Abstract
Abstract For a ∈ (0, 1/2], r ∈ (0, 1), let 𝒦a (r) (𝒦(r)) be the generalized (complete) elliptic integral of the first kind. In the article, we prove some monotonicity properties of certain combination of functions involving 𝒦a (r), and thus establish its two sharp inequalities, which extend and improve some well-known results of 𝒦(r).
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