An optimal power mean inequality for the complete elliptic integrals

✍ Scribed by Miao-Kun Wang; Yu-Ming Chu; Ye-Fang Qiu; Song-Liang Qiu

• Book ID: 104001086
• Publisher: Elsevier Science
• Year: 2011
• Tongue: English
• Weight: 214 KB
• Volume: 24
• Category: Article
• ISSN: 0893-9659
• DOI: 10.1016/j.aml.2010.12.044

✦ Synopsis

a b s t r a c t

In this work, we prove that M p (K(r), E (r)) > π /2 for all r ∈ (0, 1) if and only if p ≥ -1/2, where M p (x, y) denotes the power mean of order p of two positive numbers x and y, and K(r) and E (r) denote the complete elliptic integrals of the first and second kinds, respectively.