Geodesic
PROOFS OF THREE OPEN INEQUALITIES WITH POWER-EXPONENTIAL FUNCTIONS
CIRTOAJE, VASILE1
1 Department of Automatic Control and Computers, University of Ploiesti, Romania
Journal of nonlinear sciences and its applications, Tome 4 (2011) no. 2, p. 130-137.

Voir la notice de l'article provenant de la source International Scientific Research Publications

The main aim of this paper is to give a complete proof to the open inequality with power-exponential functions
$a^{ea} + b^{eb} \geq a^{eb} + b^{ea},$
which holds for all positive real numbers a and b. Notice that this inequality was proved in [1] for only $a \geq b \geq \frac{1}{ e}$ and $\frac{1}{ e} \geq a \geq b$.In addition, other two open inequalities with power-exponential functions are proved, and three new conjectures are presented.
DOI : 10.22436/jnsa.004.02.05
Classification : 26D10
Keywords: Open inequality, Power-exponential function, AM-GM inequality.

CIRTOAJE, VASILE 1

1 Department of Automatic Control and Computers, University of Ploiesti, Romania 
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CIRTOAJE, VASILE. PROOFS OF THREE OPEN INEQUALITIES WITH POWER-EXPONENTIAL FUNCTIONS. Journal of nonlinear sciences and its applications, Tome 4 (2011) no. 2, p. 130-137. doi : 10.22436/jnsa.004.02.05. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.004.02.05/

[1] Cirtoaje, V. On some inequalities with power-exponential functions, J. Ineq. Pure Appl. Math., 10 (1), Art. 21, [http://www.emis.de/journals/JIPAM/article 1077.html?sid=1077]., 2009

[2] V. Cirtoaje Art of Problem Solving Forum, [http://www.artofproblemsolving. com/Forum/viewtopic.php?t=327116]. , , 2010

[3] Cirtoaje, V.; A. Zeikii Art of Problem Solving Forum, [http:// www.artofproblemsolving.com/Forum/viewtopic.php?t=118722]., , 2006

[4] Matejicka, L. Solution of one conjecture on inequalities with power-exponential functions, J. Ineq. Pure Appl. Math., 10 (3), [http://www. emis.de/journals/JIPAM/article1128.html?sid=1128]., 2009

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