Log-concavity of infinite product generating functions

Abstract In the 1970s Nicolas proved that coefficients $$p_d(n)$$ pd(n) defined by generating function $$\begin{aligned} \sum _{n=0}^{\infty } p_d(n) \, q^n = \prod _{n=1}^{\infty \left( 1- q^n\right) ^{-n^{d-1}} \end{aligned}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">∑n=0∞pd(n)qn=∏n=1∞1-qn-nd-1 are log-concave for $$d=1$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">d=1 . Recently, Ono, Pujahari, and Rolen have extended result to $$d=2$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">d=2 Note $$p_1(n)=p(n)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">p1(n)=p(n) is partition $$p_2(n)=\mathrm{pp}\left( n\right) $$ xmlns:mml="http://www.w3.org/1998/Math/MathML">p2(n)=ppn number of plane partitions. this paper, we invest in properties general d Let $$n \ge 6$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">n≥6 Then almost n divisible 3 strictly log-convex otherwise.