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Integral analog of one generalization of the Hardy inequality and its applications

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Abstract

Under certain conditions on continuous functions μ, λ, a, and f, we prove the inequality

$$\int\limits_0^y {\mu (x)\lambda (x)f\left( {\frac{{\int_0^x {\lambda (t)a(t)dt} }}{{\int_0^x {\lambda (t)dt} }}} \right)dx \leqslant K\int\limits_0^y {\mu (x)\lambda (x)f(a(x))} dx,} y \leqslant \infty ,$$

and describe its application to the investigation of the problem of finding conditions under which Laplace integrals belong to a class of convergence.

References

  1. 1.

    G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities [Russian translation], Inostrannaya Literatura, Moscow (1948).

  2. 2.

    O. M. Mulyava, “On convergence classes of Dirichlet series,” Ukr. Mat. Zh., 51, No. 11, 1485–1494 (1999).

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 9, pp. 1271–1275, September, 2006.

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Mulyava, O.M. Integral analog of one generalization of the Hardy inequality and its applications. Ukr Math J 58, 1441–1447 (2006). https://doi.org/10.1007/s11253-006-0143-0

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Keywords

  • Continuous Function
  • Convex Function
  • Russian Translation
  • Maximum Point
  • Mathematical Journal