@article {Alzer2005,
 author="Horst Alzer",
 title={A functional inequality for the polygamma functions},
 journal="Bull. Austral. Math. Soc.",
 fjournal={Bulletin of the Australian Mathematical Society},
 volume="72",
 year="2005",
 number="3",
 pages="455--459",
 issn="0004-9727",
 coden="ALNBAB",
 language="English",
 date="25th July, 2005",
 classmath="26D15, 33B15",
 publisher={AMPAI, Australian Mathematical Society},
 url="http://www.austms.org.au/Publ/Bulletin/V72P3/723-5216-Alzer/index.shtml",
 acknowledgement={I thank the referee for helpful comments.},
 abstract={ Let $$ \Delta _n(x)=\frac {x^{n+1}}{n!}\bigl |\psi ^{(n)}(x)\bigr | \quad {(x>0; \, n\in \mathbf {N})},$$ where $\psi $ denotes the logarithmic derivative of Euler's gamma function. We prove that the functional inequality $$ \Delta _n(x)+\Delta _n(y) < 1+\Delta _n(z), \quad {x^r+y^r=z^r,} $$ holds if and only if $0<r\leq 1$. And, we show that the converse is valid if and only if $r<0$ or $r\geq n+1$. }
}