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In this article, we give sharp two-sided bounds for the generalized Jensen functional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>J</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>h</mi><mo>;</mo></mrow></mrow></semantics></math></inline-formula><i><b>p</b></i>,<i><b>x</b></i>). Assuming convexity/concavity of the generating function <i>h</i>, we give exact bounds for the generalized quasi-arithmetic mean <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>h</mi><mo>;</mo></mrow></mrow></semantics></math></inline-formula><i><b>p</b></i>,<i><b>x</b></i>). In particular, exact bounds are determined for the generalized power means in terms from the class of Stolarsky means. As a consequence, some sharp converses of the famous Hölder’s inequality are obtained.