Bounds for the Combinations of Neuman-Sándor, Arithmetic, and Second Seiffert Means in terms of Contraharmonic Mean - Zai-Yin He; Wei-Mao Qian; Yun-Liang Jiang; Ying-Qing Song; Yu-Ming Chu (Abstract and Applied Analysis)
We give the greatest values r1, r2 and the least values s1, s2 in (1/2, 1) such that the double inequalities C(r1a+(1-r1)b,r1b+(1-r1)a)<αA(a,b)+(1-α)T(a,b)<C(s1a+(1-s1)b,s1b+(1-s1)a) and C(r2a+(1-r2)b,r2b+(1-r2)a)<αA(a,b)+(1-α)M(a,b)<C(s2a+(1-s2)b,s2b+(1-s2)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.