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King-s-inequality-if-a-b-n-m-then-a-n-b-m-2-1-b-2-m-n-a-b-2-n-




Question Number 160644 by HongKing last updated on 03/Dec/21
King′s inequality  if   a≤b   ;   n≤m   then:  ((a^n  + b^m )/2) ≥ (((1 + b)/2))^(m-n)  ∙ (((a + b)/2))^n
$$\mathrm{King}'\mathrm{s}\:\mathrm{inequality} \\ $$$$\mathrm{if}\:\:\:\mathrm{a}\leqslant\mathrm{b}\:\:\:;\:\:\:\mathrm{n}\leqslant\mathrm{m}\:\:\:\mathrm{then}: \\ $$$$\frac{\mathrm{a}^{\boldsymbol{\mathrm{n}}} \:+\:\mathrm{b}^{\boldsymbol{\mathrm{m}}} }{\mathrm{2}}\:\geqslant\:\left(\frac{\mathrm{1}\:+\:\mathrm{b}}{\mathrm{2}}\right)^{\boldsymbol{\mathrm{m}}-\boldsymbol{\mathrm{n}}} \:\centerdot\:\left(\frac{\mathrm{a}\:+\:\mathrm{b}}{\mathrm{2}}\right)^{\boldsymbol{\mathrm{n}}} \\ $$

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