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Question Number 161162 by metamorfose last updated on 13/Dec/21

$$f:]0,+\mathrm{\infty }[\to ]0,+\mathrm{\infty }[isconvexfunction$$$$forn⩾2aninteger,prove:$$$${(f{\left(1\right)}^{f\left(1\right)}f{\left(2\right)}^{f\left(2\right)}...f{\left(n\right)}^{f\left(n\right)})}^{\frac{1}{f\left(1\right)+f\left(2\right)+...+f\left(n\right)}}+{(f\left(1\right)f\left(2\right)...f\left(n\right))}^{\frac{1}{n}}⩽f\left(1\right)+f\left(n\right)$$

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