Citations Previous Up Next Article From References: 2 From Reviews: 2 MR0003430 (2,218e) 27.0X Schmidt, Erhard Über die Ungleichung, welche die Integrale über eine Potenz einer Funktion und über eine andere Potenz ihrer Ableitung verbindet. (German) Math. Ann. 117, (1940). 301–326 Let z(t) be absolutely continuous in 0 ≤ t ≤ λ and have at least one zero there. The author proves that, for any values of a > 0 and b ≥ 1, 1/a ( Z λ Z λ b )1/b dz 1 b−1 1 dt |z|a dt ≤H , λb−1 , dt λ 0 a b 0 ( Z λ Z λ b )1/b dz b−1 1 dt , log |z| dt ≤ log 1/G λb−1 dt λ 0 b 0 where G(u) = eu u−u Γ(1 + u), H(u, v) = G(u + v)/(G(u)G(v)). It is proved that these inequalities remain true also in limiting cases a = ∞, b = ∞, and that they are the best possible. The functions for which the inequality sign must be replaced by the equality sign are completely determined. The proof is based on an elegant reduction of the problem to the consideration of the area of the curve |ξ|a + η b/(b−1) = 1, −1 ≤ ξ ≤ 1; η ≥ 0. Reviewed by J. D. Tamarkin c Copyright American Mathematical Society 1941, 2006
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