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I don't think the question has a meaningful answer unless the OP specifies a class of inequalities he has in mind. The problem is that almost any mathematical statement can be restated as an inequality.

Take, for instance, the fundamental theorem of algebra. It is equivalent to the inequality
"the number of roots of a non-constant polynomial with complex coefficients is greater than zero". Over ten different proofs of this inequality are discussed in this threadthis thread. It seems that none of them has anything to do with positivity, convexity or entropy arguments.

I don't think the question has a meaningful answer unless the OP specifies a class of inequalities he has in mind. The problem is that almost any mathematical statement can be restated as an inequality.

Take, for instance, the fundamental theorem of algebra. It is equivalent to the inequality
"the number of roots of a non-constant polynomial with complex coefficients is greater than zero". Over ten different proofs of this inequality are discussed in this thread. It seems that none of them has anything to do with positivity, convexity or entropy arguments.

I don't think the question has a meaningful answer unless the OP specifies a class of inequalities he has in mind. The problem is that almost any mathematical statement can be restated as an inequality.

Take, for instance, the fundamental theorem of algebra. It is equivalent to the inequality
"the number of roots of a non-constant polynomial with complex coefficients is greater than zero". Over ten different proofs of this inequality are discussed in this thread. It seems that none of them has anything to do with positivity, convexity or entropy arguments.

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Andrey Rekalo
Andrey Rekalo
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I don't think the question has a meaningful answer unless the OP specifies a class of inequalities he has in mind. The problem is that almost any mathematical statement can be restated as an inequality.

Take, for instance, the fundamental theorem of algebra. It is equivalent to the inequality
"the number of roots of a non-constant polynomial with complex coefficients is greater than zero". Over ten different proofs of this inequality are discussed in this thread. It seems that none of them has anything to do with positivity, convexity or entropy arguments.