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Correction to: Results Math. 75:180 (2020) https://doi.org/10.1007/s00025-020-01308-y
There is an inadvertent mistake in the statement of Theorem 3.2 and its proof. The authors are indebted to Ioan Raşa for pointing out the mistake in the statement of Theorem 3.2, which has led us to revise the proof.
The theorem should be
Let \(n,q\in \mathbb {N}\). If f defined on \([0,\infty )\) and such that \(|f(x)|\le C(1+x)^{\gamma }\), for some \(C,\gamma >0\), is a q-monotone function there, then for all \(x,y\in [0,\infty )\),
Let \(n,m,q\in \mathbb {N}\), and denote \(\mathbf {\nu }:=(\nu _1,\dots ,\nu _q)\in ( \mathbb {N}_0)^q\). By (3.2) it follows that
Hence, as is done in (2.2), for \(0\le x,y<\infty \) and any sequence \( (a_k)_{k=0}^\infty \), we have
Let
Then
Note that \(\left. \left( \partial /\partial z\right) ^j(1-uz)^{-m} \right| _{z=-1}=j!m_{m,j}(u)\ge 0\), for \(j,m=0,1,2,\ldots \), and \(0\le u<\infty \). Hence
Finally, we may rewrite the last line in (3.5), as we did in (3.3),
and Theorem 3.2 follows by taking \(a_m:=\int _0^1f\left( \frac{m+\alpha t}{n+\alpha }\right) dt\). \(\square \)
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Abel, Ulrich, Leviatan, Dany: An extension of Raşa’s conjecture to \(q\)-monotone functions. Results Math. 75(180), 1–13 (2020)
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Ulrich Abel and Dany Leviatan contributed equally to this work.
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Abel, U., Leviatan, D. Correction to: An Extension of Raşa’s Conjecture to q-Monotone Functions. Results Math 78, 122 (2023). https://doi.org/10.1007/s00025-023-01872-z
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DOI: https://doi.org/10.1007/s00025-023-01872-z