The discussion revolves around proving an inequality related to a real-valued function that satisfies the functional equation f(x+y) = f(x) . f(y) for all real numbers x and y. The specific inequality to prove is f((x + y) / 2) ≤ 1/2 (f(x) + f(y)).
Participants are exploring various approaches, including the use of AM-GM and the strategy of assuming the result to derive familiar inequalities. There is engagement with different lines of reasoning, but no consensus has been reached on a specific method or solution.
Some participants question the assumptions underlying the problem, particularly regarding the positivity of the function values involved in the AM-GM application.
Buffu said:Use AM-GM, ##{A+B\over 2} \ge \sqrt{AB}## where ##A,B >0##.
IssacNewton said:Songoku, when you are asked to prove something, sometimes, its wise to assume the result which you have been asked to prove. And you have both, the hypothesis and the conclusion. And with it, you try to see where does this lead to. Lot of times, this leads to some other familiar result. And then you can work backwards from that familiar result. This is one of those situations. Here if you assume the result, then using the hypothesis, you reach the result ##(a-b)^2 \geqslant 0##, if you let ##a = f(\frac{x}{2})## and ##b = f(\frac{y}{2})##. But now ##(a-b)^2 \geqslant 0## is a familiar result, which is true for all ##(a-b) \in \mathbb{R}## . Now you can work backwards.