Advertisement

Vietnam Journal of Mathematics

, Volume 43, Issue 4, pp 663–675 | Cite as

A Functional Generalization of Trapezoid Inequality

  • Silvestru S. Dragomir
Article
  • 90 Downloads

Abstract

We show in this paper amongst others that, if \(f:[a,b] \rightarrow \mathbb {R}\) is absolutely continuous on [a,b] and \(\varPhi :\mathbb {R} \rightarrow \mathbb {R}\) is convex (concave) on \(\mathbb {R}\) then
$$\begin{array}{@{}rcl@{}} &&\varPhi \left(\frac{(x-a) f(a) +(b-x)f(b)}{b-a}-\frac{1}{b-a}{{\int}_{a}^{b}}f(t) dt\right)\\ &&\leq (\geq)\frac{x-a}{(b-a)^{2}}{{\int}_{a}^{b}}\varPhi [f(a) -f(t)] dt+\frac{b-x}{(b-a)^{2}}{{\int}_{a}^{b}}\varPhi [f(b)-f(t)] dt \end{array} $$
for any x∈[a,b].

Natural applications for power and exponential functions are provided as well. Bounds for the Lebesgue p-norms are also given.

Keywords

Absolutely continuous functions Convex functions Integral inequalities Trapezoid inequality Jensen’s inequality Lebesgue norms 

Mathematic Subject Classification (2010)

26D15 25D10 

References

  1. 1.
    Cerone, P., Dragomir, S.S.: Trapezoidal-type rules from an inequalities point of view. In: Anastassiou, G. (ed.) Handbook of Analytic Computational Methods in AppliedMathematics, pp. 65–134. CRC Press, NY (2000)Google Scholar
  2. 2.
    Cerone, P., Dragomir, S.S. Trapezoidal-type rules from an inequalities point of view. In: Anastassiou, G (ed.) : Handbook of analytic computational methods in applied mathematics, pp 65–134. CRC Press, NY (2000)Google Scholar
  3. 3.
    Dragomir, S.S.: The Ostrowski’s integral inequality for mappings of bounded variation. Bull. Aust. Math. Soc. 60, 495–508 (1999)zbMATHCrossRefGoogle Scholar
  4. 4.
    Dragomir, S.S.: An inequality improving the first Hermite–Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products. J. Ineq. Pure Appl. Math 3, 31 (2002)Google Scholar
  5. 5.
    Dragomir, S.S.: An inequality improving the second Hermite–Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products. J. Ineq. Pure Appl. Math. 3, 35 (2002)Google Scholar
  6. 6.
    Kechriniotis, A.I., Assimakis, N.D.: Generalizations of the trapezoid inequalities based on a new mean value theorem for the remainder in Taylor’s formula. J. Inequal. Pure Appl. Math. 7, 90 (2006)MathSciNetGoogle Scholar
  7. 7.
    Liu, Z.: Some inequalities of perturbed trapezoid type. J. Inequal. Pure Appl. Math. 7, 47 (2006). electronicGoogle Scholar
  8. 8.
    Mercer, A.McD.: On perturbed trapezoid inequalities. J. Inequal. Pure Appl. Math. 7, 118 (2006)MathSciNetGoogle Scholar
  9. 9.
    Ujević, N.: Error inequalities for a generalized trapezoid rule. Appl. Math. Lett. 19, 32–37 (2006)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015

Authors and Affiliations

  1. 1.Mathematics, College of Engineering & ScienceVictoria UniversityMelbourne CityAustralia
  2. 2.School of Computational & Applied MathematicsUniversity of the WitwatersrandJohannesburgSouth Africa

Personalised recommendations