\(\left( p,q\right) \)-Hermite–Hadamard inequalities and \(\left( p,q\right) \)-estimates for midpoint type inequalities via convex and quasi-convex functions

  • Mehmet KuntEmail author
  • İmdat İşcan
  • Necmettin Alp
  • Mehmet Zeki Sarıkaya
Original Paper


In this paper, we prove the correct \(\left( p,q\right) \)-Hermite–Hadamard inequality, some new \(\left( p,q\right) \)-Hermite–Hadamard inequalities, and generalized \(\left( p,q\right) \)-Hermite–Hadamard inequality. By using the left hand part of the correct \(\left( p,q\right) \)-Hermite–Hadamard inequality, we have a new equality. Finally using the new equality, we give some \(\left( p,q\right) \)-midpoint type integral inequalities through \(\left( p,q\right) \)-differentiable convex and \(\left( p,q\right) \)-differentiable quasi-convex functions. Many results given in this paper provide extensions of others given in previous works.


Hermite–Hadamard inequality Midpoint type inequality q-integral inequalities \((p, q)\)-integral \((p, q)\)-derivative \((p, q)\)-integration Convexity Quasi-convexity 

Mathematics Subject Classification

34A08 26A51 26D15 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no competing interests.

Author contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.


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Copyright information

© Springer-Verlag Italia 2017

Authors and Affiliations

  • Mehmet Kunt
    • 1
    Email author
  • İmdat İşcan
    • 2
  • Necmettin Alp
    • 3
  • Mehmet Zeki Sarıkaya
    • 3
  1. 1.Department of Mathematics, Faculty of SciencesKaradeniz Technical UniversityTrabzonTurkey
  2. 2.Department of Mathematics, Faculty of Sciences and ArtsGiresun UniversityGiresunTurkey
  3. 3.Department of Mathematics, Faculty of Science and ArtsDüzce UniversityDüzceTurkey

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