\u039f \u0391\u03c1\u03b9\u03b8\u03bc\u03b7\u03c4\u03b9\u03ba\u03cc\u03c2 \u03bc\u03ad\u03c3\u03bf\u03c2 \u03b4\u03cd\u03bf \u03ae \u03c0\u03b5\u03c1\u03b9\u03c3\u03c3\u03bf\u03c4\u03ad\u03c1\u03c9\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03ce\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf \u03ac\u03b8\u03c1\u03bf\u03b9\u03c3\u03bc\u03ac \u03c4\u03bf\u03c5\u03c2 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03bf \u03c0\u03bb\u03ae\u03b8\u03bf\u03c2 \u03c4\u03bf\u03c5\u03c2: \\( \\frac{a+b}{2} , \\frac{a+b+c}{3},\\ldots , \\frac{a_1 + a_2 + \\cdots + a_n}{n}\\)<\/p>\n\n\n\n
\u03b5\u03bd\u03ce \u03bf \u03b3\u03b5\u03c9\u03bc\u03b5\u03c4\u03c1\u03b9\u03ba\u03cc\u03c2 \u03bc\u03ad\u03c3\u03bf\u03c2 \u03b1\u03c6\u03bf\u03c1\u03ac \u03c3\u03c4\u03bf \u03b3\u03b9\u03bd\u03cc\u03bc\u03b5\u03bd\u03cc \u03c4\u03bf\u03c5\u03c2: \\( \\sqrt{ab}, \\sqrt[3]{a\\cdot b \\cdot c}, \\ldots , \\sqrt[n]{a_1 \\cdot a_2 \\cdots a_n}\\)<\/p>\n\n\n\n
\u03cc\u03c3\u03bf \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf \u03c0\u03bb\u03ae\u03b8\u03bf\u03c2 \u03c4\u03c9\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03ce\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 \u03ba\u03b1\u03b9 \u03b7 \u03c4\u03ac\u03be\u03b7 \u03c4\u03b7\u03c2 \u03c1\u03af\u03b6\u03b1\u03c2 \u03ba\u03b1\u03b9 \u03c0\u03c1\u03bf\u03c6\u03b1\u03bd\u03ce\u03c2 \u03b8\u03b1 \u03bf\u03c1\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03b3\u03b9\u03b1 \u03bc\u03b7 \u03b1\u03c1\u03bd\u03b7\u03c4\u03b9\u03ba\u03bf\u03cd\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03cd\u03c2.<\/p>\n\n\n\n
\u03a3\u03c5\u03bd\u03b4\u03ad\u03bf\u03bd\u03c4\u03b1\u03b9 \u03bc\u03b5\u03c4\u03b1\u03be\u03cd \u03c4\u03bf\u03c5\u03c2 \u03bc\u03b5 \u03c4\u03b7\u03bd \u03b1\u03bd\u03b9\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1: \\(\\sqrt{ab} \\leq \\frac{a+b}{2}\\)<\/p>\n\n\n\n
\u03ba\u03b1\u03b9 \u03b3\u03b5\u03bd\u03b9\u03ba\u03b5\u03cd\u03bf\u03bd\u03c4\u03b1\u03c2 \u03b3\u03b9\u03b1 n \u03c4\u03bf \u03c0\u03bb\u03ae\u03b8\u03bf\u03c2 \u03cc\u03c1\u03bf\u03c5\u03c2 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5: \\(\\sqrt[n]{a_1 \\cdot a_2 \\cdots a_n} \\leq \\frac{a_1 + a_2 +\\cdots + a_n}{n}\\)<\/p>\n\n\n\n
<\/p>\n\n\n\n
\u0388\u03c7\u03bf\u03c5\u03bc\u03b5 \u03ae\u03b4\u03b7 \u03b4\u03b5\u03b9 \u03b1\u03c0\u03cc\u03b4\u03b5\u03b9\u03be\u03b7 \u03b3\u03b9\u03b1 \u03b4\u03cd\u03bf \u03cc\u03c1\u03bf\u03c5\u03c2 \u03c3\u03c4\u03b1 \u03c0\u03c1\u03bf\u03b7\u03b3\u03bf\u03cd\u03bc\u03b5\u03bd\u03b1, \u03b7 \u03bf\u03c0\u03bf\u03af\u03b1 \u03b1\u03bd\u03ac\u03b3\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03b7 \u03b2\u03b1\u03c3\u03b9\u03ba\u03ae \u03b9\u03b4\u03b9\u03cc\u03c4\u03b7\u03c4\u03b1 \u03c4\u03c9\u03bd \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ce\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03ce\u03bd \\(a^2 \\geq 0 \\) .<\/p>\n\n\n\n
\u03a3\u03c5\u03b3\u03ba\u03b5\u03ba\u03c1\u03b9\u03bc\u03ad\u03bd\u03b1 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 \\( a,b \\geq 0\\) \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: \\(\\sqrt{ab} \\leq \\frac{a+b}{2} \\Leftrightarrow 2 \\sqrt{ab} \\leq a+b \\Leftrightarrow 4ab \\leq a^2 + b^2 + 2ab \\Leftrightarrow a^2 + b^2 – 2ab \\geq 0 \\Leftrightarrow (a-b)^2 \\geq 0\\)<\/p>\n\n\n\n
\u0398\u03b1 \u03c0\u03c1\u03bf\u03c3\u03c0\u03b1\u03b8\u03ae\u03c3\u03bf\u03c5\u03bc\u03b5 \u03bd\u03b1 \u03b5\u03c6\u03b1\u03c1\u03bc\u03cc\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b7\u03bd \u03b1\u03bd\u03b9\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1 \u0391\u03c1\u03b9\u03b8\u03bc\u03b7\u03c4\u03b9\u03ba\u03bf\u03cd – \u0393\u03b5\u03c9\u03bc\u03b5\u03c4\u03c1\u03b9\u03ba\u03bf\u03cd \u03bc\u03ad\u03c3\u03bf\u03c5 \u03b3\u03b9\u03b1 \u03c4\u03b7\u03bd \u03b1\u03c0\u03cc\u03b4\u03b5\u03b9\u03be\u03b7.<\/p>\n\n\n\n
\u038c\u03c0\u03c9\u03c2 \u03c3\u03b5 \u03cc\u03bb\u03b5\u03c2 \u03c4\u03b9\u03c2 \u03b1\u03c0\u03bf\u03b4\u03b5\u03af\u03be\u03b5\u03b9\u03c2 \u03b1\u03bd\u03b9\u03c3\u03bf\u03c4\u03ae\u03c4\u03c9\u03bd \u03bc\u03b5 \u03c7\u03c1\u03ae\u03c3\u03b7 \u03ac\u03bb\u03bb\u03c9\u03bd \u03b3\u03bd\u03c9\u03c3\u03c4\u03ce\u03bd \u03b1\u03bd\u03b9\u03c3\u03bf\u03c4\u03ae\u03c4\u03c9\u03bd \u03b7 \u03b4\u03cd\u03c3\u03ba\u03bf\u03bb\u03b7 \u03b4\u03b9\u03b5\u03c1\u03b3\u03b1\u03c3\u03af\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bd\u03b1 \u03b2\u03c1\u03b5\u03b8\u03b5\u03af \u03c0\u03ce\u03c2 \u03b8\u03b1 \u03b5\u03c6\u03b1\u03c1\u03bc\u03bf\u03c3\u03c4\u03b5\u03af \u03b1\u03c5\u03c4\u03ae. \u0393\u03b9’ \u03b1\u03c5\u03c4\u03cc \u03c4\u03bf \u03bb\u03cc\u03b3\u03bf \u03c0\u03c1\u03bf\u03c3\u03c0\u03b1\u03b8\u03bf\u03cd\u03bc\u03b5 \u03bd\u03b1 \u03c3\u03c5\u03b3\u03ba\u03c1\u03af\u03bd\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b9\u03c2 \u03b4\u03cd\u03bf \u03b1\u03bd\u03b9\u03c3\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2 \u03b3\u03b9\u03b1 \u03bb\u03af\u03b3\u03bf\u03c5\u03c2 \u03cc\u03c1\u03bf\u03c5\u03c2 \u03b1\u03c1\u03c7\u03b9\u03ba\u03ac \u03ba\u03b1\u03b9 \u03ad\u03c0\u03b5\u03b9\u03c4\u03b1 \u03bd\u03b1 \u03ba\u03ac\u03bd\u03bf\u03c5\u03bc\u03b5 \u03b3\u03b5\u03bd\u03b9\u03ba\u03ae \u03b5\u03c6\u03b1\u03c1\u03bc\u03bf\u03b3\u03ae.<\/p>\n\n\n\n
\u0398\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 \\(A = \\sqrt{a_1 ^2 + a_2 ^2}, B = \\sqrt{b_1 ^2 + b_2 ^2 }\\) \u03ba\u03b1\u03b9 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b7 \u03b6\u03b7\u03c4\u03bf\u03cd\u03bc\u03b5\u03bd\u03b7 \u03bd\u03b1 \u03b3\u03c1\u03ac\u03c6\u03b5\u03c4\u03b1\u03b9 \u03b4\u03b9\u03b1\u03b4\u03bf\u03c7\u03b9\u03ba\u03ac: \\( (a_1 b_1 + a_2 b_2 )^2 \\leq (a_1 ^2 + a_2 ^2) \\cdot (b_1 ^2 + b_2 ^2 ) \\Leftrightarrow (a_1 b_1 + a_2 b_2 ) \\leq \\sqrt{(a_1 ^2 + a_2 ^2)} \\cdot \\sqrt{(b_1 ^2 + b_2 ^2 )} \\Leftrightarrow (a_1 b_1 + a_2 b_2 ) \\leq A \\cdot B \\)<\/p>\n\n\n\n
\u038c\u03bc\u03c9\u03c2 \\(a_1 b_1 = \\sqrt{a_1 ^2 \\cdot b_1 ^2 }\\)<\/p>\n\n\n\n
\u03ba\u03b1\u03b9 \u03b1\u03c0\u03bf \u03b1\u03bd\u03b9\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1 \u0391\u03c1\u03b9\u03b8\u03bc\u03b7\u03c4\u03b9\u03ba\u03bf\u03cd – \u0393\u03b5\u03c9\u03bc\u03b5\u03c4\u03c1\u03b9\u03ba\u03bf\u03cd \u03bc\u03ad\u03c3\u03bf\u03c5 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5: \\(a_1 b_1 = \\sqrt{a_1 ^2 \\cdot b_1 ^2 } \\leq \\frac{a_1 ^2 + b_1 ^2}{2}\\)<\/p>\n\n\n\n
\u03ba\u03b1\u03b9 \\( a_2 b_2 = \\sqrt{a_2 ^2 \\cdot b_2 ^2 } \\leq \\frac{a_2 ^2 + b_2 ^2}{2}\\)
\u03bf\u03c0\u03cc\u03c4\u03b5: \\( \\frac{a_1 b_1 + a_2 b_2}{A \\cdot B} \\leq \\frac{1}{2} \\left( \\frac{a_1 ^2 }{A^2} + \\frac{b_1 ^2 }{B^2}\\right) +\\frac{1}{2} \\left( \\frac{a_2 ^2 }{A^2} + \\frac{b_2 ^2 }{B^2}\\right) = \\frac{1}{2} \\left( \\frac{a_1 ^2 + a_2 ^2}{A^2} + \\frac{b_1 ^2 + b_2 ^2}{B^2} \\right) = \\frac{1}{2}(1+1) = 1 \\Leftrightarrow \\)<\/p>\n\n\n\n
\u03b4\u03b7\u03bb\u03b1\u03b4\u03ae: \\( \\frac{a_1 b_1 + a_2 b_2}{A\\cdot B} \\leq 1 \\Leftrightarrow a_1 b_1 + a_2 b_2 \\leq A\\cdot B = \\sqrt{a_1 ^2 + a_2 ^2}\\sqrt{b_1 ^2 + b_2 ^2}\\)<\/p>\n\n\n\n
\u0397 \u03bf\u03c0\u03bf\u03af\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b7 \u03b6\u03b7\u03c4\u03bf\u03cd\u03bc\u03b5\u03bd\u03b7.
\u0397 \u03b1\u03c0\u03cc\u03b4\u03b5\u03b9\u03be\u03b7 \u03b3\u03b9\u03b1 n \u03cc\u03c1\u03bf\u03c5\u03c2 \u03bc\u03c0\u03bf\u03c1\u03b5\u03af \u03bd\u03b1 \u03b3\u03af\u03bd\u03b5\u03b9 \u03b5\u03cd\u03ba\u03bf\u03bb\u03b1 \u03bc\u03b5 \u03bc\u03af\u03b1 \u03c4\u03c1\u03bf\u03c0\u03bf\u03c0\u03bf\u03af\u03b7\u03c3\u03b7, \u03b8\u03ad\u03c4\u03bf\u03bd\u03c4\u03b1\u03c2 \\(A = \\sqrt{a_1 ^2 + a_2 ^2 + \\cdots a_n ^2},\\ B = \\sqrt{b_1 ^2 + b_2 ^2 + \\cdots + b_n ^2}\\)<\/p>\n\n\n\n
\u0397 \u03b1\u03bd\u03b9\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1 \u0391\u03c1\u03b9\u03b8\u03bc\u03b7\u03c4\u03b9\u03ba\u03bf\u03cd – \u0393\u03b5\u03c9\u03bc\u03b5\u03c4\u03c1\u03b9\u03ba\u03bf\u03cd \u03bc\u03ad\u03c3\u03bf\u03c5 \u039f \u0391\u03c1\u03b9\u03b8\u03bc\u03b7\u03c4\u03b9\u03ba\u03cc\u03c2 \u03bc\u03ad\u03c3\u03bf\u03c2 \u03b4\u03cd\u03bf \u03ae \u03c0\u03b5\u03c1\u03b9\u03c3\u03c3\u03bf\u03c4\u03ad\u03c1\u03c9\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03ce\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf \u03ac\u03b8\u03c1\u03bf\u03b9\u03c3\u03bc\u03ac \u03c4\u03bf\u03c5\u03c2 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03bf \u03c0\u03bb\u03ae\u03b8\u03bf\u03c2 \u03c4\u03bf\u03c5\u03c2: \\( \\frac{a+b}{2} , \\frac{a+b+c}{3},\\ldots , \\frac{a_1 + a_2 + \\cdots + a_n}{n}\\) \u03b5\u03bd\u03ce \u03bf \u03b3\u03b5\u03c9\u03bc\u03b5\u03c4\u03c1\u03b9\u03ba\u03cc\u03c2 \u03bc\u03ad\u03c3\u03bf\u03c2 \u03b1\u03c6\u03bf\u03c1\u03ac \u03c3\u03c4\u03bf \u03b3\u03b9\u03bd\u03cc\u03bc\u03b5\u03bd\u03cc \u03c4\u03bf\u03c5\u03c2: \\( \\sqrt{ab}, \\sqrt[3]{a\\cdot b \\cdot c}, \\ldots , \\sqrt[n]{a_1 \\cdot a_2 \\cdots a_n}\\) \u03cc\u03c3\u03bf … \u0394\u03b9\u03b1\u03b2\u03ac\u03c3\u03c4\u03b5 \u03c0\u03b5\u03c1\u03b9\u03c3\u03c3\u03cc\u03c4\u03b5\u03c1\u03b1 “\u0397 \u03b1\u03bd\u03b9\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1 Cauchy-Schwarz \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03b1\u03bd\u03b9\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1 \u0391\u03c1\u03b9\u03b8\u03bc\u03b7\u03c4\u03b9\u03ba\u03bf\u03cd – \u0393\u03b5\u03c9\u03bc\u03b5\u03c4\u03c1\u03b9\u03ba\u03bf\u03cd \u03bc\u03ad\u03c3\u03bf\u03c5”<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[3],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/posts\/5541"}],"collection":[{"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/comments?post=5541"}],"version-history":[{"count":4,"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/posts\/5541\/revisions"}],"predecessor-version":[{"id":5545,"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/posts\/5541\/revisions\/5545"}],"wp:attachment":[{"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/media?parent=5541"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/categories?post=5541"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/tags?post=5541"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
*Lohwater, Arthur (1982), ”Introduction to Inequalities”, Online e-book in PDF fomat.
*Wu H.H., Wu S., Various proofs of the Cauchy-Schwarz inequality, [\u03b5\u03b4\u03ce<\/a>].<\/p>\n\n\n\n<\/h2>\n","protected":false},"excerpt":{"rendered":"