Bulletin of the Australian Mathematical Society Bull. Austral. Math. Soc. 0004-9727 ALNBAB 2005 72 3 10.wxyz/CV72P3 http://www.austms.org.au/Publ/Bulletin/V72P3/ Horst Alzer 13 February 2006 2006 2 13 455 459 723-5216-Alzer-2005 10.wxyz/C2005V72P3p455 http://www.austms.org.au/Publ/Bulletin/V72P3/723-5216-Alzer/ 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 $r\geq n+1$. 26D15, 33B15 I thank the referee for helpful comments. M. Abramowitz and I.A. Stegun (eds.) Dover Publications Inc. MR208797 M. Abramowitz and I.A. Stegun (eds.); \textit{Handbook of mathematical functions with formulas and mathematical tables} (Dover Publications Inc., New York, 1966). H. Alzer Aequationes Math. MR1820816 H. Alzer; Mean-value inequalities for the polygamma functions, \textit{Aequationes Math.} \textbf{61} (2001), pp.~151--161. H. Alzer Forum Math. MR2039096 H. Alzer; Sharp inequalities for the digamma and polygamma functions, \textit{Forum Math.} \textbf{16} (2004), pp.~181--221. R. Askey J. Math. Anal. Appl. MR316769 R. Askey; Gr\"unbaum's inequality for Bessel functions, \textit{J. Math. Anal. Appl.} \textbf{41} (1973), pp.~122-124. W. Gautschi Tricomi's ideas and contemporary applied mathematics Atti Convegni Lincei 147 Accad. Naz. Lincei MR1737497 W. Gautschi; The incomplete gamma function since Tricomi, in \textit{Tricomi's ideas and contemporary applied mathematics}, Atti Convegni Lincei 147 (Accad. Naz. Lincei, Rome, 1998), pp.~203--237. F.A. Grünbaum J. Math. Anal. Appl. MR316768 F.A. Gr\"unbaum; A new kind of inequality for Bessel functions, \textit{J. Math. Anal. Appl.} \textbf{41} (1973), pp.~115--121.