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On estimating a function on an infinite interval. (English. Russian original) Zbl 0790.26002

Sov. Math., Dokl. 43, No. 2, 449-450 (1991); translation from Dokl. Akad. Nauk SSSR 317, No. 1, 559-560 (1991).
In an earlier paper [Dokl. Akad. Nauk SSSR 315, No. 6, 1294-1297 (1990; Zbl 0739.26002)] the author presented a necessary and sufficient condition for an \(r\)-times differentiable function \(f\) on \(x\geq x_ 0\geq 1\) to stabilize as \(x\to\infty\) to a polynomial \(P(x)= \sum^{r- 1}_{j=0} a_ j x^ j\) of degree not greater than \(r-1\), which led to the convergence of the integral \[ \int^{+\infty}_{x_ 0} dt_ 1 \int^{+\infty}_{t_ 1} dt_ 2\cdots \int^{+\infty}_{t_{r-1}} f^{(r)} (t_ r)dt_ r.{(*)} \] In the present paper, the author obtains the following main result: If the \(r\)-times differentiable function \(f\) on \(x\geq x_ 0\) has a convergent integral \((*)\) and stabilizes as \(x\to\infty\) to a polynomial \(P(x)\), then, for any sequence of indices satisfying the Pólya condition [G. Pólya, Z. Angew. Math. Mech. 11, 445-449 (1931; Zbl 0003.27302)] the following inequality holds: \[ | f(x)|\leq c\left(\sum^ k_{\mu=1} | f^{(i_ \mu)}(x_ 0)|+ \sum^ \ell_{\nu=1}| a_{j_ \nu}|+ \sum^ r_{m=1} | J^{+\infty}_{x_ 0} (m) f^{(r)}|\right)(1+x)^{r-1}+| J^{+\infty}_ x(r)f^{(r)}|, \] \(x\geq x_ 0\), where the constant \(c\) does not depend on \(f\), \((a_ j)\) are the coefficients of the polynomial \(P\) and \(J^{+\infty}_ x(r) f^{(r)}= \int^{+\infty}_ x dt_ 1 \int^{+\infty}_{t_ 1} dt_ 2\cdots \int^{+\infty}_{t_{r-1}} f^{(r)}(t_ r)dt_ r\). This result and some of its consequences are only announced without proofs.

MSC:

26A12 Rate of growth of functions, orders of infinity, slowly varying functions
26D10 Inequalities involving derivatives and differential and integral operators
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