Ukrainian Mathematical Journal

, Volume 67, Issue 11, pp 1721–1738 | Cite as

Generalizations of Steffensen’s Inequality by Lidstone’s Polynomials

  • J. Pečarić
  • A. Perušić
  • K. Smoljak

We obtain generalizations of Steffensen’s inequality by using Lidstone’s polynomials. Furthermore, the functionals associated with the obtained generalizations are used to generate n-exponentially and exponentially convex functions, as well as the new Stolarsky-type means.


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  1. 1.
    R. P. Agarwal and P. J. Y. Wong, Error Inequalities in Polynomial Interpolation and Their Applications, Kluwer, Dordrecht, etc. (1993).Google Scholar
  2. 2.
    K. E. Atkinson, An Introduction to Numerical Analysis, 2nd ed., Wiley, New York, etc. (1989).Google Scholar
  3. 3.
    R. P. Boas, “Representation of functions by Lidstone series,” Duke Math. J., 10, 239–245 (1943).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    J. Jakšetić and J. Pečarić, “Exponential convexity method,” J. Convex Anal., 20, No. 1, 181–197 (2013).MathSciNetzbMATHGoogle Scholar
  5. 5.
    G. J. Lidstone, “Notes on the extension of Aitken’s theorem (for polynomial interpolation) to the Everett types,” Proc. Edinburgh Math. Soc., 2, No. 2, 16–19 (1929).Google Scholar
  6. 6.
    J. Pečarić and J. Perić, “Improvement of the Giaccardi and the Petrović inequality and related Stolarsky-type means,” An. Univ. Craiova Ser. Mat. Inform., 39, No. 1, 65–75 (2012).MathSciNetzbMATHGoogle Scholar
  7. 7.
    J. E. Pečarić, F. Proschan, and Y. L. Tong, “Convex functions, partial orderings, and statistical applications,” Math. Sci. Eng., 187 (1992).Google Scholar
  8. 8.
    H. Poritsky, “On certain polynomial and other approximations to analytic functions,” Trans. Amer. Math. Soc., 34, No. 2, 274–331 (1932).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    I. J. Schoenberg, “On certain two-point expansions of integral functions of exponential type,” Bull. Amer. Math. Soc., 42, No. 4, 284–288 (1936).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    J. F. Steffensen, “On certain inequalities between mean values and their application to actuarial problems,” Skand. Aktuarietidskr., 82–97 (1918).Google Scholar
  11. 11.
    D. V. Widder, The Laplace Transform, Princeton Univ. Press, New Jersey (1941).Google Scholar
  12. 12.
    D. V. Widder, “Completely convex functions and Lidstone series,” Trans. Amer. Math. Soc., 51, 387–398 (1942).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    J. M. Whittaker, “On Lidstone series and two-point expansions of analytic functions,” Proc. London Math. Soc., 36, 451–469 (1933-1934).Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • J. Pečarić
    • 1
  • A. Perušić
    • 2
  • K. Smoljak
    • 1
  1. 1.University of ZagrebZagrebCroatia
  2. 2.University of RijekaRijekaCroatia

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