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Omni-MATH / Find all positive real numbers λ such that for all integers n≥2 and all positive real numbers a₁,a₂,⋯,a_(n) with a₁+a₂+⋯+a_(n)=n, the following inequality holds: …
Problem
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problem
Find all positive real numbers such that for all integers and all positive real numbers with , the following inequality holds:
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Plain-text mathematical notation (without MathML)
Find all positive real numbers λ such that for all integers n≥2 and all positive real numbers a₁,a₂,⋯,a_(n) with a₁+a₂+⋯+a_(n)=n, the following inequality holds: ∑_(i=1)^(n)(1)/(a_(i))−λ∏_(i=1)^(n)(1)/(a_(i))≤n−λ.
Original LaTeX notation
Find all positive real numbers $\lambda$ such that for all integers $n\geq 2$ and all positive real numbers $a_1,a_2,\cdots,a_n$ with $a_1+a_2+\cdots+a_n=n$, the following inequality holds:
$\sum_{i=1}^n\frac{1}{a_i}-\lambda\prod_{i=1}^{n}\frac{1}{a_i}\leq n-\lambda$.Discussion
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