Question

5. (a) Show that if the functions f and g are log-convex, f+g is also log-convex. Give a counter example to show that this is not true for log-concave functions (Hint: log(f +g)log(elogf +elogs). Show that this is convex by the second-order test for convexity.) (Hint: Use the definition of log-convex functions.) (Note: Harmonic mean of a,b is defined as T^T.) b) Suppose f is convex, g is non-decreasing and log-convex. Show that h(x) g(f(x)) is log-convex. (c) Show that the harmonic mean of two log-concave functions is log-concave

Answer 1

(a) Here f and g are log-convex function As f is log convex So, log (f (x)) is convex. g is log convex as log (g (x)) is convex Now, log (f + g) = log (e^log f + e^log g) Now, we know as log f is convex then e^log f is convex, and as log g is convex then e^log g is convex. Here log (e^log f + e^log g) is convex log f (x) = f_1 (say) log g (x) = f_2 (say) Let h (x) = log (e^f_1 + e^f_2) nabla^2 h (x) = 1/1^T Z diag (2) - 1/1^T Z)^2 Z Z^T Z_1 = e^f_1, Z_2 = e^f_2 To show, nabla^2 h (x) greaterthanorequalto 0. We must verify that v^T nabla^2 h (x) greaterthanorequalto 0 Forall v. v^T nabla^2 h (x) v = (sigma^2_k = 1 z_k v_k^2) (sigma^2_k = 1 z_k) - (sigma^2_k = 1 v_k z_k)^2/(sigma^2_k = 1 z_k)^2 greaterthanorequalto 0