Sharp bounds for the lemniscatic mean by the one-parameter geometric and quadratic means

• Published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Vol. 116; no. 1
• Main Authors: Xu, Hui-Zuo, Qian, Wei-Mao, Chu, Yu-Ming
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2022; Springer Nature B.V
• Subjects: Applications of Mathematics; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Original Paper; Parameters; Theoretical; One-parameter mean; 26E60; Geometric mean; 33E05; Arc lemniscate function; Lemniscatic mean; Quadratic mean
• ISSN: 1578-7303; 1579-1505
• DOI: 10.1007/s13398-021-01162-9

In the article, we present the best possible parameters α 1 , α 2 , α 3 , α 4 , β 1 , β 2 , β 3 and β 4 on the interval (0, 1) such that the double inequalities G α 1 ( a , b ) < L M GA a , b < G β 1 ( a , b ) , G α 2 ( a , b ) < L M AG a , b < G β 2 ( a , b ) , Q α 3 ( a , b ) < L M AQ a , b < Q β 3 ( a , b ) and Q α 4 ( a , b ) < L M QA a , b < Q β 4 ( a , b ) hold for a , b > 0 with a ≠ b , where G p ( a , b ) and Q p ( a , b ) are respectively the one-parameter geometric and quadratic means, L M GA ( a , b ) , L M AG ( a , b ) , L M AQ ( a , b ) and L M QA ( a , b ) are four lemniscatic means of a and b . As applications, some new bounds for the arc lemniscate functions are given.