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Omni-MATH / Find the smallest real constant α such that for all positive integers n and real numbers 0=y₀< y₁<⋯<y_(n), the following inequality holds: …

Problem

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problem

Find the smallest real constant α\alpha such that for all positive integers nn and real numbers 0=y0<0=y_{0}< y1<⋯<yny_{1}<\cdots<y_{n}, the following inequality holds: α∑k=1n(k+1)3/2yk2−yk−12≥∑k=1nk2+3k+3yk\alpha \sum_{k=1}^{n} \frac{(k+1)^{3 / 2}}{\sqrt{y_{k}^{2}-y_{k-1}^{2}}} \geq \sum_{k=1}^{n} \frac{k^{2}+3 k+3}{y_{k}}.
Plain-text mathematical notation (without MathML)
Find the smallest real constant α such that for all positive integers n and real numbers 0=y₀< y₁<⋯<y_(n), the following inequality holds: α∑_(k=1)^(n)((k+1)^(3/2))/(√(y_(k)²−y_(k−1)²))≥∑_(k=1)^(n)(k²+3k+3)/(y_(k)).
Original LaTeX notation
Find the smallest real constant $\alpha$ such that for all positive integers $n$ and real numbers $0=y_{0}<$ $y_{1}<\cdots<y_{n}$, the following inequality holds: $\alpha \sum_{k=1}^{n} \frac{(k+1)^{3 / 2}}{\sqrt{y_{k}^{2}-y_{k-1}^{2}}} \geq \sum_{k=1}^{n} \frac{k^{2}+3 k+3}{y_{k}}$.

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Source and history

task_id
2c3693dc-a221-5674-a8a0-ac6bc19c9840
task_key
test--2c3693dc-a221-5674-a8a0-ac6bc19c9840
task_revision_id
1
config / split
/ test

Official source

initial import