1 Introduction and main results

In this paper, we study Turán inequalities \(p_n(x)^2 -p_{n-1}(x) \, p_{n+1}(x) \ge 0\) for families of polynomials \(\{p_n(x)\}_n\) attached to arithmetic functions.

Our work is motivated by a recent result by Ono et al. [8, 13]. Ono et al. proved the log-concavity conjecture ([9], Conjecture 1) for plane partitions \({{\textrm{pp}}}\left( n\right) \) for \(n >11\). Twenty-four years ago, Nicolas [12] had already proved the log-concavity property for the partition numbers p(n) for \(n >25\). This result was reproved by DeSalvo and Pak [5]. For an introduction to partition numbers and plane partition numbers, we refer to Andrews’ book [1]. Further, to study the concept of log-concavity and related topics, Brenti [2] and Stanley [14, 15] are suitable references.

This paper is also a significant generalization of our previous result in [8]. Let \(a_d(n):= \frac{1}{n} \sum _{k=1}^n \left( \sum _{\ell \vert k} \ell ^d \right) \, a_d(n-k)\), with \(a_d(0)=1\). Let \(n \ge 6\) be fixed. Then the sequence \(\{a_d(n)\}_n\) is log-concave at n for almost all \( d\in {\mathbb {N}}\) if and only if \(n \equiv 0 \pmod {3}\). Note that \(p(n)= a_1(n)\) and \({{\textrm{pp}}}(n)= a_2(n)\). The quantities p(n) and \({{\textrm{pp}}}\left( n\right) \) are induced by certain arithmetic functions. This leads to the following generalization.

Definition

Let \({\mathbb {D}}\) be the set of all double sequences \(\{g_d(n)\}_{d,n \ge 1}\) with normalization \(g_d(1)=1\), such that \(\sum _{n=1}^{\infty } g_d(n) \, q^{n-1}\) is regular at \(q=0\) with radius of convergence R, and

$$\begin{aligned} 0 \le g_{d}\left( n\right) -n^{d} \le g_{1} \left( n\right) \, \left( n-1\right) ^{d-1}, \end{aligned}$$

for all d and n.

We investigate sequences of polynomials \(\{P_n^{g_d}(x)\}_n\), defined by the recurrence relation:

$$\begin{aligned} P_n^{g_d}(x) := \frac{x}{n} \, \sum _{k=1}^{n} g_d(k) \, P_{n-k}^{g_d}(x), \qquad \text {with } P_0^{g_d}(x):=1. \end{aligned}$$

We have the generating series

$$\begin{aligned} \sum _{n=0}^{\infty } P_n^{g_d}(x) \, q^n = {{\textrm{exp}}}\left( x \sum _{n=1}^{\infty } g_d(n) \frac{q^n}{n} \right) = \prod _{n=1}^{\infty } \left( 1 - q^n \right) ^{-x \, f_d(n)}, \end{aligned}$$

where \( n \, f_d(n) = \sum _{\ell \mid n } \mu (\ell ) \, g_d(n/ \ell )\) with \(\mu \) the Moebius function. Examples for \(g_d(n)\) are \(\{\sigma _d(n)\}\) and \(\{\psi _d(n)\}\), where \(\sigma _d(n)= \sum _{\ell \mid n} \ell ^d\) and \(\psi _d(n)= n^d\). Turán’s inequality of \(\{P_n^{g_d}(x)\}\) at n for a subset of \({\mathbb {R}}\) is defined by

$$\begin{aligned} \Delta _{n}^{g_d}(x) := \left( P_n^{g_d}(x) \right) ^2 - P_{n-1}^{g_d}(x) \, P_{n+1}^{g_d}(x) \ge 0. \end{aligned}$$

Let \(x_0\) be fixed, we call \(P_n^{g_d}(x_0)\) log-concave at n if \(\Delta _{n}^{g_d}(x_0)\ge 0\).

We note that the partition function and the plane partition function satisfy \(p(n)= P_{n}^{\sigma _1}(1)\) and \({{\textrm{pp}}}\left( n\right) = P_n^{\sigma _2}(1)\). Let \(E^{g_d}\) be the set of all \(n \in {\mathbb {N}}\) with \(\Delta _{n}^{g_d}(1)<0\), denoted as strictly log-convex.

Nicolas [12] proved that the partition function p(n) is log-concave for almost all n. The set of exceptions is given by \(E^{\sigma _{1}}= \left\{ 2k +1 \, : \, 0 \le k \le 12\right\} \). Ono et al. [13] proved that \(E^{\sigma _2}= \{1,3,5,7,9,11\}\). Numerical investigations [8] for \(n \le 10^5\) indicate that \(E^{\sigma _3}= \{1,3,5,7\}\), \(E^{\sigma _4}= E^{\sigma _5}=\{1,5\}\). Surprisingly, \(E^{\sigma _{20}}\) has at least 10 elements. We believe that the general and clean patterns associated to double sequences in \( {\mathbb {D}}\) are displayed by \(g_d(n)= \psi _d(n)\) (see Table 1).

Table 1 Exceptions for \(g_d\left( n\right) =n^{d}\), \(1\le d\le 18\) and \(1\le n\le 14 \)
Full size table

In our main result, we capture the impact of the residue of n divided by 3 and the range of the argument of the \(\Delta _n^{g_d}(x)\).

Theorem 1.1

Let \(\{g_d(n)\}\) be a double sequence in \({\mathbb {D}}\). Let \(n \ge 6\). Moreover let

$$\begin{aligned} \Delta _{n}^{g_d}(x)= \left( P_n^{g_d}(x) \right) ^2 - P_{n-1}^{g_d}(x) \, P_{n+1}^{g_d}(x) \ge 0, \end{aligned}$$
(1.1)

be the Turán inequality.

  1. (a)

    Let \(0 \le x < 2- \frac{12}{n+4}\). Then (1.1) holds true for almost all d if and only if n is divisible by 3.

  2. (b)

    Let \(n \not \equiv 2 \pmod {3}\) and \(x \ge 0\). Then (1.1) holds true for almost all d if and only if n is divisible by 3.

The case \(g_{d}\left( n\right) = \sigma _{d} \left( n\right) \) and \(x=1\) leads to the results obtained in [8], Theorem 1.2 and Theorem 1.3. An explicit analysis of the bounds obtained in the proof of Theorem 1.1 leads to the following:

Theorem 1.2

Let \(\left\{ g_{d} \left( n\right) \right\} \) be a double sequence in \( {\mathbb {D}}\). Let \(n \ge 3\) and \(n \ne 5\). Let R be the radius of convergence of \(\sum _{n=1}^{\infty } g_1(n) \, \frac{q^n}{n}\). For each x, let r(x) be chosen with \( 0< r(x) < R\) and \(P_n^{g_{1} }(x) \le r(x)^{-n}\) for all n. Then we have the following properties.

  1. (i)

    Let \(n \equiv 0 \pmod {3}\) and \(x >0\). Then \(\Delta _n^{g_d}(x) \ge 0\) for \(d \ge d_{0}\left( n,x\right) \), where

    $$\begin{aligned} d_{0}\left( n,x\right) =1+\frac{2n}{3\ln \left( 9/8\right) } \left( \ln \left( n/3\right) - \ln \left( x\right) -3\ln \left( r(x) \right) \right) . \end{aligned}$$
  2. (ii)

    Let \(n \equiv 1 \pmod {3}\) and \(x >0\). Then \(\Delta _n^{g_d}(x) < 0\) for \(d \ge d_{0 }\left( n,x\right) \), where

    $$\begin{aligned} d_{0} \left( n,x\right) =1+\frac{2n}{3\ln \left( 9/8\right) } \left( \ln \left( \frac{n-1}{3}\right) -\frac{ 2n+1}{ 2n} \ln \left( x\right) -3\ln \left( r(x) \right) \right) . \end{aligned}$$
  3. (iii)

    Let \(n \equiv 2 \pmod {3}\) and \(0<x<2-\frac{12}{n+4}\). Then \(\Delta _n^{g_d}(x) < 0\) for \(d \ge d_{0 }\left( n,x\right) \), where

    $$\begin{aligned} d_{0} \left( n,x\right){} & {} = 1+\frac{1}{ \ln \left( 9/8\right) }\left( -\min \left\{ 0,\ln \left( \frac{n-2}{3n+3} \left( \frac{1}{x}+\frac{1}{2}\right) -\frac{1}{3}\right) \right\} \right. \\{} & {} \left. \quad + \, \frac{n-2}{3}\ln \left( \frac{n-2}{3}\right) - \frac{n+1}{3}\ln \left( x\right) -n\ln \left( r(x) \right) \right) . \end{aligned}$$

Remark

  1. (a)

    The positive real number r(x) always exists due to Cauchy–Hadamard’s theorem.

  2. (b)

    Let \(g_d(n)= n^d\). Then \(\Delta _4^{g_5}(x)\) has sign changes for positive real x, since there are two positive, real zeros \(\alpha _1 < \alpha _2\).

2 Records

Let \(g_d(n)= \sigma _d(n)\). Then for \(d=1\) and \(d=2\), complete results for the log-concavity \(\Delta _n^{g_d}(1) \ge 0\), including the explicit \(E^{\sigma _d}\), are provided by Nicolas [12] and Ono–Pujahari–Rolen [13]. For \(n\le 10^{5}\) and \(d \le 8\), further results have been obtained by Heim–Neuhauser [8].

Let \(g_d(n)= \psi _d(n)\). In Table 2, we have displayed the results for \(1 \le d \le 9\). In this paper, we prove the analogue to Nicolas’ result and give some numerical evidence for the case \(d=2\), which is for \(\sigma _{2}\left( n \right) \) the log-concavity for plane partitions.

Table 2 Properties of \(\{ P_n^{\psi _d}(1)\}\)
Full size table

More generally, let \(\{g_d(n)\}\) be a double sequence in \( {\mathbb {D}}\). Then \(\Delta _1^{g_d}(1)\) is always negative, since

$$\begin{aligned} \Delta _1^{g_d}(x) = \frac{x}{2} \, \left( x - g_d(2)\right) , \end{aligned}$$

and \(g_d(2) \ge 2\). This explains the results for \(n=1\) at \(x=1\).

For \(x >0\), we have \(\Delta _{1}^{\psi _{d}}\left( x\right) \ge 0 \) if and only if \(x \ge 2^d\). Let \(n \ge 2\). Table 3 records our results for Turán inequalities for small d.

Table 3 Turán inequalities in the d aspect
Full size table
Fig. 1
figure 1

The zeros of \(\Delta _n^{\psi _2}(x)\) with the largest positive real part for \(1 \le n \le 40\), blue labels the real zeros and red the non-real zeros (Color figure online)

Full size image

In the case \(d=5\), a new feature appears (see Fig. 2). Let \(3 \le n \le 100\) then there are exactly two simple positive zeros. Their position implies \(\Delta _3^{\psi _5}(1)>0\), \(\Delta _4^{\psi _5}(1)<0\), and \(\Delta _n^{\psi _5}(1)>0\) for \( 5 \le n \le 100\). We expect that this holds true for all \(n \ge 5\).

3 Basic formulas

Let g be a normalized arithmetic function. Let

$$\begin{aligned} P_n^g(x):= \frac{x}{n} \sum _{k=1}^n g(k) \, P_{n-k}^g(x), \qquad \text {with } P_{0}^{g}\left( x\right) =1. \end{aligned}$$

Then \(P_n^g(x)\) are polynomials of degree n. We refer to [10] for a detailed study of these polynomials. For example \(P_1^g(x)=x\) and \(P_2^g(x) = x/2 \, (x +g(2))\).

3.1 Coefficients of \(P_n^g(x)\)

Let

$$\begin{aligned} P_n^g(x) = \sum _{k=0}^n A_{n,k}^g \,\, x^k. \end{aligned}$$

Then \(A_{0,0}^g=1\). Let \(n \ge 1\) then \(A_{n,0}^g=0\), \(A_{n,1}^g= g(n)\, / \, n\) and \(A_{n,n}^g = 1 \, / \, n!\). We also have [10] for \(1 \le m <n\) and \(n-m=1,2,3\):

$$\begin{aligned} A_{n,n-1}^{g}= & {} \frac{1}{n!} \,\, g\left( 2\right) \left( {\begin{array}{c}n\\ 2\end{array}}\right) ,\\ A_{n,n-2}^{g}= & {} \frac{1}{n!} \,\, \left[ 3\left( g\left( 2\right) \right) ^{2}\left( {\begin{array}{c}n\\ 4\end{array}}\right) +2g\left( 3\right) \left( {\begin{array}{c}n\\ 3\end{array}}\right) \right] ,\\ A_{n,n-3}^{g}= & {} \frac{1}{n!} \,\, \left[ 15\left( g\left( 2\right) \right) ^{3}\left( {\begin{array}{c}n\\ 6\end{array}}\right) +20g\left( 2\right) g\left( 3\right) \left( {\begin{array}{c}n\\ 5\end{array}}\right) +6g\left( 4\right) \left( {\begin{array}{c}n\\ 4\end{array}}\right) \right] . \end{aligned}$$

Lemma 3.1

Let g be a normalized arithmetic function and

$$\begin{aligned} \Delta _{n}^g(x):= P_n^g(x)^2 - P_{n-1}^g(x) \, P_{n+1}^g(x). \end{aligned}$$

Then \(\Delta _1^g(x) = \frac{x}{2} \left( x-g(2) \right) \) and \(\Delta _2^g(x) = \frac{x^2}{12} \left( x^2 + 3 \, g(2)^2 - 4 \, g(3)\right) \). Further,

$$\begin{aligned} \Delta _3^g(x)= & {} \frac{ x^{2}}{144} \bigg (x^{ 4} + 3 g\left( 2\right) x^{ 3} + \left( 9 \left( g\left( 2\right) \right) ^{2} - 8 g\left( 3\right) \right) x^{ 2} + ( -9 \left( g\left( 2\right) \right) ^{3}\\{} & {} + 24 g\left( 3\right) g\left( 2\right) - 1 8 g\left( 4\right) ) x + \left( - 1 8 g\left( 4\right) g\left( 2\right) + 16 \left( g\left( 3\right) \right) ^{2}\right) \bigg ). \end{aligned}$$
Fig. 2
figure 2

The zeros of \(\Delta _n^{\psi _5}(x)\) with the largest positive real part for \(1 \le n \le 40\), blue labels the real zeros and red the non-real zeros (Color figure online)

Full size image

This follows from the explicit form of the polynomials. We have

$$\begin{aligned} P_3^g(x)= & {} \frac{x}{6} \left( x^2 + 3 \, g(2)\, x + 2 \, g(3) \right) ,\\ P_4^g(x)= & {} \frac{x}{24} \left( x^3 + 6 \, g(2) \, x^2 + \left( 8 \, g(3) + 3 \, g(2)^2 \right) \, x + 6 \, g(4) \right) . \end{aligned}$$

3.2 Properties of \(\Delta _n^g(x)\)

Let us establish the following notation:

$$\begin{aligned} \Delta _n^g(x) = \sum _{k=0}^{2n } D_{n,k}^g \,\, x^k. \end{aligned}$$

In contrast to \(P_n^g(x)\), the coefficients of \(\Delta _n^g(x)\) are not always non-negative in general. Nevertheless, we have \(\Delta _n^g(0)=0\) and the important asymptotic property

$$\begin{aligned} \lim _{x \rightarrow \infty } \Delta _n^g(x) = \infty . \end{aligned}$$

This follows from \(D_{n,2n}^g = \frac{1}{(n!)^2 \, (n+1)}\). Let \(n \ge 2\). We can always factor out \(x^2\) and still have polynomials, since \(D_{n,0}^g = D_{n,1}^g =0\). The new constant term is given by \(D_{n,2}^g\), which does not need to be non-negative:

$$\begin{aligned} D_{n,2}^g= & {} \frac{1}{n^2} \, \left[ g(n)^2 - \frac{n^2}{n^2-1} \, g(n-1) \, g(n+1) \right] ,\\ D_{n,3}^g= & {} 2\frac{g\left( n\right) }{n^{2}}\sum _{k=1}^{n-1}\frac{g\left( k\right) g\left( n-k\right) }{k}-\frac{g\left( n-1\right) }{n -1}\sum _{k=1}^{n}\frac{g\left( n+1-k\right) g\left( k\right) }{2\left( n+1-k\right) k}\\{} & {} -\frac{g\left( n+1\right) }{n +1}\sum _{k=1}^{n-2}\frac{g\left( n-1-k\right) g\left( k\right) }{2\left( n-1-k\right) k}. \end{aligned}$$

3.3 Special cases

Let \(\{g_d(n)\}\) be a double sequence in \( {\mathbb {D}}\). We have \(\Delta _1^{g_d}(x) = x \, (x-g_d(2))/2\). Thus, \(\Delta _1^{g_d}(x) = 0\) if \(x=0\) or \(x= g_d(2)\). Thus, \(\Delta _1^{g_d}(x)>0\) if and only if \(x \not \in [0, g_d(2)]\). This implies that \(\Delta _1^{g_d}(x)<0\) for \(x\in \left( 0,g_{d}\left( 2\right) \right) \) and all \(d \in {\mathbb {N}}\). The case \(n=2\) is still directly accessible. We have \(\Delta _2^{g_d}(x) = 0\) if \(x=0\) or \(x^2= 4\, g_d(3)-3 \,\left( g_{d}\left( 2\right) \right) ^2\). We consider \(4\, g_d(3)-3 \,\left( g_{d}\left( 2\right) \right) ^2 \ge 0\) and \(x \ne 0\). Let \(g_{d} = \psi _{d} \) or \(g_{d}=\sigma _{d}\). Then \(\Delta _2^{\psi _d}(x) >0\) for \(d \in {\mathbb {N}}\), especially \(\Delta _2^{\psi _1}(x) =x^4/12\).

4 Proof of Theorem 1.1

Our strategy is to utilize the well-known formula ([11, Sect. 4.7]):

$$\begin{aligned} P_{n}^{g_{d}}\left( x\right) = \sum _{k\le n}\sum _{\begin{array}{c} m_{1},\ldots ,m_{k}\ge 1 \\ m_{1}+\ldots +m_{k}=n \end{array}} \frac{1}{k!} \, \frac{g _{d}\left( m_{1}\right) \cdots g _{d}\left( m_{k}\right) }{m_{1}\cdots m_{k}} \,\, x^k. \end{aligned}$$

4.1 Lower and upper bounds

In [8, Sect. 3], we have obtained lower and upper bounds for \(P_n^{\sigma _d}(1)\). The invented proof method can be generalized in a straightforward manner to obtain the following result for all double sequences in \( {\mathbb {D}}\) and the associated polynomials for \(x >0 \).

Proposition 4.1

Let the double sequence \(\{g_d(n)\}_{d,n \in {\mathbb {N}}}\) be an element of \( {\mathbb {D}}\). Let \(n \ge 3\) and \(x >0\). Then we have for all \(d\ge 1\) the following upper and lower bounds.

Let \(n \equiv 0 \pmod {3}\) and \(n^{\prime }:= n/3\). Then

$$\begin{aligned} \frac{3^{\left( d-1\right) n'}}{\left( n^{\prime } \right) !} \,\, x^{n^{\prime }} \, < \, P_n^{g_d}(x) \, \le \, 3^{\left( d-1\right) n^{\prime }} P_n^{g_1}(x). \end{aligned}$$

Further, let \(n \equiv 1 \pmod {3}\) and \(n^{\prime }:= (n-4)/3\). Then

$$\begin{aligned} \frac{\left( 4 \cdot 3^{n'}\right) ^{d-1}}{\left( n^{\prime } \right) !} \, \left( x^{n^{\prime }+1} + \frac{x^{n^{\prime }+2}}{2} \right) \, < \, P_n^{g_d}(x) \, \le \, \left( 4 \cdot 3^{n^{\prime }}\right) ^{d-1} \,\, P_n^{g_1}(x). \end{aligned}$$

Further, let \(n \equiv 2 \pmod {3}\) and \(n' := (n-2)/3\). Then

$$\begin{aligned} \frac{\left( 2\cdot 3^{n'}\right) ^{d-1}}{\left( n' \right) !} \,\, x^{n'+1} \, < \, P_n^{g_d}(x) \, \le \, \left( 2\cdot 3^{n'}\right) ^{d-1} P_n^{g_1}(x). \end{aligned}$$

Additionally, let \(n \equiv 2 \pmod {3}\) and \(n \ge 8\). Let \(n':= (n-2)/3\). Then

$$\begin{aligned} P_n^{g_d}(x) \le \frac{\left( 2\cdot 3^{n'}\right) ^{d-1}}{\left( n'\right) !} \, x^{n'+1}+ \left( 16\cdot 3^{n'-2 }\right) ^{d-1} \, P_n^{g_1}(x). \end{aligned}$$

4.2 Proof of Theorem 1.1

For \(x=0\), the inequality (1.1) holds certainly true. Therefore, let \(x>0\). We apply Proposition 4.1.

4.2.1 The case \(n\equiv 0 \pmod {3}\)

In the first step, we apply Proposition 4.1. This leads to

$$\begin{aligned} \frac{\left( P_{n}^{g_{d} }\left( x\right) \right) ^{2}}{P_{n-1}^{g_{d} }\left( x\right) P_{n+1}^{g_{d} }\left( x\right) } > \frac{x^{2n/3}}{\left( \left( n/3\right) !\right) ^{2}P_{n-1}^{g_{1} }\left( x\right) P_{n+1}^{g_{1} }\left( x\right) }\left( \frac{9}{8}\right) ^{d-1}. \end{aligned}$$

We choose \(r(x)>0\), such that \(P_n^{g_1}(x) \le r(x)^{-n}\) for all n. Let

$$\begin{aligned} d_{0}=d_{0}\left( n,x\right) =1+\frac{2n}{3\ln \left( 9/8\right) } \left( \ln \left( n/3\right) -\ln \left( x\right) -3\ln \left( r\left( {x}\right) \right) \right) . \end{aligned}$$

Then \(\left\{ P_{n}^{g_{d}}\left( x\right) \right\} _{n}\) is strictly log-concave at n for \(d \ge d_0\), since

$$\begin{aligned} \frac{\left( P_{n}^{g_{d}}\left( x\right) \right) ^{2}}{P_{n-1}^{g_{d}}\left( x\right) P_{n+1}^{g_{d}}\left( x\right) }> \frac{x^{2n/3}}{\left( n/3\right) ^{2n/3} \left( r\left( {x}\right) \right) ^{-2n}}\left( \frac{9}{8}\right) ^{d-1} \ge 1. \end{aligned}$$

4.2.2 The case \(n\equiv 1 \pmod {3}\)

In the first step, we apply Proposition 4.1. This leads to

$$\begin{aligned} \frac{\left( P_{n}^{g_{d} }\left( x\right) \right) ^{2}}{P_{n-1}^{g_{d} }\left( x\right) P_{n+1}^{g_{d} }\left( x\right) } \, <\, \left( \left( \frac{n-1}{3} \right) ! \right) ^{2}\,\, \left( P_{n}^{g_{1}}\left( x\right) \right) ^{2} \,\, x^{- \left( 2n+ 1\right) /3} \,\, \left( \frac{8}{9}\right) ^{d-1}. \end{aligned}$$

We choose \(r(x)>0\), such that \(P_n^{g_1}(x) \le r(x)^{-n}\) for all n. Let

$$\begin{aligned} d_{0} =d_{0} \left( n,x\right) =1+\frac{ 2n}{3\ln \left( 9/8\right) }\left( \ln \left( \frac{n-1}{3}\right) -3 \ln \left( r\left( {x}\right) \right) -\frac{ 2n+1}{ 2n} \ln \left( x\right) \right) . \end{aligned}$$

Then the sequence \(\left\{ P_{n}^{g_{d}}\left( x\right) \right\} _{n}\) is strictly log-convex at n for \(d \ge d_{0 }\), since

$$\begin{aligned} \frac{\left( P_{n}^{g_{d}}\left( x\right) \right) ^{2}}{P_{n-1}^{g_{d}}\left( x\right) P_{n+1}^{g_{d}}\left( x\right) } < \left( \frac{n-1}{3}\right) ^{ 2n /3}\left( r\left( {x}\right) \right) ^{-2n} x ^{- \left( 2n+1\right) /3 }\left( \frac{8 }{9 }\right) ^{d-1} \le 1. \end{aligned}$$

4.2.3 The case \(n\equiv 2 \pmod {3}\)

This final case involves some additional considerations. Again we first apply Proposition 4.1 and obtain

$$\begin{aligned}{} & {} \left( P_{n}^{g_{d} }\left( x\right) \right) ^{2} -P_{n-1}^{g_{d} }\left( x\right) P_{n+1}^{g_{d} }\left( x\right) \\{} & {} \quad < \left( \frac{\left( 2\cdot 3^{\left( n-2\right) /3}\right) ^{d-1} x^{\left( n+1\right) /3}}{\left( \left( n-2\right) /3\right) !}+ \left( 16\cdot 3^{\left( n-8\right) /3}\right) ^{d-1}P_{n}^{g_{1} }\left( x\right) \right) ^{2}\\{} & {} \qquad {}-\frac{ \left( 4\cdot 3^{\left( n-5\right) /3}\right) ^{d-1}\left( x^{\left( n-2\right) /3}+x^{\left( n+1\right) /3}/2\right) }{ \left( \left( n-5 \right) /3\right) !}\frac{3^{\left( d-1\right) \left( n+1\right) /3} x^{\left( n+1\right) /3}}{\left( \left( n+1\right) /3\right) !}\\{} & {} \quad \le \left( \frac{\left( 2\cdot 3^{\left( n-2\right) /3 }\right) ^{d-1}x^{\left( n+1\right) /3}}{\left( \left( n-2\right) /3\right) !}\right) ^{2} \left( 1-\frac{\left( n-2\right) /3}{\left( n+1\right) /3}\left( \frac{1}{x}+\frac{1}{2}\right) \right. \\{} & {} \qquad + 2\left( 8 \cdot 3^{ -2 }\right) ^{d-1}x^{-\left( n+1\right) /3}P_{n}^{g_{1}}\left( x\right) \left( \left( n-2\right) /3\right) !\\{} & {} \qquad \left. +\left( \left( \left( n-2\right) /3\right) !\left( 8 \cdot 3^{ -2 }\right) ^{d-1}x^{-\left( n+1\right) /3}P_{n}^{g_{1}}\left( x\right) \right) ^{2}\right) . \end{aligned}$$

The last inequality can only be not larger than zero if \(0<x<2-\frac{12}{n+4}\). We choose \(r(x)>0\) such that \(P_n^{g_1}(x) \le r(x)^{-n}\) for all n. Then

$$\begin{aligned}{} & {} \left( \frac{n-2}{3}\right) !\left( \frac{8 }{9 }\right) ^{d-1}x^{-\left( n+1\right) /3}P_{n}^{g_{1}}\left( x\right) \\{} & {} < \left( \frac{n-2}{3}\right) ^{\left( n-2\right) /3}\left( \frac{8 }{9 }\right) ^{d-1}x^{-\left( n+1\right) /3} \left( r\left( {x }\right) \right) ^{-n} \le 1, \end{aligned}$$

for

$$\begin{aligned} d\ge 1+\frac{1}{\ln \left( 9/8\right) }\left( \frac{n-2}{3} \ln \left( \frac{n-2}{3}\right) -\frac{n+1}{3}\ln \left( x\right) -n\ln \left( r\left( {x}\right) \right) \right) . \end{aligned}$$

Let \(d_{0} =d_{0} \left( n,x\right) \) be defined by

$$\begin{aligned}{} & {} 1+\frac{1}{\ln \left( 9/8\right) }\left( -\min \left\{ 0,\ln \left( \frac{n-2}{3n+3}\left( \frac{1}{x}+\frac{1}{2}\right) -\frac{1}{3}\right) \right\} \right. \\{} & {} \quad \left. {}+\frac{n-2}{3} \ln \left( \frac{n-2}{3}\right) -\frac{n+1}{3}\ln \left( x\right) -n\ln \left( r\left( {x}\right) \right) \right) . \end{aligned}$$

Then the sequence \(\{P_n^{g_d}(x)\}_d\) is strictly log-convex for all \(d \ge d_{0} \).

5 Turán inequalities

Let \(\{g_d(n)\}\) be a double sequence in \( {\mathbb {D}}\). We are interested in finding the set of positive real numbers, such that \(\Delta _n^{g_d}(x) \ge 0\), with special emphasis on the behavior at \(x=1\).

In [7], a conjecture for \(\Delta _n^{\sigma _1}(x)\) was stated, which generalized a conjecture of Chern–Fu–Tang [4] related to integers \(x \ge 2\). The Chern–Fu–Tang conjecture was proven by Bringmann et al. [3]. Recently, a second conjecture [9] was proposed for \(\Delta _n^{\sigma _2}(x)\). We have shown for \(x=1\), the case of plane partitions, that \(\Delta _n^{\sigma _2}(1)>0\) for almost all n. Finally, Ono et al. [13] have proven that \(\Delta _n^{\sigma _2}(1)>0\) for all \(n \ge 12\). We now show that \(\Delta _{n}^{\psi _{1}}\left( x\right) \ge 0\) for all \(x \in {\mathbb {R}}\). This is the first case where a full result on Turán inequalities is obtained for a double sequence in \( {\mathbb {D}}\) with d fixed.

5.1 \(\Delta _n^{\psi _1}(x) \ge 0\)

We have \(\psi _1(n)=1\). The polynomials \(P_n^{\psi _1}(x)\) had been studied in [6] and had been found intimately related with the \(\alpha \)-associated Laguerre polynomials. We have \(P_n^{\psi _1}(x)= \frac{x}{n} L_{n-1}^{(1)}(-x)\), where

$$\begin{aligned} \sum _{n=0}^{\infty } L_n^{(\alpha )}(x) \, t^n = \frac{1}{(1-t)^{ \alpha +1 }}\,\, \textrm{e}^{-x \frac{t}{1-t}}, \qquad \alpha > -1. \end{aligned}$$

The Laguerre polynomials of degree n are given by \(L_n(x)= L_n^{(0)}(x)\). It is known that \(\alpha \)-associated Laguerre polynomials for \(\alpha \ge 0\) satisfy:

$$\begin{aligned} \left( L_n^{(\alpha )}(x)\right) ^2 - L_{n-1}^{(\alpha )}(x) \, L_{n+1}^{(\alpha )}(x) = \sum _{k=0}^{n-1} \frac{ \left( {\begin{array}{c}\alpha + n-1\\ n-k\end{array}}\right) }{ n \, \left( {\begin{array}{c}n\\ k\end{array}}\right) } \left( L_k^{(\alpha -1)}(x)\right) ^2 >0. \end{aligned}$$

These Turán inequalities are not sufficient to prove \(\Delta _n^{\psi _1}(x) \ge 0\). We have to show for all \(x \in {\mathbb {R}}\) that

$$\begin{aligned} \Delta _{n}^{\psi _1}\left( x\right) = \frac{1}{n^2} \left( L_{n-1}^{(1)}(-x) \right) ^2 - \frac{1}{n^2-1} L_{n-2}^{(1)}(-x) \,\, L_{n}^{(1)}(-x)\ge 0. \end{aligned}$$

Szegő [16] proved in 1948, that \(L_n^{(\alpha )}(x) / L_n^{(\alpha )}(0)\) satisfies Turán inequalities, where \( L_n^{(1)}(0)= n+1\). This proves our claim.

5.2 Challenges

We propose three open questions.

5.2.1 Log-concavity of partition and plane partition numbers

Reprove the results of Nicolas [12] and Ono et al. [13] on the log-concavity of the partition numbers and the plane partition numbers utilizing the zero distribution of the polynomials \(\left\{ P_{n}^{\sigma _{d}}\left( x\right) \right\} \) for \(d=1\) and \(d=2\).

5.2.2 Turán inequalities \(\Delta _{n}^{\psi _d}(x) \ge 0\)

Based on our numerical investigations on the zeros of \(\Delta _n^{\psi _d}(x)\), we believe that it is very likely for \(2 \le d \le 4\) that \(\Delta _n^{\psi _d}(x) \ge 0\) for \(n \ge 2\) and \(x \in {\mathbb {R}}\). Prove this observation.

5.2.3 The case \(n \equiv 0 \pmod {3}\)

The following problem was presented at the Conference: 100 Years of Mock Theta Functions at Vanderbilt University in 2022 (organized by Rolen and Wagner). Prove that \(\Delta _n^{\psi _d}\left( x\right) \ge 0\) for all \(n \equiv 0 \pmod {3}\) and all \(d \in {\mathbb {N}}\).