| Summary: | We prove that the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula> defined by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mn>6</mn><msup><mi>s</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>s</mi><mo>+</mo><mn>5</mn></mrow><mrow><mn>3</mn><msup><mi>s</mi><mn>2</mn></msup><mrow><mo>(</mo><mn>2</mn><mi>s</mi><mo>+</mo><mn>3</mn><mo>)</mo></mrow></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mi>ψ</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow><mn>2</mn></mfrac><mo>−</mo><mfrac><mrow><mi>σ</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mrow><mn>2</mn><msup><mi>s</mi><mn>5</mn></msup></mrow></mfrac><mo>,</mo><mi>s</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, is strictly increasing with the sharp bounds <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>σ</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo><</mo><mfrac><mn>49</mn><mn>120</mn></mfrac></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula> is Nielsen’s beta function and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>ψ</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is the trigamma function. Furthermore, we prove that the two functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>↦</mo><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mi>μ</mi></mrow></msup><mfenced separators="" open="[" close="]"><mi>β</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>−</mo><mfrac><mrow><mn>6</mn><msup><mi>s</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>s</mi><mo>+</mo><mn>5</mn></mrow><mrow><mn>3</mn><msup><mi>s</mi><mn>2</mn></msup><mrow><mo>(</mo><mn>2</mn><mi>s</mi><mo>+</mo><mn>3</mn><mo>)</mo></mrow></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mi>ψ</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><mn>49</mn><mi>μ</mi></mrow><mrow><mn>240</mn><msup><mi>s</mi><mn>5</mn></msup></mrow></mfrac></mfenced><mo>,</mo><mo> </mo><mi>μ</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></mrow></semantics></math></inline-formula> are completely monotonic for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. As an application, double inequality for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula> involving <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>ψ</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is obtained, which improve some recent results.
|
|---|