4
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These two simple expressions fail to simplify (Mma 11.1):

FullSimplify[Power[x, n] < Power[y, n], Assumptions -> 0 < x < y && n >= 1]
FullSimplify[1 < Power[y/x, n], Assumptions -> 0 < x < y && n >= 1]

Logarithmic transformation helps; this works:

Simplify[Log /@ (Power[x, n] < Power[y, n]), Assumptions -> 0 < x < y && n >= 1]

Can I force simplification of the first two?

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    $\begingroup$ FullSimplify[1 < Power[a, n], Assumptions -> a > 1 && n >= 1] doesn't simplify either. No need for 3 variables. $\endgroup$ – user202729 Jul 1 '18 at 11:24
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    $\begingroup$ Or even FullSimplify[1 < Power[a, a], Assumptions -> a > 1]. $\endgroup$ – user202729 Jul 1 '18 at 11:24

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