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関数方程式のあやしい世界

 関数方程式のあやしい世界

関数方程式の闇

Yoriyuki Yamagata

April 13, 2019
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  1. ͸ղ ଞʹղ͸ʁ f(x) + f(y) = f(x + y +

    2f(xy)) f: ℝ≥0 → ℝ≥0 f(x) = 0 and f(x) = x
  2. ิ୊̍ f͕୯ࣹͳΒ͹f(x)=√x f(x) + f(y) + f(1) = f(x +

    y + 1 + 2f(y) + 2f(xy + x + 2xf(y)) f(x) + f(y) + f(1) = f(x + y + 1 + 2f(xy) + 2f(x) + 2f(y)) ূ໌ɿ f͕୯ࣹΑΓ f(xy + x + 2xf(y)) = f(xy) + f(x) = f(xy + x + 2f(x2y)) ∴ f(x2y) = xf(y) ∴ f(x) = f(1) x
  3. ఆཧͷূ໌ ΋͋͠ΔaͰf(a) = 0ͳΒ͹ f(2a) ≤ f(2a + 2f(a2)) =

    2f(a) = 0 f͸ݮগ͠ͳ͍͔Βɺf͸߃౳తʹ̌ɻ ͦ͏Ͱͳ͍ͱ͢Δɻิ୊̎ͷূ໌ͱಉ༷ʹ ೚ҙͷy>xʹ͍ͭͯɺ͋Δt͕͋ͬͯy-x = t + f(tx) f(y) = f(x + t + f(tx)) = f(x) + f(t) > f(x) Αͬͯf͸୯ௐ૿େɻิ୊̍ΑΓ