Abstract
For \(k,l\in \mathbf {N}\), let
We prove that the inequality
is valid for all natural numbers k and l. The sign of equality holds if and only if \(k=l=1\). This complements a result of Vietoris, who showed that
An immediate corollary is that
The constant bounds are sharp.
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References
W. Raab, Die Ungleichungen von Vietoris, Monatsh. Math. 98 (1984), 311–322.
L. Vietoris, Über gewisse die unvollständige Betafunktion betreffende Ungleichungen, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 191 (1982), 85–92.
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The research of Man Kam Kwong is supported by the Hong Kong Government GRF Grant PolyU 50003/12P and the Hong Kong Polytechnic University Grant G-UC22 and G-UA10.
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Alzer, H., Kwong, M.K. Inequalities for combinatorial sums. Arch. Math. 108, 601–607 (2017). https://doi.org/10.1007/s00013-017-1024-5
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DOI: https://doi.org/10.1007/s00013-017-1024-5