Bounds for the Combinations of Neuman-Sándor, Arithmetic, and Second Seiffert Means in terms of Contraharmonic Mean. (25th March 2013) - He, Zai-Yin; Qian, Wei-Mao; Jiang, Yun-Liang; Song, Ying-Qing; Chu, Yu-Ming
Abstract : We give the greatest values r 1, r 2 and the least values s 1, s 2 in (1/2, 1) such that the double inequalities C ( r 1 a + ( 1 - r 1 ) b, r 1 b + ( 1 - r 1 ) a ) < α A ( a, b ) + ( 1 - α ) T ( a, b ) < C ( s 1 a + ( 1 - s 1 ) b, s 1 b + ( 1 - s 1 ) a ) and C ( r 2 a + ( 1 - r 2 ) b, r 2 b + ( 1 - r 2 ) a ) < α A ( a, b ) + ( 1 - α ) M ( a, b ) < C ( s 2 a + ( 1 - s 2 ) b, s 2 b + ( 1 - s 2 ) a ) hold for any α ∈ ( 0, 1 ) and all a, b > 0 with a ≠ b, where A ( a, b ), M ( a, b ), C ( a, b ), and T ( a, b ) are the arithmetic, Neuman-Sándor, contraharmonic, and second Seiffert means of a and b, respectively.