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Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">K</mi><mo>(</mo><mi>r</mi><mo>)</mo></mrow></semantics></math></inline-formula> be the complete elliptic integral of the first kind. Then, the inequality <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi mathvariant="script">K</mi><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><mo>/</mo><mi>π</mi><mo>></mo><msup><mo form="prefix">tanh</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><mo>/</mo><msup><mo form="prefix">sin</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mfenced open="(" close=")"><mi>r</mi></mfenced></mrow></semantics></math></inline-formula> holds for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>. This conclusion does not match those in the existing literature.