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In this paper, the definition of the harmonically <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mi>r</mi></mrow></semantics></math></inline-formula>-<i>h</i>-convex function is given, and its important properties are discussed. Jensen type inequality, Hermite–Hadamard type inequalities and Fejér type inequalities for harmonically <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mi>r</mi></mrow></semantics></math></inline-formula>-<i>h</i>-convex functions are also established. In addition, some numerical examples are given to verify the accuracy of the results.