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In particular, we conclude that this proofs give affirmatively answers to three, until now, open problems (conjectures~4.4, 2.1 and 2.2) posed by C{\\^i}rtoaje in the following two works:\n  \"{\\it J. Inequal. Pure Appl. Math.} 10, Article 21, 2009\" and \"{\\it J. Nonlinear Sci. Appl.} 4:2:130-137, 2011\". Moreover, we present a new proof of the inequality $a^{ra}+b^{rb}\\ge a^{rb}+b^{ra}$ for all positive real numbers $a$ and $b$ and $r\\in [0,e]$. 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