HOSSAIN, SARMAD
(2023)
Geometric and Calculus Proofs of Some Inequalities.
At Right Angles.
pp. 99102.
ISSN 25821873
Abstract
There are many visual and calculusbased proofs of πe < eπ and ba < ab(for e ≤ a < b). The aim of this paper is to give geometric and calculusbased proofs of these inequalities. We show in addition that ba > ab for 0 < a < b ≤ e and ab ≤ 1 < ba for 0 < a ≤ 1 < b.
To show πe < eπ , Nelson ([2]) uses the fact that the curve y = ex/e lies above the line y = x, while Nakhli ([1]) uses the fact that the curve y = ln x x attains the global maximum at the point e.
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