Aequationes mathematicae

, Volume 75, Issue 3, pp 276–288 | Cite as

A pair of functional inequalities characterizing polynomials and Bernoulli numbers

  • Dorota KrassowskaEmail author
  • Tomasz Małolepszy
  • Janusz Matkowski


We show that if a function \(f : {\mathbb{R}} \rightarrow {\mathbb{R}}\) continuous at least at one point satisfies the pair of functional inequalities
$$f(x + a) \leq f(x) + \sum\limits^k_{j=0} \alpha_jx^j,$$
$$f(x + b) \leq f(x) + \sum\limits^k_{j=0} \beta_jx^j,$$
and the constants a, b, α i , β i (i = 0, 1,…, k) fulfil some general algebraic conditions, then f must be a polynomial. An explicit formula for the solution, involving Bernoulli numbers, is given.

Mathematics Subject Classification (2000).

Primary 39B72 26D15 secondary 46E30 


Functional inequality polynomial Kronecker’s theorem Bernoulli number 


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Copyright information

© Birkhäuser Verlag AG 2008

Authors and Affiliations

  • Dorota Krassowska
    • 1
    Email author
  • Tomasz Małolepszy
    • 1
  • Janusz Matkowski
    • 1
    • 2
  1. 1.Faculty of Mathematics, Informatics and EconometricsUniversity of Zielona GóraZielona GóraPoland
  2. 2.Institute of MathematicsSilesian UniversityKatowicePoland

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