Some properties of the gamma and psi functions, with applications

Abstract

In this paper, some monotoneity and concavity properties of the gamma, beta and psi functions are obtained, from which several asymptotically sharp inequalities follow. Applying these properties, the authors improve some well-known results for the volume <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega Subscript n"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\Omega _n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the unit ball <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Superscript n Baseline subset-of double-struck upper R Superscript n"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>B</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo> ⊂ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">B^n\subset \mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the surface area <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega Subscript n minus 1"> <mml:semantics> <mml:msub> <mml:mi> ω </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\omega _{n-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the unit sphere <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript n minus 1"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">S^{n-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and some related constants.