Let 2 C X be an open convex set. A function f: R++ is logarithmically...

 

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Let 2 C X be an open convex set. A function f: → R++ is logarithmically log(f(r)) is convex (resp. concave). Prove that conver (resp. concave) if 1. f is logarithmically convex (resp. concave) iff f((1-A)r+Ay) ≤ f(x)-f(y) (resp. f((1- A) + Ay) 2 f(x)-f(y)), for every r.ye X and every A = [0, 1]. 2. If f is logarithmically convex (resp. concave), then f is convex (resp. concave). [Hint: note that p(f(a)) is convex if f is convex and is increasing and convex] 3. By induction on ne N, that, if f is logarithmically convex (resp. concave), then f(x) II f(x) (resp. 2) for every r, X, i = 1,...,n and (A)sign ER such that 14 = 1. II, for every r ER++ 4. The generalized arithmetic-mean inequality, that is, and A, € [0, 1] with = 1. (Hint: use the characterization at point 1. and then point 3.] Let 2 C X be an open convex set. A function f: → R++ is logarithmically log(f(r)) is convex (resp. concave). Prove that conver (resp. concave) if 1. f is logarithmically convex (resp. concave) iff f((1-A)r+Ay) ≤ f(x)-f(y) (resp. f((1- A) + Ay) 2 f(x)-f(y)), for every r.ye X and every A = [0, 1]. 2. If f is logarithmically convex (resp. concave), then f is convex (resp. concave). [Hint: note that p(f(a)) is convex if f is convex and is increasing and convex] 3. By induction on ne N, that, if f is logarithmically convex (resp. concave), then f(x) II f(x) (resp. 2) for every r, X, i = 1,...,n and (A)sign ER such that 14 = 1. II, for every r ER++ 4. The generalized arithmetic-mean inequality, that is, and A, € [0, 1] with = 1. (Hint: use the characterization at point 1. and then point 3.]

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ANSWER 1 If f is logarithmically convex then f1 A Ay fxu for every x y X and every A 01 2 If f is lo... View the full answer