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This gives inequality (1.1).
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This gives inequalities (2.14) with inequality sign.
(4.6) Substitution of (g_{k} (t)) in Eq. (4.6) gives inequality (4.2) immediately.
Let (i=n-p-1), (j=n-r-1), then (3.8) gives inequality (1.8) and its equality condition.
Let (i=n-p-1) and (j=n-r-1), then (3.12) gives inequality (1.10) and its equality condition.
Hence, the given inequality is satisfied.
end{aligned}Hence, the given inequality is satisfied.
Otherwise, this point violates one or more of the given inequalities.
It is easy to see that this gives back inequality (2.3) upon letting p → +∞ by a change of variables a n p ↦ a n and setting L = M.
Intuitively, if a given inequality in life expectancy is inevitable, then it is not unjust.
First, it is possible to relate inequality indices to social welfare functions, so as to give inequality indices a more transparent ethical content.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com