Title: Remarks on a paper by Chao-Ping Chen and Feng Qi
Abstract:In a recent paper, Chao-Ping Chen and Feng Qi (2005) established sharp upper and lower bounds for the sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/M...In a recent paper, Chao-Ping Chen and Feng Qi (2005) established sharp upper and lower bounds for the sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript n Baseline colon equals StartFraction 1.3 ellipsis left-parenthesis 2 n minus 1 right-parenthesis Over 2.4 ellipsis 2 n EndFraction"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo>:=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>1.3</mml:mn> <mml:mo>…</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>2.4</mml:mn> <mml:mo>…</mml:mo> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:mfrac> </mml:mrow> <mml:annotation encoding="application/x-tex">P_{n}:=\frac {1.3\ldots (2n-1)}{2.4\ldots 2n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that their result follows easily from a theorem of G. N Watson published in 1959. We also show that the main result of Chen and Qi’s paper is a special case of a more general inequality which admits a very short proof.Read More