It is the weighted arithmetic mean - geometric mean inequality, and can be proved using Jensen's Inequality. See here.mertcan said:Hi, in this link https://math.stackexchange.com/questions/2605752/proof-log-convex-implies-convexity , there is a theorem that labelled with yellow. I do not know the name of theorem and would like to know it's derivation. Could you please help me?
Niharika said:Yeah, sure it is a weighted arithmetic mean of the functions involved. If you want its detailed explanation then prefer NCERT books.
The Proof of Log Convexity Theorem is a mathematical proof that shows the relationship between the logarithm of a function and its convexity. It states that if a function f(x) is log-convex, then its logarithm log(f(x)) is a convex function.
The Proof of Log Convexity Theorem is important because it allows us to determine the convexity of a function without having to directly analyze its second derivative. This makes it a useful tool in optimization and economics, where convexity is a desirable property for many functions.
The Proof of Log Convexity Theorem is derived using the definition of convexity, which states that a function is convex if its graph lies above any of its tangent lines. By taking the logarithm of both sides of this definition and using the properties of logarithms, we can show that a log-convex function satisfies this definition.
The Proof of Log Convexity Theorem has many applications in economics, such as in utility theory where it is used to show that the logarithm of a utility function is convex. It is also used in finance to analyze the convexity of option pricing models. Additionally, the theorem has applications in optimization problems, where convexity is a desirable property for objective functions.
One limitation of the Proof of Log Convexity Theorem is that it only applies to functions that are log-convex. It does not provide information about the convexity of functions that are not log-convex. Additionally, the theorem assumes that the function is differentiable, which may not always be the case in real-world applications.