| Summary: | In this article, we use quantum integrals to derive Hermite–Hadamard inequalities for preinvex functions and demonstrate their validity with mathematical examples. We use the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>q</mi><msub><mi>ϰ</mi><mn>2</mn></msub></msup></semantics></math></inline-formula>-quantum integral to show midpoint and trapezoidal inequalities for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>q</mi><msub><mi>ϰ</mi><mn>2</mn></msub></msup></semantics></math></inline-formula>-differentiable preinvex functions. Furthermore, we demonstrate with an example that the previously proved Hermite–Hadamard-type inequality for preinvex functions via <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>q</mi><msub><mi>ϰ</mi><mn>1</mn></msub></msub></semantics></math></inline-formula>-quantum integral is not valid for preinvex functions, and we present its proper form. We use <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>q</mi><msub><mi>ϰ</mi><mn>1</mn></msub></msub></semantics></math></inline-formula>-quantum integrals to show midpoint inequalities for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>q</mi><msub><mi>ϰ</mi><mn>1</mn></msub></msub></semantics></math></inline-formula>-differentiable preinvex functions. It is also demonstrated that by considering the limit <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>→</mo><msup><mn>1</mn><mo>−</mo></msup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mfenced separators="" open="(" close=")"><msub><mi>ϰ</mi><mn>2</mn></msub><mo>,</mo><msub><mi>ϰ</mi><mn>1</mn></msub></mfenced><mo>=</mo><mo>−</mo><mi>η</mi><mfenced separators="" open="(" close=")"><msub><mi>ϰ</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ϰ</mi><mn>2</mn></msub></mfenced><mo>=</mo><msub><mi>ϰ</mi><mn>2</mn></msub><mo>−</mo><msub><mi>ϰ</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula> in the newly derived results, the newly proved findings can be turned into certain known results.
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