RACSAM DOI 10.1007/s13398-017-0402-y ORIGINAL PAPER ( p, q )-Hermite–Hadamard inequalities and ( p, q )-estimates for midpoint type inequalities via convex and quasi-convex functions Mehmet Kunt 1 · ˙ Imdat ˙ I¸ scan 2 · Necmettin Alp 3 · Mehmet Zeki Sarıkaya 3 Received: 4 November 2016 / Accepted: 2 May 2017 © Springer-Verlag Italia 2017 Abstract In this paper, we prove the correct ( p, q )-Hermite–Hadamard inequality, some new ( p, q )-Hermite–Hadamard inequalities, and generalized ( p, q )-Hermite–Hadamard inequal- ity. By using the left hand part of the correct ( p, q )-Hermite–Hadamard inequality, we have a new equality. Finally using the new equality, we give some ( p, q )-midpoint type inte- gral inequalities through ( p, q )-differentiable convex and ( p, q )-differentiable quasi-convex functions. Many results given in this paper provide extensions of others given in previous works. Keywords Hermite–Hadamard inequality · Midpoint type inequality · q -integral inequalities · ( p, q )-integral inequalities · ( p, q )-derivative · ( p, q )-integration · Convexity · Quasi-convexity Mathematics Subject Classification 34A08 · 26A51 · 26D15 B Mehmet Kunt mkunt@ktu.edu.tr ˙ Imdat ˙ I¸ scan imdati@yahoo.com; imdat.iscan@giresun.edu.tr Necmettin Alp placenn@gmail.com Mehmet Zeki Sarıkaya sarikayamz@gmail.com 1 Department of Mathematics, Faculty of Sciences, Karadeniz Technical University, 61080 Trabzon, Turkey 2 Department of Mathematics, Faculty of Sciences and Arts, Giresun University, 28200 Giresun, Turkey 3 Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Turkey