SEQUENCES WITH INEQUALITIES

Abstract

Abstract We consider infinite sequences of positive numbers. The connection between log-concavity and the Bessenrodt–Ono inequality has been the focus of several papers. It has applications in the white noise distribution theory and combinatorics. We improve a recent result by Benfield and Roy and show that for the sequence of partition numbers $$\{p(n)\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>p</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> , Nicolas’ log-concavity result implies the result by Bessenrodt and Ono towards $$p(n) \, p(m) &gt; p(n+m)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> <mml:mspace/> <mml:mi>p</mml:mi> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> <mml:mo>&gt;</mml:mo> <mml:mi>p</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . We provide several examples. Benfield and Roy gave a conjecture related to $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> -ary partition numbers. We prove a part of this conjecture.