| Summary: | In this work, we obtain some new integral inequalities of the Hermite–Hadamard–Fejér type for operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><msub><mi>ω</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ω</mi><mn>2</mn></msub></mfenced></semantics></math></inline-formula>-preinvex functions. The bounds for both left-hand and right-hand sides of the integral inequality are established for operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><msub><mi>ω</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ω</mi><mn>2</mn></msub></mfenced></semantics></math></inline-formula>-preinvex functions of the positive self-adjoint operator in the complex Hilbert spaces. We give the special cases to our results; thus, the established results are generalizations of earlier work. In the last section, we give applications for synchronous (asynchronous) functions.
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