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I've been reading some papers and found this term "Sharp constant" being used in inequalities frequently. Can anyone provide me a detailed meaning of this term ? I couldn't find proper resources to learn about this.

For example, statements are along the following lines:

$ f(x) \leq C g(x) , x \in \mathbb{R} $ where C is a sharp constant.

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1 Answer

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The term sharp means we can find a best bound, that cannot be improved by a better number. I.e., to say a function is bounded is to say there exists an $M$ such that $|f(x)|\le M$. To say a particular number is sharp, say $|f(x)|\le 3$, means that there is no number smaller than 3 we could put there and have it be true.

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