On approximating the quasi-arithmetic mean

• Published in: Journal of inequalities and applications Vol. 2019; no. 1; pp. 1 - 12
• Main Authors: Zhao, Tie-Hong, Zhou, Bu-Chuan, Wang, Miao-Kun, Chu, Yu-Ming
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 20.02.2019; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Applications of Mathematics; Arithmetic; Arithmetic mean; Contra-harmonic mean; Geometric mean; Harmonic mean; Mathematics; Mathematics and Statistics; Quasi-arithmetic mean; Quasi-arithmetic mean; 26E60; Geometric mean; Arithmetic mean; Contra-harmonic mean; 33C05; Harmonic mean
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-019-1991-0

In this article, we prove that the double inequalities α 1 [ 7 C ( a , b ) 16 + 9 H ( a , b ) 16 ] + ( 1 − α 1 ) [ 3 A ( a , b ) 4 + G ( a , b ) 4 ] < E ( a , b ) < β 1 [ 7 C ( a , b ) 16 + 9 H ( a , b ) 16 ] + ( 1 − β 1 ) [ 3 A ( a , b ) 4 + G ( a , b ) 4 ] , [ 7 C ( a , b ) 16 + 9 H ( a , b ) 16 ] α 2 [ 3 A ( a , b ) 4 + G ( a , b ) 4 ] 1 − α 2 < E ( a , b ) < [ 7 C ( a , b ) 16 + 9 H ( a , b ) 16 ] β 2 [ 3 A ( a , b ) 4 + G ( a , b ) 4 ] 1 − β 2 hold for all a , b > 0 with a ≠ b if and only if α 1 ≤ 3 / 16 = 0.1875 , β 1 ≥ 64 / π 2 − 6 = 0.484555 … , α 2 ≤ 3 / 16 = 0.1875 and β 2 ≥ ( 5 log 2 − log 3 − 2 log π ) / ( log 7 − log 6 ) = 0.503817 … , where E ( a , b ) = ( 2 π ∫ 0 π / 2 a cos 2 θ + b sin 2 θ d θ ) 2 , H ( a , b ) = 2 a b / ( a + b ) , G ( a , b ) = a b , A ( a , b ) = ( a + b ) / 2 and C ( a , b ) = ( a 2 + b 2 ) / ( a + b ) are the quasi-arithmetic, harmonic, geometric, arithmetic and contra-harmonic means of a and b , respectively.