Paper's Title:

Necessary and Sufficient Conditions for Cyclic Homogeneous Polynomial Inequalities of Degree Four in Real Variables

Author(s):

Vasile Cirtoaje and Yuanzhe Zhou

Department of Automatic Control and Computers
University of Ploiesti
Romania.
vcirtoaje@upg-ploiesti.ro.
 

High School Affiliated to Wuhan University, China

Abstract:

In this paper, we give two sets of necessary and sufficient conditions that the inequality f₄(x,y,z) ≥ 0 holds for any real numbers x,y,z, where f₄(x,y,z) is a cyclic homogeneous polynomial of degree four. In addition, all equality cases of this inequality are analysed. For the particular case in which f₄(1,1,1)=0, we get the main result in [3]. Several applications are given to show the effectiveness of the proposed methods.

Paper's Title:

On the Inequality

Author(s):

A. Coronel and F. Huancas

Departamento de Ciencias Básicas,
Facultad de Ciencias, Universidad del Bío-Bío, Casilla 447,
Campus Fernando May, Chillán, Chile.

acoronel@roble.fdo-may.ubiobio.cl

Departamento Académico de Matemática,
Facultad de Ciencias Físicas y Matemáticas,
Universidad Nacional Pedro Ruiz Gallo, Juan XIII s/n,
Lambayeque, Perú

fihuanca@gmail.com
 

Abstract:

In this paper we give a complete proof of for all positive real numbers a, b and c. Furthermore, we present another way to prove the statement for

Paper's Title:

The Best Upper Bound for Jensen's Inequality

Author(s):

Vasile Cirtoaje

Department of Automatic Control and Computers
University of Ploiesti
Romania.

vcirtoaje@upg-ploiesti.ro.

Abstract:

In this paper we give the best upper bound for the weighted Jensen's discrete inequality applied to a convex function f defined on a closed interval I in the case when the bound depends on f, I and weights. In addition, we give a simpler expression of the upper bound, which is better than existing similar one.

Paper's Title:

Some Homogeneous Cyclic Inequalities of Three Variables of Degree Three and Four

Author(s):

TETSUYA ANDO

Department of Mathematics and Informatics,
 Chiba University, Chiba 263-8522, JAPAN

ando@math.s.chiba-u.ac.jp

Abstract:

We shall show that the three variable cubic inequality
    t² (a³+b³+c³) + (t⁴-2t)(ab²+bc²+ca²)
            ≥ (2t³-1)(a²b+b²c+c²a) + (3t⁴-6t³+3t²-6t+3)abc
holds for non-negative a, b, c, and for any real number t.
We also show some similar three variable cyclic quartic inequalities.
 

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