Mean Value Inequalities for the Digamma Function

Pub Date : 2023-03-01 DOI:10.1007/s10476-023-0206-6

H. Alzer, M. K. Kwong

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Abstract

Let ψ be the digamma function and let L(a,b) = (b − a)/log(b/a) be the logarithmic mean of a and b. We prove that the inequality

$$\left( * \right)\,\,\,\,\,\,\,\,\,\,\,\,{\kern 1pt} \left( {b - a} \right)\psi \left( {\sqrt {ab} } \right) < \left( {L\left( {a,b} \right) - a} \right)\psi \left( a \right) + \left( {b - L\left( {a,b} \right)} \right)\psi \left( b \right)$$

holds for all real numbers a and b with b > a ≥ α₀. Here, α₀ ≈ 0.56155 is the only positive solution of

$$5{\psi ^\prime }\left( x \right) + 3x{\psi ^{\prime \prime }}\left( x \right) = 0.$$

The constant lower bound α₀ is best possible. This refines a result of Chu, Zhang and Tang, who showed that (*) is valid for b > a ≥ 2. Moreover, we prove that the following companion to (*) holds for all a and b with b > a > 0,

$$\left( {L\left( {a,b} \right) - a} \right)\psi \left( a \right) + \left( {b - L\left( {a,b} \right)} \right)\psi \left( b \right) < \left( {b - a} \right)\psi \left( {{{a + b} \over 2}} \right).$$

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二函数的均值不等式

设ψ为digamma函数，设L（a，b）=（b−a）/log（b/a）为a和b的对数平均值；\left（｛L\left（｛a，b｝\right）-a｝\right\psi\left；a≥α0。这里，α0≈0.56155是$$5｛\psi^\prime｝\left（x\right）+3x｛\pisi^｛\prime）｝\lift（x\ right）=0的唯一正解。$$常数下界α0是最可能的。这改进了Chu、Zhang和Tang的结果，他们证明（*）对于b>；a≥2。此外，我们证明了（*）的以下伴随对所有a和b都成立，其中b>；a>；0，$$\left（｛L\left（｛a，b｝\right）-a｝\right\psi\left（a\right）+\left；\left（｛b-a｝\right）\psi\left（｛｛a+b｝\over 2｝｝\right.）$$

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