Inequalities for Generalized Logarithmic Means

For p∈ℝ, the generalized logarithmic mean Lp of two positive numbers a and b is defined as Lp(a,b)=a, for a=b, LP(a,b)=[(bp+1−ap+1)/(p+1)(b−a)]1/p , for a≠b, p≠−1, p≠0, LP(a,b)=(b−a)/(log⁡b−log⁡a), for a≠b, p=−1, and LP(a,b)=(1/e)(bb/aa)1/(b−a) , for a≠b, p=0. In this paper, we prove that G(a,b)+H(a,b)⩾2L−7/2(a,b),A(a,b)+H(a,b)⩾2L−2(a,b), and L−5(a,b)⩾H(a,b) for all a,b>0, and the constants −7/2,−2, and −5 cannot be improved for the corresponding inequalities. Here A(a,b)=(a+b)/2=L1(a,b),G(a,b)=ab=L−2(a,b), and H(a,b)=2ab/(a+b) denote the arithmetic, geometric, and harmonic means of a and b, respectively