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In this work, we obtained new results relating the generalized atom-bond connectivity index with the general Randić index. Some of these inequalities for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>B</mi><msub><mi>C</mi><mi>α</mi></msub></mrow></semantics></math></inline-formula> improved, when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula>, known results on the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>B</mi><mi>C</mi></mrow></semantics></math></inline-formula> index. Moreover, in order to obtain our results, we proved a kind of converse Hölder inequality, which is interesting on its own.