Nagaraja, K. M.; Lokesha, V.; Spadmanabhan, S. A simple proof on strengthening and extension of inequalities. (English) Zbl 1173.26329 Adv. Stud. Contemp. Math., Kyungshang 17, No. 1, 97-103 (2008). The main result of the paper is that among the arithmetic mean \(A,\) the power exponential mean \(Z,\) and the contraharmonic mean \(C,\) we have the inequalities: \(A\leq Z\leq C\) thus, if \(a,b>0,\) then \[ \frac{a+b}{2}\leq a^{\frac{a}{a+b}}b^{\frac{b}{a+b}}\leq \frac{a^{2}+b^{2}}{ a+b}. \] Similar inequalities for means in \(n\) variables are also proved. Reviewer: Gheorghe Toader (Cluj-Napoca) Cited in 1 ReviewCited in 2 Documents MSC: 26E60 Means 26D10 Inequalities involving derivatives and differential and integral operators 26D15 Inequalities for sums, series and integrals Keywords:arithmetic mean; power exponential mean; contra harmonic mean; inequality PDFBibTeX XMLCite \textit{K. M. Nagaraja} et al., Adv. Stud. Contemp. Math., Kyungshang 17, No. 1, 97--103 (2008; Zbl 1173.26329)