Bounds of the logarithmic mean

Sharp Two Parameter Bounds for the Logarithmic Mean and the Arithmetic–geometric Mean of Gauss

For fixed s 1 and t1,t2 ∈ (0,1/2) we prove that the inequalities G(t1a + (1− t1)b,t1b+(1− t1)a)A1−s(a,b) > AG(a,b) and G(t2a+(1− t2)b,t2b+(1− t2)a)A1−s(a,b) > L(a,b) hold for all a,b > 0 with a = b if and only if t1 1/2− √ 2s/(4s) and t2 1/2− √ 6s/(6s) . Here G(a,b) , L(a,b) , A(a,b) and AG(a,b) are the geometric, logarithmic, arithmetic and arithmetic-geometric means of a and b , respectively....

Sharp Bounds by the Generalized Logarithmic Mean for the Geometric Weighted Mean of the Geometric and Harmonic Means

Optimal One–parameter Mean Bounds for the Convex Combination of Arithmetic and Logarithmic Means

We find the greatest value p1 = p1(α) and the least value p2 = p2(α) such that the double inequality Jp1 (a,b) <αA(a,b)+(1−α)L(a,b) < Jp2 (a,b) holds for any α ∈ (0,1) and all a,b > 0 with a = b . Here, A(a,b) , L(a,b) and Jp(a,b) denote the arithmetic, logarithmic and p -th one-parameter means of two positive numbers a and b , respectively. Mathematics subject classification (2010): 26E60.

Complete Monotonicity of Logarithmic Mean

In the article, the logarithmic mean is proved to be completely monotonic and an open problem about the logarithmically complete monotonicity of the extended mean values is posed.

Optimal Lower Generalized Logarithmic Mean Bound for the Seiffert Mean

Ying-Qing Song, Wei-Mao Qian, Yun-Liang Jiang, and Yu-Ming Chu 1 School of Mathematics and Computation Sciences, Hunan City University, Yiyang, Hunan 413000, China 2 School of Distance Education, Huzhou Broadcast and TV University, Huzhou, Zhejiang 313000, China 3 School of Information & Engineering, Huzhou Teachers College, Huzhou, Zhejiang 313000, China Correspondence should be addressed to Y...