Abstract
We generalize several monotonicity and convexity properties as well as sharp inequalities for the complete elliptic integrals to the complete p-elliptic integrals.
Highlights
2 Lemmas In order to prove our main results, we need several formulas and lemmas, which we present
5 Results and discussion In the article, we present the monotonicity and convexity properties and provide the sharp bounds for the functions r
The obtained results are the generalization of the well-known results on the classical complete elliptic integrals given in [32, 33]
Summary
The well-known complete elliptic integrals K(r) and E(r) [12,13,14,15] of the first and second kinds are respectively defined by. The complete p-elliptic integrals Kp(r) and Ep(r) [16, 17] of the first and second kinds are respectively defined by. Inequality (3.1) holds for all r ∈ (0, 1) if and only if α ≤ 0, and β ≥ 2p/[(p – 1)πp] – ((p – 1)πp/(2p) follows from (3.6) and (3.7) together with the monotonicity and convexity of fp(r) on (0, 1). The desired results in Theorem 3.2 follow from (3.11) and (3.16) together with the monotonicity and convexity of gp(r) on the interval (0, 1). Corollary 3.4 follows from Theorem 3.2 and (3.20)
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