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Some new quantum Trapezoid and quantum Midpoint type inequalities for preinvex functions
  • Muhammad Aamir Ali,
  • Hüseyin BUDAK
Muhammad Aamir Ali
Nanjing Normal University School of Mathematical Sciences
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Hüseyin BUDAK
Düzce University
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Abstract

In this article, we derive Hermite-Hadamard inequalities for preinvex functions using the quantum integrals and show their validation with mathematical examples. We prove midpoint and trapezoidal inequalities for q^{ϰ₂}-differentiable preinvex functions via q^{ϰ₂}-quantum integrals. Moreover, we show with an example that the already proved inequality of Hermite-Hadamard type for preinvex functions via q_{ϰ₁}-quantum integrals is not valid for preinvex functions and we give its correct version. We prove the midpoint inequalities for q_{ϰ₁}-differentiable preinvex functions via q_{ϰ₁}-quantum integrals. It is also shown that the newly proved results transformed into some known results by considering the limit q→1⁻ and η(ϰ₂,ϰ₁)=-η(ϰ₁,ϰ₂)=ϰ₂-ϰ₁ in the newly derived results.