Approximations for the complete elliptic integral of the second Kind

• Published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Vol. 115; no. 2
• Main Authors: Qian, Wei-Mao, Wang, Miao-Kun, Xu, Hui-Zuo, Chu, Yu-Ming
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2021; Springer Nature B.V
• Subjects: Applications of Mathematics; Integrals; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Original Paper; Theoretical; 26E60; Complete elliptic integrals; Seiffert-like means; 33E05; Bivariate means; Inequality
• ISSN: 1578-7303; 1579-1505
• DOI: 10.1007/s13398-021-01031-5

In this paper, we present the best possible parameters α 1 , α 2 , α 3 , α 4 , β 1 , β 2 , β 3 , β 4 ∈ R such that α 1 H ( x , y ) + 1 - α 1 L ( x , y ) < 1 V ( x , y ) < β 1 H ( x , y ) + 1 - β 1 L ( x , y ) , α 2 H ( x , y ) + 1 - α 2 P ( x , y ) < 1 V ( x , y ) < β 2 H ( x , y ) + 1 - β 2 P ( x , y ) , α 3 H ( x , y ) + 1 - α 3 N S ( x , y ) < 1 V ( x , y ) < β 3 H ( x , y ) + 1 - β 3 N S ( x , y ) , α 4 H ( x , y ) + 1 - α 4 T ( x , y ) < 1 V ( x , y ) < β 4 H ( x , y ) + 1 - β 4 T ( x , y ) hold for all x , y > 0 with x ≠ y , where H ( x ,  y ), G ( x ,  y ), L ( x ,  y ), A ( x ,  y ), NS ( x ,  y ), P ( x ,  y ) and T ( x ,  y ) are respectively the harmonic, geometric, logarithmic, arithmetic, Neuman-Sándor, and first and second Seiffert means of two distinct positive numbers x and y , and V ( x , y ) = π G 2 ( x , y ) / 2 ∫ 0 π / 2 A 2 ( x , y ) cos 2 φ + G 2 ( x , y ) sin 2 φ d φ is a new Seiffert-like mean. As applications, some new inequalities for the complete elliptic integral of the second kind are given.