A stronger inequality of Cîrtoaje's one with power exponential functions

Volume 8, Issue 3, pp 224--230 http://dx.doi.org/10.22436/jnsa.008.03.06

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Authors

Mitsuhiro Miyagi - General Education, Ube National College of Technology, Tokiwadai 2-14-1, Ube, Yamaguchi 755-8555, Japan. Yusuke Nishizawa - General Education, Ube National College of Technology, Tokiwadai 2-14-1, Ube, Yamaguchi 755-8555, Japan.

Abstract

In this paper, we will show that \(a^{2b} + b^{2a} + r (ab(a - b))^2 \leq 1 \) holds for all \(0 \leq a\) and \(0 \leq b\) with \(a + b = 1\) and all \(0 \leq r \leq\frac{1}{2}\). This gives the first example of a stronger inequality of \(a^{2b} +b^{2a} \leq 1\).

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Keywords

• Power-exponential function
• monotonically decreasing function
• monotonically increasing function.

MSC

•  26D10

References

• [1] A. Coronel, F. Huancas , On the inequality \(a^{2a} +b^{2b} +c^{2c} \geq a^{2b} +b^{2c} +c^{2a}\) , Aust. J. Math. Anal. Appl., 9 (2012), 5 pages.


• [2] V. Cîrtoaje, On some inequalities with power-exponential functions, JIPAM. J. Inequal. Pure Appl. Math., 10 (2009), 6 pages.


• [3] V. Cîrtoaje, Proofs of three open inequalities with power-exponential functions, J. Nonlinear Sci. Appl., 4 (2011), 130-137.


• [4] L. Matejicka, Proof of one open inequality, J. Nonlinear Sci. Appl., 7 (2014), 51-62.


• [5] M. Miyagi, Y. Nishizawa, Proof of an open inequality with double power-exponential functions, J. Inequal. Appl., 2013 (2013), 11 pages.


• [6] M. Miyagi, Y. Nishizawa, A short proof of an open inequality with power-exponential functions, Aust. J. Math. Anal. Appl., 11 (2014), 3 pages.