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By (6), we know that the new variable ξ is strictly increasing with respect to x and y, respectively.
Then T is increasing with respect to (xin P).
(i) First, they are monotonically increasing with respect to.
We proved that is monotonically increasing with respect to and.
Therefore, γ is monotone increasing with respect to α < 1.
Since, it is clear that is increasing with respect to for.
(ii) Since, it is clear that is increasing with respect to for. .
The function (a ( x,y ) ) is monotonically increasing with respect to the variable x.
It is well known that is continuous and increasing with respect to for fixed and.
In addition, (B_{M}) is increasing with respect to (q_{M}).
By noting that, it implies that is increasing with respect to.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com