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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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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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    $\begingroup$ In a sense, its like the supremum or infimum for our expressions right ? $\endgroup$ – pikachuchameleon Sep 8 '15 at 4:47
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    $\begingroup$ Pretty much so, yes. Generally in math, first some bound is found, and then the goal is to find a sharp bound, i.e. the sup/inf/best. $\endgroup$ – Alan Sep 8 '15 at 4:49
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    $\begingroup$ no problem. If this answered your question, you should click the check mark to accept it :) $\endgroup$ – Alan Sep 8 '15 at 4:55

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