AN OPTIMAL INEQUALITIES CHAIN FOR BIVARIATE MEANS

作者：Yang, Zhen Hang^*; Chu, Yu Ming

来源：Journal of Mathematical Inequalities, 2015, 9(2): 331-343.

DOI：10.7153/jmi-09-27

• Logarithmic mean
• first Seiffert mean
• Neuman-Sandor mean
• identric mean
• power-exponential mean
• exponential-geometric mean

Let p is an element of R, M be a bivariate mean, and M-p be defined by M-p(a, b) = M-1/p(a(p), b(p)) (p not equal 0) and M-0(a, b) = lim(p -> 0)M(p)(a, b). In this paper, we prove that the sharp inequalities L-2(a, b) < P(a, b) < NS1/2(a, b) < He(a, b) < A(2/3)(a, b) < I(a, b) < Z(1/3)(a, b) < Y-1/2(a, b) hold for all a, b > 0 with a not equal b, where L(a, b) = (a- b)/(loga - logb), P(a, b) = (a - b)/[2arcsin((a- b)/(a + b))], NS(a, b) =(a- b)/[2arcsinh ((a- b)/(a + b))], He(a, b) =(a + root ab+ b)/3, A(a, b) = (a + b)/2, I(a, b) = 1/e(a(a)/b(b))(1/)((a-b)), Z(a, b) = a(a/(a+b)) b(b/(a+b)) and Y(a, b) = I(a, b) e(1-ab/L2(a, b)) are respectively the logarithmic, first Seiffert, Neuman-Sandor, Heronian, arithmetic, identric, power-exponential and exponential-geometric means of a and b.

• 出版日期2015-6
• 单位湖南城市学院