Mean Value Inequalities for the Digamma Function

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,

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