Best possible inequalities between generalized logarithmic mean and weighted geometric mean of geometric, square-root, and root-square means

Best Possible Inequalities between Generalized Logarithmic Mean and Classical Means

and Applied Analysis 3 Theorem B. For all positive real numbers a and b with a/ b, we have √ G a, b A a, b < √ L a, b I a, b

Some weighted operator geometric mean inequalities

In this paper, using the extended Holder- -McCarthy inequality, several inequalities involving the α-weighted geometric mean (0<α<1) of two positive operators are established. In particular, it is proved that if A,B,X,Y∈B(H) such that A and B are two positive invertible operators, then for all r ≥1, ‖X^* (A⋕_α B)Y‖^r≤‖〖(X〗^* AX)^r ‖^((1-α)/2) ‖〖(Y〗^* AY)^r ‖^((1-α)/2) ‖〖(X〗^* BX)^r ‖^(α/2) ‖〖(Y...

Best Possible Inequalities among Harmonic, Geometric, Logarithmic and Seiffert Means

In this paper, we find the greatest value α and the least values β , p , q and r in (0,1/2) such that the inequalities L(αa+ (1−α)b,αb+ (1−α)a) < P(a,b) < L(βa + (1− β)b,βb + (1− β)a) , H(pa + (1− p)b, pb + (1− p)a) > G(a,b) , H(qa+ (1− q)b,qb +(1− q)a) > L(a,b) , and G(ra+(1− r)b,rb+(1− r)a) > L(a,b) hold for all a,b > 0 with a = b . Here, H(a,b) , G(a,b) , L(a,b) and P(a,b) denote the harmoni...

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

Improved logarithmic-geometric mean inequality and its application

In this short note, we present a refinement of the logarithmic-geometric mean inequality. As an application of our result, we obtain an operator inequality associated with geometric and logarithmic means.