Minimul unei expresii cu logaritmi

[Razvan Balan]

Fie \( $m,n,p$ \) naturale diferite de 0 cu \( a,b,c \in (0,1) \) sau in \( (0,\infty) \). Sa se afle minimul expresiei \( E=(log_ab)^m+(log_bc)^n+(log_ca)^p. \)

[Marius Mainea]

\( E=\begin{array}{cc}\underbrace{\frac{(log_ab)^m}{np}+....+\frac{(log_ab)^m}{np}}\\np\ ori\end{array} +\begin{array}{cc}\underbrace{\frac{(log_bc)^n}{mp}+...+\frac{(log_bc)^n}{mp}}\\mp\ ori\end{array} +\begin{array}{cc}\underbrace{\frac{(log_ca)^p}{mn}+...+\frac{(log_ca)^p}{mn}}\\mn\ ori\end{array} \ge\sqrt[mn+np+pm]{\frac{1}{m^{n+p}\cdot n^{m+p}\cdot p^{m+n}}} \)