I'm currently studying functional analysis and in the preliminary chapter the author gives the following inequality:
AM-GM Inequality:
Let $x,y>0$ and $0< \lambda <1$. Then $$x^\lambda y^{1-\lambda} \leq \lambda x+ (1-\lambda)y$$
But I know the AM-GM Inequality of two positive numbers $x$ and $y$ is $$\sqrt{xy} \leq \frac{x+y}{2}$$
But the first version is different. What is the difference between these two versions? Actually the latter one is the special case of the first one, namely, putting $\lambda =1/2$. Its ok.
So, why the author gives that name for the first one? Is there anything special about the first one?
Any help? and thanks in advance
Taking the logarithm, your inequality is equivalent to $$\lambda \ln(x) + (1-\lambda) \ln(y) \leq \ln (\lambda x + (1-\lambda)y)$$
which expresses the fact that $\ln$ is concave.
The classical AM-GM inequality expresses this concavity in the special case where you are considering the middle point between $x$ and $y$. But of course the concavity is true for every convex combination of $x$ and $y$.