Let u,v∈RNu,v\in \mathbb{R}^{N}, f:RN→Rf:\mathbb{R}^{N}\rightarrow \mathbb{R} given by f(u)=12uTAu+2∑ni=1cosh(ui)−bTuf(u)=\frac{1}{2}u^{T}Au + 2\sum_{i=1}^{n}cosh(u_{i})-b^{T}u. Show that ff is uniformly convergent on RN\mathbb{R}^{N}. That is, show:

λf(u)+(1−λ)f(v)−f(λu+(1−λ)v)>12λ(1−λ)||u−v||2A\lambda f(u)+(1-\lambda )f(v)-f(\lambda u+(1-\lambda )v) >\frac{1}{2}\lambda (1-\lambda )||u-v||_{A}^{2}

for some 0<λ<10<\lambda <1, where ‖u‖A=uTAu\left \|u \right \|_{A}=u^{T}Au for a symmetric positive definite matrix AA. ================= ================= =================