Editing Digamma function (section)

==Inequalities==
When {{math|''x'' > 0}}, the function
:<math>\ln x - \frac{1}{2x} - \psi(x)</math>
is completely monotonic and in particular positive.  This is a consequence of [[Bernstein's theorem on monotone functions]] applied to the integral representation coming from Binet's first integral for the gamma function.  Additionally, by the convexity inequality <math>1 + t \le e^t</math>, the integrand in this representation is bounded above by <math>e^{-tz}/2</math>.  {{not a typo|Consequently}}
:<math>\frac{1}{x} - \ln x + \psi(x)</math>
is also completely monotonic.  It follows that, for all {{math|''x'' > 0}},
:<math>\ln x - \frac{1}{x} \le \psi(x) \le \ln x - \frac{1}{2x}.</math>
This recovers a theorem of Horst Alzer.<ref>{{cite journal |jstor=2153660 |url=https://www.ams.org/journals/mcom/1997-66-217/S0025-5718-97-00807-7/S0025-5718-97-00807-7.pdf|title=On Some Inequalities for the Gamma and Psi Functions |last1=Alzer |first1=Horst |journal=Mathematics of Computation |year=1997 |volume=66 |issue=217 |pages=373–389 |doi=10.1090/S0025-5718-97-00807-7 }}</ref>  Alzer also proved that, for {{math|''s'' ∈ (0, 1)}},
:<math>\frac{1 - s}{x + s} < \psi(x + 1) - \psi(x + s),</math>

Related bounds were obtained by Elezovic, Giordano, and Pecaric, who proved that, for {{math|''x'' > 0 }},
:<math>\ln(x + \tfrac{1}{2}) - \frac{1}{x} < \psi(x) < \ln(x + e^{-\gamma}) - \frac{1}{x},</math>
where <math>\gamma=-\psi(1)</math> is the [[Euler–Mascheroni constant]].<ref>{{cite journal |doi=10.7153/MIA-03-26|title=The best bounds in Gautschi's inequality |year=2000 |last1=Elezović |first1=Neven |last2=Giordano |first2=Carla |last3=Pečarić |first3=Josip |journal=Mathematical Inequalities & Applications |issue=2 |pages=239–252 |doi-access=free }}</ref>  The constants (<math>0.5</math> and <math>e^{-\gamma}\approx0.56</math>) appearing in these bounds are the best possible.<ref>{{cite journal | arxiv=0902.2524 | doi=10.1515/anly-2014-0001 | title=Sharp inequalities for the psi function and harmonic numbers | year=2014 | last1=Guo | first1=Bai-Ni | last2=Qi | first2=Feng | journal=Analysis | volume=34 | issue=2 | s2cid=16909853 }}</ref>

The [[mean value theorem]] implies the following analog of [[Gautschi's inequality]]: If {{math|''x'' > ''c''}}, where {{math|''c'' ≈ 1.461}} is the unique positive real root of the digamma function, and if {{math|''s'' > 0}}, then
:<math>\exp\left((1 - s)\frac{\psi'(x + 1)}{\psi(x + 1)}\right) \le \frac{\psi(x + 1)}{\psi(x + s)} \le \exp\left((1 - s)\frac{\psi'(x + s)}{\psi(x + s)}\right).</math>
Moreover, equality holds if and only if {{math|''s'' {{=}} 1}}.<ref>{{cite journal |doi=10.1016/j.jmaa.2013.05.045 |doi-access=free|title=Exponential, gamma and polygamma functions: Simple proofs of classical and new inequalities |year=2013 |last1=Laforgia |first1=Andrea |last2=Natalini |first2=Pierpaolo |journal=Journal of Mathematical Analysis and Applications |volume=407 |issue=2 |pages=495–504 }}</ref>

Inspired by the harmonic mean value inequality for the classical gamma function, Horzt Alzer and Graham Jameson proved, among other things, a harmonic mean-value inequality for the digamma function:

<math> -\gamma \leq \frac{2 \psi(x) \psi(\frac{1}{x})}{\psi(x)+\psi(\frac{1}{x})} </math> for <math>x>0</math>

Equality holds if and only if <math>x=1</math>.<ref>{{cite journal |last1=Alzer |first1=Horst |last2=Jameson |first2=Graham |s2cid=41966777 |year=2017 |title=A harmonic mean inequality for the digamma function and related results |journal=[[Rendiconti del Seminario Matematico della Università di Padova]] |pages=203–209 |doi=10.4171/RSMUP/137-10 |volume=70 |issue=201 |issn=0041-8994 |lccn=50046633 |oclc=01761704|url=https://eprints.lancs.ac.uk/id/eprint/136736/1/5d0aee750965cd339d8a0965d80de4c18b68.pdf }}</ref>