Paper's Title:

Boundness of the Power Exponential Function a^2b +b^2a

Author(s):

Yusuke Nishizawa, Yukuhito Yasuda

Faculty of Education, Saitama University,
Shimo-okubo 255, Sakura-ku, Saitama-city, Saitama,
Japan.
E-mail: ynishizawa@mail.saitama-u.ac.jp 

Faculty of Education, Saitama University,
Shimo-okubo 255, Sakura-ku, Saitama-city, Saitama,
Japan.
E-mail: y.yasuda.694@ms.saitama-u.ac.jp
 

Abstract:

In this paper, we consider the boundness of the power exponential function a^2b +b^2a for nonnegative real numbers a and b. The author proved that the function has the maximum value 1 for a + b = ½, but it is no known that the minimum value for a + b = ½. In this paper, we give the new proof of the function has the maximum value 1 and show that a^2b +b^2a > 0.989905 for a + b = ½.

Paper's Title:

Rational Expressions of Arithmetic and Geometric Means for the Sequence n^p_{n ∈ N} and the Geometric Progression

Author(s):

M. Kinegawa, S. Miyamoto and Y. Nishizawa

Faculty of Education, Saitama University,
Shimo-okubo 255, Sakura-ku, Saitama-city, Saitama,
Japan.
E-mail: m.kinegawa.645@ms.saitama-u.ac.jp
 
Faculty of Education, Saitama University,
Shimo-okubo 255, Sakura-ku, Saitama-city, Saitama,
Japan.
E-mail: s.miyamoto.245@ms.saitama-u.ac.jp

Faculty of Education, Saitama University,
Shimo-okubo 255, Sakura-ku, Saitama-city, Saitama,
Japan.
E-mail: ynishizawa@mail.saitama-u.ac.jp

Abstract:

In this paper, we consider the arithmetic and geometric means for the sequence n^p_{n ∈ N} and the geometric progression. We obtain the results associated with the rational expressions of the means.

Paper's Title:

Three Inequalities Associated with Rado Inequality

Author(s):

Rin Miyao, Yusuke Nishizawa, Keigo Takamura

Faculty of Education,
Saitama University,
Shimo-okubo 255, Sakura-ku, Saitama-city, Saitama,
Japan.
E-mail: r.miyao.242@ms.saitama-u.ac.jp

Faculty of Education,
Saitama University,
Shimo-okubo 255, Sakura-ku, Saitama-city, Saitama,
Japan.
E-mail: ynishizawa@mail.saitama-u.ac.jp

Faculty of Education,
Saitama University,
Shimo-okubo 255, Sakura-ku, Saitama-city, Saitama,
Japan.
E-mail: k.takamura.442@ms.saitama-u.ac.jp

Abstract:

In this short note we estimate three inequalities associated with Rado inequality and show the refinement and reverse of Arithmetic mean- Geometric mean inequality.

Paper's Title:

A Short Proof of an Open Inequality with Power-Exponential Functions

Author(s):

Mitsuhiro Miyagi and Yusuke Nishizawa

General Education, Ube National College of Technology,
Tokiwadai 2-14-1, Ube,
Yamaguchi 755-8555,
Japan

E-mail: miyagi@ube-k.ac.jp   
yusuke@ube-k.ac.jp  

Abstract:

V. Cīrtoaje conjectured that a^3b + b^3a + ( (a -b)/2 )⁴ ≤ 2 holds for all nonnegative numbers a and b with a +b =2. In this short note, we give a proof of the Cīrtoaje's conjecture with power-exponential functions.

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