Optimal Inequalities between Harmonic, Geometric, Logarithmic, and Arithmetic-Geometric Means

We find the least values p, q, and s in (0, 1/2) such that the inequalities H(pa+(1 − p)b, pb+(1 − p)a)>AG(a,b), G(qa+(1−q)b, qb+(1−q)a)>AG(a,b), and L(sa+(1−s)b,sb+(1−s)a)> AG(a,b) hold for all a,b>0 with a≠b, respectively. Here AG(a,b), H(a,b), G(a,b), and L(a,b) denote the arithmetic-geometric, harmonic, geometric, and logarithmic means of two positive numbers a and b, respectively

Distances between power spectral densities

We present several natural notions of distance between spectral density functions of (discrete-time) random processes. They are motivated by certain filtering problems. First we quantify the degradation of performance of a predictor which is designed for a particular spectral density function and then it is used to predict the values of a random process having a different spectral density. The logarithm of the ratio between the variance of the error, over the corresponding minimal (optimal) variance, produces a measure of distance between the two power spectra with several desirable properties. Analogous quantities based on smoothing problems produce alternative distances and suggest a class of measures based on fractions of generalized means of ratios of power spectral densities. These distance measures endow the manifold of spectral density functions with a (pseudo) Riemannian metric. We pursue one of the possible options for a distance measure, characterize the relevant geodesics, and compute corresponding distances.Comment: 16 pages, 4 figures; revision (July 29, 2006) includes two added section

On two new means of two arguments III

In this paper authors establish the two sided inequalities for the following two new means $$X=X(a,b)=Ae^{G/P-1},\quad Y=Y(a,b)=Ge^{L/A-1}.$$ As well as many other well known inequalities involving the identric mean $I$ and the logarithmic mean are refined from the literature as an application.Comment: 14. arXiv admin note: substantial text overlap with arXiv:1509.0197

Two Inequalities and Two Means

The paper presents geometric derivations of Jensen's and Hermite-Hadamard's inequality.Jensen's inequality is further involved to a concept of quasi-arithmetic means. Hermite-Hadamard's inequality is applied to compare the basic quasi-arithmetic means