Book: Inegalitati; Authors:L.Panaitopol,V. Bandila,M.Lascu
i already gave one more from this book
Let x,yε R. Prove that if m and n ε N and of the same parity we have:
a]\frac{x^m+y^m}{2} \frac{x^n+y^n}{2} \leq \frac{x^{m+n}+y^{m+n}}{2}
b] if x,y ε R prove that
\frac{x+y}{2} \frac{x^2+y^2}{2} \frac{x^3+y^3}{2} \leq \frac{x^6+y^6}{2}
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3 Answers
1assuming\ x \ge y\\ pairs\ (x^m,y^m)\ and (x^n,y^n)\ are\ similarly\ arranged\\ so\ from\ chebyshev's\ inequality\ we\ have\\ \frac{x^{m+n}+y^{m+n}}{2}\ge (\frac{x^m+y^m}{2})(\frac{x^n+y^n}{2})
1[2]\ from\ the\ first\ question\ take\ m=n=3\\ \frac{x^6+y^6}{2} \ge (\frac{x^3+y^3}{2})(\frac{x^3+y^3}{2})\ \ \ \\ again\ from\ the\ first\ question\ taking\ m=1,n=2\\ \frac{x^3+y^3}{2} \ge (\frac{x+y}{2})(\frac{x^2+y^2}{2})\\ substituting\ in\ the\ first\ inequality\ we\ get\\ \frac{x^6+y^6}{2} \ge (\frac{x^3+y^3}{2})(\frac{x^2+y^2}{2})( \frac{x+y}{2})
