v_k

❖❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ 1

ku_ku±

1

kv_kv = 0✱ ♦✉ s❡❥❛✱ u = ±

ku_k

kv_kv✱ ♦✉ ❞❡

♠♦❞♦ ❡q✉✐✈❛❧❡♥t❡✱x1 =λy1✱ . . .✱ xn=λyn✳

❉❡♠♦♥str❛çã♦ ✺✳

❈♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♣♦r f(u, v) = _hu, v_i✱ ❡♠ q✉❡ u = (x1, x2, . . . , xn) ❡

v = (y1, y2, . . . , yn)❀ s✉❥❡✐t❛ às ❝♦♥❞✐çõ❡s g(u, v) = n

X

i=1

x2_i = 1 ❡ h(u, v) =

n

X

i=1

y2_i = 1✳

❯t✐❧✐③❛r❡♠♦s ♦s ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦ ✈❛❧♦r ♠á①✐♠♦ ❞❡f s✉❥❡✐t❛ às ❝♦♥❞✐çõ❡s ❞❛❞❛s✳

P❡r❝❡❜❛ q✉❡ ✭❛✮ ∂f

∂xi =yi ❡

∂f

∂yi =xi✱ ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ 1≤i≤n✳ ✭❜✮ ∂g

∂xi = 2xi ❡

∂g

∂yi = 0✱ ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ 1≤i≤n✳ ✭❝✮ ∂h

∂xi = 0 ❡

∂h

∂yi = 2yi✱ ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ 1≤i≤n✳ ❉❛ ✐❣✉❛❧❞❛❞❡ ∇f =λ_∇g+µ_∇h✱ t❡♠♦s

(y1, y2, . . . , yn, x1, x2, . . . , xn) = (2λx1,2λx2, . . . ,2λxn,2µy1,2µy2, . . . ,2µyn)

⇐⇒yi = 2λxi ❡ xi = 2µyi ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ 1≤i≤n✳

❉❛s ❝♦♥❞✐çõ❡s t❡♠♦s

n

X

i=1

x2_i = 1 _⇒

n

X

i=1

4µ2y_i2 = 1 _⇒ 4µ2

n

X

i=1

y_i2 = 1 _⇒ 4µ2 = 1 _⇒ µ=_±1 2✳

❡

n

X

i=1

y_i2 = 1 _⇒

n

X

i=1

4λ2x2_i = 1 _⇒ 4λ2

n

X

i=1

x2_i = 1 _⇒ 4λ2 = 1 _⇒ λ =_±1 2✳

❈❛❧❝✉❧❛♥❞♦ ♦ ✈❛❧♦r ❞❛ ❢✉♥çã♦ ♣❛r❛ ❝❛❞❛ ❝❛s♦ ✭❛✮ P❛r❛ λ=µ= 1

2✱ t❡♠♦sxi =yi ❡ ♣♦rt❛♥t♦f(u, v) =

Pn

❈❆P❮❚❯▲❖ ✸✳ ❉❊❙■●❯❆▲❉❆❉❊ ❉❊ ❈❆❯❈❍❨✲❙❈❍❲❆❘❩ ✷✻

✭❜✮ P❛r❛ λ=µ=₋1₂✱ t❡♠♦sxi =−yi ❡ ♣♦rt❛♥t♦ f(u, v) =P n

i=1(−x2i) = −1

✭❝✮ P❛r❛ ♦s ❞♦✐s ♦✉tr♦s ❝❛s♦s✱ t❡♠♦s xi = 0 ❡ yi = 0✱ ♦✉ s❡❥❛✱ f(u, v) = 0✳

P♦rt❛♥t♦ ♦ ✈❛❧♦r ♠á①✐♠♦ ❞❡f s✉❥❡✐t❛ às ❝♦♥❞✐çõ❡s ❞❛❞❛s é1❡ ♦ ♠í♥✐♠♦₋1✱ ♦❝♦rr❡♥❞♦

♦ ♣r✐♠❡✐r♦ q✉❛♥❞♦xi =yi ❡ ♦ s❡❣✉♥❞♦ q✉❛♥❞♦ xi =−yi ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ 1≤i≤n✳

❈♦♥s✐❞❡r❡ ❛❣♦r❛ ♦s ✈❡t♦r❡s a = (a1, a2, . . . , an) ❡ b = (b1, b2, . . . , bn)✳ ❚♦♠❡♠♦s xi =

ai

pPn i=1a2i

❡ yi =

bi

pPn i=1b2i

✱ ❝♦♠ ✐ss♦ r❡t♦♠❛♠♦s ❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s✱ ❡ ♣♦rt❛♥t♦

−1_≤

n

X

i=1

xiyi ≤1⇐⇒

n X i=1

xiyi

≤1⇐⇒ n X i=1

aibi

pPn i=1a2i ·

pPn i=1b2i

! ≤1 ⇐⇒ n X i=1

aibi

≤ v u u t n X i=1 a2 i · v u u t n X i=1 b2 i

❖❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱|xi|=|yi|✱ ♦✉ ❞❡ ♠♦❞♦ ❡q✉✐✈❛❧❡♥t❡✱ ai =λ′bi✳

❈❛♣ít✉❧♦ ✹

❆♣❧✐❝❛çõ❡s

✶✳ ✭❈❛s♦ ♣❛rt✐❝✉❧❛r ❞❛ ❉❡s✐❣✉❛❧❞❛❞❡ ■s♦♣❡r✐♠étr✐❝❛✮ Pr♦✈❡ q✉❡✿

✭❛✮ ❉❡♥tr❡ t♦❞♦s ♦s r❡tâ♥❣✉❧♦s ❞❡ ♣❡rí♠❡tr♦ ❞❛❞♦ P✱ ♦ ❞❡ ♠❛✐♦r ár❡❛ é ♦ q✉❛❞r❛❞♦✳ ✭❜✮ ❉❡♥tr❡ t♦❞♦s ♦s r❡tâ♥❣✉❧♦s ❞❡ ár❡❛ ❞❛❞❛ ❙✱ ♦ ❞❡ ♠❡♥♦r ♣❡rí♠❡tr♦ é ♦ q✉❛❞r❛❞♦✳

❘❡s♦❧✉çã♦

✭❛✮ ❈♦♥s✐❞❡r❡ q✉❡ ❛s ❞✐♠❡♥sõ❡s ❞❡ ✉♠ r❡tâ♥❣✉❧♦ ❞❡ ♣❡rí♠❡tr♦ P sã♦ x ❡ y✱ ❡♥tã♦ P = 2(x+y) ❡ s✉❛ ár❡❛ é S=xy✳

P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ▼● ✲ ▼❆✱ t❡♠♦s

G(x, y)_≤A(x, y) _⇔√xy_≤ x+y

2

⇔xy_≤

x+y

2

2

⇔S_≤

P

4

2

▲♦❣♦ ❛ ár❡❛ ❞❡ ✉♠ r❡tâ♥❣✉❧♦ ❞❡ ♣❡rí♠❡tr♦ P é ♠❡♥♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛ P₄2✱ ♦❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x =y✱ ♦✉ s❡❥❛✱ q✉❛♥❞♦ ♦ r❡tâ♥❣✉❧♦ ❢♦r ✉♠ q✉❛❞r❛❞♦✳

✭❜✮ ❈♦♥s✐❞❡r❡ q✉❡ ❛s ❞✐♠❡♥sõ❡s ❞❡ ✉♠ r❡tâ♥❣✉❧♦ ❞❡ ár❡❛ S sã♦ x❡y✱ ❡♥tã♦S =xy ❡ ♦ s❡✉ ♣❡rí♠❡tr♦ é P = 2(x+y)✳

❈❆P❮❚❯▲❖ ✹✳ ❆P▲■❈❆➬Õ❊❙ ✷✽

G(x, y)_≤A(x, y) _⇔√xy_≤ x+y

2

⇔4√xy_≤2(x+y)

⇔4√S _≤P

▲♦❣♦ ♦ ♣❡rí♠❡tr♦ ❞❡ ✉♠ r❡tâ♥❣✉❧♦ ❞❡ ár❡❛ S é ♠❛✐♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛ 4√S✱ ♦❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x =y✱ ♦✉ s❡❥❛✱ q✉❛♥❞♦ ♦ r❡tâ♥❣✉❧♦ ❢♦r ✉♠ q✉❛❞r❛❞♦✳

✷✳ Pr♦✈❡ q✉❡ ♣❛r❛ q✉❛✐sq✉❡r r❡❛✐s ♣♦s✐t✐✈♦s x❡ y t❡♠♦s

x+y

2

2

≤ x

2_+y2 2

♦❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x=y✳

❘❡s♦❧✉çã♦

P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ▼● ✲ ▼❆ t❡♠♦s

G x2 2 , y2 2 ≤A x2 2 , y2 2 ⇔ r x2 2 · y2 2 ≤ 1 2 x2 2 , y2 2 ⇔ xy 2 ≤

x2+y2

4

⇔ x

2_+y2

4 +

xy

2 ≤

x2+y2

4 +

x2+y2

4

⇔x

2

2 + 2x

2 y 2 + _y 2 2 ≤ x

2_+y2 2 ⇔ _x +y 2 2 ≤ x

2_+y2 2

❖❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x2

2 =

y2

2✱ ♦✉ s❡❥❛✱ x=y✳

❈❆P❮❚❯▲❖ ✹✳ ❆P▲■❈❆➬Õ❊❙ ✷✾

✸✳ ❙❡❥❛♠ a ❡b r❡❛✐s ♣♦s✐t✐✈♦s t❛✐s q✉❡ a+b = 1✱ ♣r♦✈❡ q✉❡

a+1

a 2

+

b+1

b 2

≥ 25₂

❘❡s♦❧✉çã♦

❯s❛r❡♠♦s ♦ r❡s✉❧t❛❞♦ ❞♦ ♣r♦❜❧❡♠❛ ❛♥t❡r✐♦r✱ t♦♠❛♥❞♦ x=a+ 1

a ❡ y=b+

1

b✳ a+1

a

2

+ b+ 1

b

2

2 ≥

_a+ 1

a +b+

1

b

2

2

=

_{1 +} 1

a+ 1 b 2 2 = 1 4 ·

1 + a+b

ab 2

= 1 4 ·

1 + 1

ab 2

(■)

❖r❛✱ ♣❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ▼● ✲ ▼❆ t❡♠♦s q✉❡

G(a, b)_≤A(a, b) _⇔√ab_≤ a+b

2

⇔ab_≤ 1

4

⇔ _ab1 ≥4

⇔1 + 1

ab ≥5

⇔ 1₄

1 + 1

ab 2

≥ 25₄ (■■)

❉❡ ✭■✮ ❡ ✭■■✮ t❡♠♦s a+1

a 2

+

b+1

b 2

≥ 25₂

❈❆P❮❚❯▲❖ ✹✳ ❆P▲■❈❆➬Õ❊❙ ✸✵

✹✳ ❙❡❥❛♠ a✱ b ❡ cr❡❛✐s ♣♦s✐t✐✈♦s✱ ♣r♦✈❡ q✉❡ a

b+c+ b c+a +

c a+b ≥

3 2

❘❡s♦❧✉çã♦ ✶

❈♦♥s✐❞❡r❡ ♦s ♥ú♠❡r♦s

x= a

b+c+ b c+a +

c a+b

y= b

b+c + c c+a +

a a+b

z = c

b+c+ a c+a +

b a+b

P❡r❝❡❜❛ q✉❡

y+z = b+c

b+c + c+a c+a +

a+b a+b = 3

❆❧é♠ ❞✐ss♦

x+y = a+b

b+c + b+c c+a +

c+a a+b

x+z = a+c

b+c + b+a c+a +

c+b a+b

P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ▼● ✲ ▼❆ t❡♠♦s

G

a+b b+c,

b+c c+a,

c+a a+b

≤A

a+b b+c,

b+c c+a,

c+a a+b

⇔1_≤ x+y 3

❡

G

a+c b+c,

b+a c+a,

c+b a+b

≤A

a+c b+c,

b+a c+a,

c+b a+b

⇔1_≤ x+z 3

❙♦♠❛♥❞♦ ♦s ♠❡♠❜r♦s ❞❡ss❛s ❞❡s✐❣✉❛❧❞❛❞❡s t❡♠♦s

2x+y+z _≥6 _⇔ 2x_≥3 _⇔ x_≥ 3

❈❆P❮❚❯▲❖ ✹✳ ❆P▲■❈❆➬Õ❊❙ ✸✶

⇔ _b+a _c + b

c+a + c a+b ≥

3 2

❘❡s♦❧✉çã♦ ✷

P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ t❡♠♦s p

(b+c) + (c+a) + (a+b)_·qa+b+c b+c +

a+b+c c+a +

a+b+c a+b ≥3

√

a+b+c

⇔((b+c) + (c+a) + (a+b))_· a+b+c b+c +

a+b+c c+a +

a+b+c a+b

≥9(a+b+c)

⇔2(a+b+c)_·

a b+c +

b c+a +

c a+b + 3

≥9(a+b+c)

⇔ _b+a _c + b

c+a + c

a+b + 3≥

9 2

⇔ _b+a _c + b

c+a + c a+b ≥

3 2

✺✳ ❙❡❥❛♠ a✱ b ❡ c r❡❛✐s ♣♦s✐t✐✈♦s✱ ♣r♦✈❡ q✉❡ (a+b)(b+c)(c+a) _≥ 8abc✱ ♦❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ a=b=c✳

❘❡s♦❧✉çã♦

P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ▼● ✲ ▼❆ t❡♠♦s

A(a, b)_≥G(a, b)_⇔ a+b

2 ≥

√

ab

A(a, b)_≥G(b, c)_⇔ b+c

2 ≥

√

bc

A(c, a)_≥G(c, a)_⇔ c+a

2 ≥

√

ca ▼✉❧t✐♣❧✐❝❛♥❞♦ ♠❡♠❜r♦ ❛ ♠❡♠❜r♦✱ t❡♠♦s

(a+b)(b+c)(c+a)

8 ≥abc ⇔ (a+b)(b+c)(c+a)≥8abc

❖❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦❝♦rr❡r ❛ ✐❣✉❛❧❞❛❞❡ ❡♠ ❝❛❞❛ ✉♠❛ ❞❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❞♦ t✐♣♦ ▼● ✲ ▼❆✱ ♦✉ s❡❥❛✱ a=b=c✳

❈❆P❮❚❯▲❖ ✹✳ ❆P▲■❈❆➬Õ❊❙ ✸✷

✻✳ ❙❡❥❛♠ n >1♥ú♠❡r♦s r❡❛✐s ♣♦s✐t✐✈♦s p1, p2, . . . , pn ❞❡ s♦♠❛ ✉♥✐tár✐❛✳ ▼♦str❡ q✉❡ n

X

i=1

pi+

1

pi

2

≥n3+ 2n+ 1

n

❘❡s♦❧✉çã♦

❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ ♥❛s s❡q✉ê♥❝✐❛s(

n termos

z }| {

1,1, . . . ,1)❡(p1, p2, . . . , pn)

t❡♠♦s

|p1·1 +p2·1 +. . .+pn·1| ≤

q p2

1+p22+. . .+p2n·

√

12_{+ 1}2_{+. . .+ 1}2

⇐⇒p1+p2+. . .+pn ≤

q p2

1+p22+. . .+p2n·

√

n

⇐⇒1_≤

q p2

1+p22+. . .+p2n·

√

n

⇐⇒ _n1 ≤p2₁+p2₂+. . .+p2_n

⇐⇒ _n1 ≤

n

X

i=1

p2_i ✭■✮

❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ ♥❛s s❡q✉ê♥❝✐❛s (√p1,√p2, . . . ,√pn) ❡

(√1_p

1,

1

√_p

2, . . . ,

1

√_p

n) t❡♠♦s

|√p1· 1

√_p

1

+. . .+√pn·

1

√_p

n| ≤

√

p1+. . .+pn·

r

1

p1

+. . .+ 1

pn

⇐⇒ |1 + 1 +. . .+ 1_{| ≤}√1_·

r

1

p1

+ 1

p2

+. . .+ 1

pn

⇐⇒n _≤ r

1

p1

+ 1

p2

+. . .+ 1

pn

⇐⇒n2 _≤ 1 p1

+ 1

p2

+. . .+ 1

pn

⇐⇒n2 _≤

n

X

i=1 1

pi ✭■■✮

❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ ♥❛s s❡q✉ê♥❝✐❛s(

n termos

z }| {

1,1, . . . ,1)❡(_p1₁,_p1

2, . . . ,

1

❈❆P❮❚❯▲❖ ✹✳ ❆P▲■❈❆➬Õ❊❙ ✸✸

|1_· 1

p1

+ 1_· 1

p2

+. . .+ 1_· 1

pn| ≤

√

1 + 1 +. . .+ 1_·

s

1

p2₁ +

1

p2₁ +. . .+

1

p2

n

⇐⇒ _p1

1

+ 1

p2

+. . .+ 1

pn ≤

√

1 +. . .+ 1_·

s 1 p2 1 + 1 p2 2

+. . .+ 1

p2

n

❯s❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ t❛♥s✐t✐✈❛ ❡♠ r❡❧❛çã♦ à ❞❡s✐❣✉❛❧❞❛❞❡ ✭■■✮ t❡♠♦s

⇐⇒n2 _≤√n_· s 1 p2 1 + 1 p2 2

+. . .+ 1

p2

n

⇐⇒n4 _≤n_·

1

p2₁ +

1

p2₂ +. . .+

1

p2

n

⇐⇒n3 _≤

n X i=1 1 p2 i ✭■■■✮

❙♦♠❛♥❞♦ ♠❡♠❜r♦ ❛ ♠❡♠❜r♦ ❛s ❞❡s✐❣✉❛❧❞❛❞❡s ✭■✮ ❡ ✭■■■✮ t❡♠♦s

n3+ 1

n ≤

n

X

i=1

p2i + n X i=1 1 p2 i

⇐⇒n3+ 2n+ 1

n ≤

n

X

i=1

p2_i +

n

X

i=1 1

p2_i + 2n

⇐⇒n3+ 2n+ 1

n ≤

n

X

i=1

p2_i +

n

X

i=1 1

p2i

+ 2

n

X

i=1

p2_{i ·} 1

p2i

⇐⇒n3+ 2n+ 1

n ≤

n

X

i=1

pi+

1

pi

2

❖❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦❝♦rr❡ ❛ ✐❣✉❛❧❞❛❞❡ ♥❛s três ✈❡③❡s q✉❡ ✉s❛♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✱ ♦✉ ❞❡ ♠♦❞♦ ❡q✉✐✈❛❧❡♥t❡✱p1 =p2 =. . .=pn= _n1✳

✼✳ Pr♦✈❡ q✉❡ ❞❡♥tr❡ t♦❞♦s ♦s tr✐â♥❣✉❧♦s ❞❡ ♣❡rí♠❡tr♦ ❞❛❞♦ 2p✱ ♦ ❞❡ ♠❛✐♦r ár❡❛ é ♦ ❡q✉✐❧át❡r♦✳

❘❡s♦❧✉çã♦

❙❡❥❛ S ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ABC ❞❡ ♣❡rí♠❡tr♦ ❞❛❞♦ 2p✱ ❝✉❥❛s ♠❡❞✐❞❛s ❞♦s ❧❛❞♦s ✐♥❞✐❝❛r❡♠♦s ♣♦r a✱ b ❡ c✳ ❚❡♠♦s p= a+b+c

2 ❡S =

p

❈❆P❮❚❯▲❖ ✹✳ ❆P▲■❈❆➬Õ❊❙ ✸✹

P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ▼● ✲ ▼❆ t❡♠♦s

G(p₋a, p₋b, p₋c)_≤A(p₋a, p₋b, p₋c)

⇔ p3

(p₋a)(p₋b)(p₋c)_≤ 3p−(a+b+c) 3

⇔(p₋a)(p₋b)(p₋c)_≤p 3

3

⇔p(p₋a)(p₋b)(p₋c)_≤ p 4

27

⇔pp(p₋a)(p₋b)(p₋c)_≤ p 2√₃

9

⇔S _≤ p

2√₃ 9

❖❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ p₋a =p₋b =p₋c✱ ♦✉ ❛✐♥❞❛✱ a=b =c✱

♦✉ s❡❥❛✱ q✉❛♥❞♦ ♦ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦✳

✽✳ ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ♠á①✐♠♦ ❞❛ ❢✉♥çã♦f(x) =x(1₋x)3_{✱ s❡♥❞♦x∈(0,1)✳}

❘❡s♦❧✉çã♦

◆♦ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦✱ 3x❡ 1₋x sã♦ ♥ú♠❡r♦s r❡❛✐s ♣♦s✐t✐✈♦s✳

P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ▼● ✲ ▼❆ t❡♠♦s

G(3x,1₋x,1₋x,1₋x)_≤A(3x,1₋x,1₋x,1₋x)

⇔ p4

3x(1₋x)3 _≤ 3x+ (1−x) + (1−x) + (1−x) 4

⇔x(1₋x)3 _≤ 1 3

3 4

4

⇔f(x)_≤ 27 256

❖❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱3x= 1₋x✱ ♦✉ s❡❥❛✱x= ₄1✳ ❈♦♠♦ 1_{4 ∈}D(f)✱

❡♥tã♦ ♦ ✈❛❧♦r ♠á①✐♠♦ ❞❛ ❢✉♥çã♦ é 27 256✳

❈❆P❮❚❯▲❖ ✹✳ ❆P▲■❈❆➬Õ❊❙ ✸✺

✾✳ ❉❛❞♦s n > 1 ♥ú♠❡r♦s r❡❛✐s ♥ã♦ ♥✉❧♦s x1, x2, . . . , xn✱ ❛ ♠é❞✐❛ ❤❛r♠ô♥✐❝❛ ❞❡❧❡s é ♦

♥ú♠❡r♦ r❡❛❧

H(x1, x2, . . . , xn) =

n

1

x1 +

1

x2 +· · ·+

1

xn

❙❡❥❛♠ H ❡ G ❛s ♠é❞✐❛s ❤❛r♠ô♥✐❝❛ ❡ ❣❡♦♠étr✐❝❛ ❞❡n r❡❛✐s ♣♦s✐t✐✈♦s x1, x2, . . . , xn✱

♠♦str❡ q✉❡ H _≤G❡ q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ ♦❝♦rr❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x1 =x2 =...=xn✳

❘❡s♦❧✉çã♦

P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ▼● ✲ ▼❆ t❡♠♦s

G 1 x1 , 1 x2

, . . . , 1 xn ≤A 1 x1 , 1 x2

, . . . , 1 xn ⇔ n r 1 x1 1 x2

. . . 1 xn ≤

1

x1 +

1

x2 +. . .+

1

xn n

⇔ _G1 ≤ _H1 ⇔H _≤G

❖❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ 1

x1 =

1

x2 =. . .=

1

xn✱ ♦✉ ❞❡ ♠♦❞♦ ❡q✉✐✈❛❧❡♥t❡✱ x1 =x2 =. . .=xn✳

✶✵✳ ▼♦str❡ q✉❡ ♣❛r❛ q✉❛✐sq✉❡r r❡❛✐s ♣♦s✐t✐✈♦s x1, x2, . . . , xn✱ t❡♠♦s

(x1+x2 +. . .+xn)

1

x1

+ 1

x2

+. . .+ 1

xn

≥n2

❘❡s♦❧✉çã♦ ✶

P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ▼● ✲ ▼❆ ❡ ♣❡❧❛ ♦❜t✐❞❛ ♥♦ ♣r♦❜❧❡♠❛ ❛♥t❡r✐♦r t❡♠♦s

A(x1, x2, . . . , xn)≥G(x1, x2, . . . , xn)≥H(x1, x2, . . . , xn)

⇒ x1+x2+_n. . .+xn ≥ 1 n

x1 +

1

x2 +. . .+

1

xn

⇔(x1+x2+. . .+xn)·

1

x1

+ 1

x2

+. . .+ 1

xn

≥n2