In this paper we will extend the well-known chain of inequalities involving the Pythagorean means, namely the harmonic, geometric, and arithmetic means to the more refined chain of inequalities by including the logarithmic and identric means using nothing more than basic calculus. Of course, these results are all well-known and several proofs of them and their generalizations have been given. See [1-6] for more information. Our goal here is to present a unified approach and give the proofs as corollaries of one basic theorem.

Abstract:
In this
paper, we calculate four different kinds of means—AM,
GM, HM, and GDM—to investigate the risk-return contour using Markowitz risk
minimization and Sharpe’s angle maximization models. For a given ？value (target portfolio return), the rank order of risk or
variance-covariance (υ) can change.
In the vertical segment of an efficient frontier curve, we observed v(GDM) >
v(HM) > v(GM) > v(AM). At higher k？values, the rank changes to v(GDM) > v(HM) > v(AM) > v(GM). That
is to say, ranking a portfolio using different kinds of means may well give
different rankings depending on what k value one is evaluating. It is also shown the harmonic mean should not be used
in the case of a small negative growth rate in stock prices.

Abstract:
A polynomial y = p(x) is continuous and differentiable on its domain R. Therefore, at any closed interval, the graph attains both the maximum and minimum values in the stationary points or the borders of the interval. The method commonly used to find the extremum is Calculus by using derivatives. This paper presents a method for finding the extremum of certain polynomials using inequalities of the mean based on de Alwis's work.

Abstract:
The geometric mean operator is defined by A precise two-sided estimate of the norm for , is given and some applications and extensions are pointed out.

Abstract:
A new refinement of the classical arithmetic mean and geometric mean inequality is given. Moreover, a new interpretation of the classical mean is given and this refinement theorem is generalized.

Abstract:
this study aimed to evaluate the effect of time since the adoption of the no-till system, in comparison with a native forest area and a conventional tillage area, using the distribution of soil aggregates in a distroferric red nitosol. treatments were as follows: native forest (nf), conventional tillage (ct), no-till for one year (nt1), no-till for four years (nt4), no-till for five years (nt5), and no-till for 12 years (nt12). aggregate samples were collected randomly within each treatment at depths of 0-5 and 10-15 cm. after sifting the aggregates in water they were separated into the following aggregate classes > 2 mm; < 2 mm; 2-1 mm, and < 1 mm. the adoption time in the no-till system favored soil aggregation. the mean weighted diameter (mwd) of the soil aggregates and the percentage of aggregates greater than 2 mm increased with adoption time in the no-till system at the 0-5 cm depth. the nf and nt12 treatments had higher mwd values in the 0-5 cm layer. ct had the highest percentage of aggregates smaller than 1 mm.

Abstract:
the experiment was conducted to evaluate the effect of different corn particle size, expressed as geometric mean diameter (gmd)(0.336 mm, 0.585mm, 0.856 mm and 1.12 mm) and two diet forms (mash-m and pellets-p) on performance and carcass yield of broilers from 21 to 42 days of age. m diets, produced with 0.336 mm of gmd resulted in lower feed intake (fi) (p<0.001), lower weight gain (wg) (p<0.001) and worse feed efficiency (fe) (p<0.001) than 0.336 mm p diets. m and p diets with other gmd did not show differences in performance. when particle size was evaluated itself, increments in gmd resulted a linear increase on wg and a quadratic increase on fi and fe. neither form of diet nor particle size influenced carcass and leg+drumstick yields, although breast yield decreased with m diet,0.336 mm gmd (p<0.001).

Abstract:
This paper proposes a new technique to measure the geometric mean diameter (GMD) of selected fruits and vegetables calculated from a three-dimensional (3D) image by computer vision system (CVS). From a single view of the image data a linear laser light projects onto the top of the sample through the center in order to mark the measurement points. The planar metrology and the measurement between planes are employed to calculate the width and height of the samples. Homographytransformation and cross ratio are the mathematical parameter applied to calibrate the image data to real world distance (in metric system). GMD of sample can be calculated from a single view of the image with this technique. The percentage of error of GMD obtained from CVS compared with GMD measurements using vernier calipers is approximately 0.03-5.14 depending on the shape of the objects. However, it can be concluded that this technique is worthwhile for measuring GMD of symmetrical objects.