Problem 2. [7% Let 2 C X be an open convex set. A function : 9R++ is logarithmically cones (resp. concave) if a log((x))

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Problem 2 7 Let 2 C X Be An Open Convex Set A Function 9r Is Logarithmically Cones Resp Concave If A Log X 1 (31.16 KiB) Viewed 78 times

Problem 2. [7% Let 2 C X be an open convex set. A function : 9R++ is logarithmically cones (resp. concave) if a log((x)) is convex (resp. concave). Prove that 1. S is logarithmically convex (resp. concave) iff f((1 - 1)2 + Ay) (x) *-*/(x)* (resp. /((1 - 1). + Ay) 25(x) -*/)), for every rye X and every 1 € (0,1). 2. If / is logarithmically convex (resp. concave), then f is convex (resp. concave). (Hint: note that y(x)) is convex if / is convex and y is increasing and convex] 3. By induction on ne N, that, if / is logarithmically convex (resp. concave), then (1) II-1/(x) (resp. 2) for every , e X,1 = 1,...,n and (ISIS. ER such that i = 1. 4. The generalized arithmetic-mean inequality, that is, E, AT: IT, for every € R++ and 11 € (0,1) with 1 = 1. (Hint: use the characterization at point 1. and then point 3.