| Summary: | This manuscript develops the study of reverse Hilbert-type inequalities by applying reverse Hölder inequalities on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">T</mi></semantics></math></inline-formula>. We generalize the reverse inequality of Hilbert-type with power two by replacing the power with a new power <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>></mo><mn>1</mn><mo>.</mo></mrow></semantics></math></inline-formula> The main results are proved by using Specht’s ratio, chain rule and Jensen’s inequality. Our results (when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">T</mi><mo>=</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula>) are essentially new. Symmetrical properties play an essential role in determining the correct methods to solve inequalities.
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