MathDB

Problem 3

Part of 1997 VJIMC

Problems(2)

f^(k)(0)>0 for k=1,2,...

Source: VJIMC 1997 1.3

9/30/2021
Let c1,c2,…,cnc_1,c_2,\ldots,c_n be real numbers such that c1k+c2k+…+cnk>0for all k=1,2,…c_1^k+c_2^k+\ldots+c_n^k>0\qquad\text{for all }k=1,2,\ldotsLet us put f(x)=1(1−c1x)(1−c2x)⋯(1−cnx).f(x)=\frac1{(1-c_1x)(1-c_2x)\cdots(1-c_nx)}.z∈Cz\in\mathbb C Show that f(k)(0)>0f^{(k)}(0)>0 for all k=1,2,…k=1,2,\ldots.
calculusdifferentiation
integral inequality in R^3

Source: VJIMC 1997 2.3

10/9/2021
Let u∈C2(D‾)u\in C^2(\overline D), u=0u=0 on ∂D\partial D where DD is the open unit ball in R3\mathbb R^3. Prove that the following inequality holds for all ε>0\varepsilon>0: ∫D∣∇u∣2dV≤ε∫D(Δu)2dV+14ε∫Du2dV.\int_D|\nabla u|^2dV\le\varepsilon\int_D(\Delta u)^2dV+\frac1{4\varepsilon}\int_Du^2dV.(We recall that ∇u\nabla u and Δu\Delta u are the gradient and Laplacian, respectively.)
calculusintegrationinequalitiesMultivariable Calculus