Proof of the Bessenrodt--Ono inequality by Induction

In 2016 Bessenrodt--Ono discovered an inequality addressing additive and multiplicative properties of the partition function. Generalization by several authors have been given; on partitions with rank in a given residue class by Hou--Jagadeesan and Males, on $k$-regular partitions by Beckwith--Bessenrodt, on $k$-colored partitions by Chern, Fu, Tang, and Heim--Neuhauser on their polynomization, and Dawsey--Masri on the Andrews ${\it spt}$-function. The proofs depend on non-trivial asymptotic formulas related to the circle method on one side, or a sophisticated combinatorial proof invented by Alanazi--Gagola--Munagi. We offer in this paper a new proof of the Bessenrodt--Ono inequality, which is built on a well-known recursion formula for partition numbers. We extend the proof to the result of Chern--Fu--Tang and its polynomization. Finally, we also obtain a new result.


Introduction and main results
Bessenrodt and Ono [BO16] discovered an interesting inequality for partition numbers p(n). Their discovery nudged further research and new results in several directions.
A partition is called a k-colored partition of n if each part can appear in k colors. The proof depends on results of Rademacher [Ra37] and Lehmer [Le39] on the size of the partition numbers, built on the circle method of Hardy and Ramanujan. The partition numbers are considered as the coefficients of a weakly modular form, the reciprocal of the Dedekind η-function [On03]. The modularity condition is very strong, and it would be desirable to also have other proofs.
Two proofs are given. One depends on the Bessenrodt-Ono inequality and some new ideas, the second is combinatorial and motived by the work of Alanazi, Gagola, and Munagi [AGM17].
Bessenrodt-Ono type inequalities appeared also in works by Beckwith and Bessenrodt [BB16] on k-regular partitions and Hou and Jagadeesan [HJ18] on the numbers of partitions with ranks in a given residue class modulo 3. Males [Ma20] obtained results for general t and Dawsey and Masri [DM19] obtained new results for the Andrews spt-function. The authors of this paper generalized the Chern-Fu-Tang Theorem to D'Arcais polynomials, also known as Nekrasov-Okounkov polynomials [Ne55, NO06, Ha10, CFT18, HN20, HNT20]. Let with σ(k) := d|k d and the initial condition P 0 (x) = 1. Then p −k (n) = P n (k) and p(n) = p −1 (n) = P n (1).
The proof is based on the result for 2-colored partitions [CFT18], Lehmer's [Le39] lower and upper bound on the partition numbers, and a detailed analysis of the growth of the derivative of P a,b (x).
These results are related to the work of Griffin, Ono, Rolen, and Zagier [GORZ19] on Jensen polynomials and their hyperbolicity. This includes work of Nicolas [Ni78] and De Salvo and Pak [DP15] on the log-concavity of the partition function p(n) for n > 25, and results and a conjecture of Chen, Jia, and Wang [CJW19] for the higher order Turán inequalities. Related to their work on the Bessenrodt-Ono inequality, Chern, Fu, and Tang, came up with a subtle and explicit conjecture on k-colored partitions. The positivity of the discriminant in the case of degree 2 is equal to the log-concavity of the considered sequences coded in the Jensen polynomials.
Conjecture 2 (Heim, Neuhauser 2020). Let a > b ≥ 0 be integers. Then for all x ≥ 2: Based on a recently obtained exact formula of Rademacher type (based on the circle method) for P n (x) with x > 0 and n > x 24 obtained by Iskander, Jain, and Talvola [IJT20], new strong estimates on the Bessel function, the determination of the main term of P a,b (x), and some sophisticated computer calculation, Bringmann, Kane, Rolen, and Tripp [BKRT20] were able to proof the conjecture of Chern, Fu, and Tang. They essentially proved the conjecture for x = 2, 3, 4 and applied a result of Hoggar on the convolution of log-concave sequences. They further proved that the conjecture of Heim and Neuhauser is true The crux of these methods is that one needs for the general case Rademacher type formulas, and has to check the conjecture for each x for finitely many cases. In the discrete case for the Bessenrodt-Ono inequality there is a combinatorial proof available, this was also requested in [BKRT20], see concluding remarks (5), for the Chern-Fu-Tang conjecture.
In this paper we offer a new proof for the Bessenrodt-Ono inequality for partition numbers. Ingredients are the well-known recurrence property: and an elementary upper bound of σ(n) and lower bound of p(n). The proof also perfectly fits to the Bessenrodt-Ono inequality for k-colored partitions p −k (n) and D'Arcais polynomials P n (k). With slight modification and including some extra considerations, we obtain new proofs of Theorem 1.2 of Chern-Fu-Tang and Theorem 1.3. Finally, we obtain the following Theorem, to give evidence that the proof method offered in this paper also gives an extension of Theorem 1.3.
It is hoped that the results of this paper lead to a new proof of the former Conjecture 1.2 of Chern-Fu-Tang [BKRT20] and to a proof of the Conjecture 2.

New Proof of the Bessenrodt-Ono inequality
We estimate the divisor sum function σ(n) and the partition numbers p(n) with the following upper and lower bounds: The upper bound for σ(n) follows easily by integral comparison. We have a strict upper bound for n > 1. There are exactly n−1 k−1 ways to represent n as a sum of exactly k positive integers. Thus, we have at most k! compositions representing the same partition. For a generalization we refer to Section 3 and the relation to associated Laguerre polynomials.
Proof. Let A := 2 and B := 10. Let n ≥ B. We say the statement S(n) is true if for all partitions n = a + b with a, b ≥ A: By symmetry the claim can be reduced to all pairs (a, b) with A ≤ b ≤ a. We assume that n > N 0 > 1 and S(m) are true for all B ≤ m ≤ n − 1. For B ≤ ℓ ≤ N 0 we show S(ℓ) by a direct computer calculation with PARI/GP. Note, it is sufficient to prove S(n) for fixed A ≤ b ≤ a with a + b = n. We have introduced the constants A, B and N 0 to make the generalization of the given proof in our applications transparent. It will turn out that we can chose N 0 = 2184.
We utilize the recurrence (1.8) and obtain for p a,b (n) = L + R the expressions: We show that p a,b (n) > 0. Further, we will refine the right sum R into 2.2. Right sum R. The dominant term is related to k = 1 appearing in the right sum R. Note that the induction hypothesis cannot be applied in general to all terms. Therefore, we decompose the right sum R into three parts. Let 2.2.1. The sum R 1 . The first sum related to k = 1 is simplified by the induction hypothesis: Thus, we obtain the lower bound: (2.10) The second sum, using again the induction hypothesis, can be estimated from below with 0. This will be sufficient for our purpose: R 2 > 0.

Final step.
Putting everything together leads to In the last step we used the property (2.2). For a → ∞ we can immediately observe that this is positive since the sum is a polynomial of degree 4 in a which grows faster than a 2 (1 + ln (2a)). In fact the expression (2.17) is positive for all a ≥ 1093. Note that if a < 1093 then a + b ≤ 2a ≤ 2184 = N 0 . Therefore, we have shown that p a,b (n) > 0, which proves the Theorem.

Applications
We extend the proof method presented in Section 2 to prove the Bessenrodt-Ono inequality for k-colored partitions and its extension to D'Arcais polynomials. Let k ∈ N. Then P n (k) is equal to the k-colored partition number and p(n) = P n (1). We define Before we start, we fix the following lower bound for the D'Arcais polynomials P n (x). Let x and α be real numbers with x ≥ 0 and α > −1. Let L (α) n (x) be the αassociated Laguerre polynomial. Then P n (x) ≥ x n L (1) n−1 (−x). We refer to [HLN19]. This implies 3.1. Bessenrodt-Ono for x > 3 and arbitrary a and b. We first prove that P a,b (x) > 3 is true for all a and b ∈ N. Since there are no restrictions on a and b, the proof will be straightforward.
Proposition 3.1. Let a and b be positive integers with a, b ≥ 1 and x a real number with x > 3. Then P a,b (x) > 0.
Proof. We follow the proof by induction of the Bessenrodt-Ono inequality presented in Section 2. Let n ≥ 2. The statement S (n) is true if for all partitions n = a + b with a, b ≥ 1 holds P a,b (x) > 0 for x > 3. Let 1 ≤ b ≤ a. Let n > N 0 and S(m) be true for all 2 ≤ m ≤ n − 1. We show that it is sufficient to put N 0 = 14. Note that S(n) is true for all 1 ≤ a, b ≤ 14 (see Table 3). Let P a,b (x) = L + R with L and R defined as in (2.4) and (2.5), where we have to substitute p(n) by P n (x).
3.1.1. Left sum L. The left sum satisfies ln (a + b)) .

Right sum R.
We take care about the dominating term for k = 1. Similar to (2.8) with we study R = R 1 + R 2 + R 3 . By the induction hypothesis we get R 2 > 0. And since we have no extra condition on a and b we also get R 3 ≥ 0. Thus, only R 1 attached to k = 1 contributes and leads to

Final step.
Putting everything together leads to In the last step we used the property (3.2) and that x > 3. We obtain that the expression (3.4) is positive for all a ≥ 12. Now 200 (1 + ln (20)) < 800 and P 8 (3) = 810. Therefore, (3.3) is positive already for a ≥ 8. Since the leading coefficient 1 a!b! − 1 (a+b)! of P a,b (x) is positive, we only have to check the largest real zero of all remaining P a,b (x). This was done for 2 ≤ b+a ≤ N 0 = 14 with PARI/GP (Table 3). Note that in the case of P 1,1 (x) = (x − 3) x 2 the largest real zero is exactly 3. 3.2. The 2-colored partitions. Chern-Fu-Tang ([CFT18], Theorem 1.2) proved the following result.
Theorem 3.2. Let a and b be positive integers with a, b ≥ 1 and n = a + b ≥ 5. Then P a,b (2) > 0.
Proof. We have A = 1, B = 5, and k 0 = a − max {5 − b, 1} + 1. Let n ≥ 5 and S(n) be the statement: P a,b (n) > 0 for all a, b ≥ 1 with n = a+b. A numerical calculation with PARI/GP shows that S(m) is true for all 5 ≤ m ≤ N 0 = 28. We prove S(n) by induction on n. Let n = a + b > N 0 and 1 ≤ b ≤ a. Let P a,b (2) = L + R. Then Further, We have R 2 ≥ 0 by the induction hypothesis and R 3 ≥ 0 for b > 3. Moreover From the induction hypothesis and Table 2 we see that this is non-negative for a ≥ 2 or b ≥ 2. For a = 1 = b we obtain (P 1 (2)) 2 − P 2 (2) = −1. This leads to P a,b (2) > b P b (2) 2 a 2 −3 a 2 (1 + ln (a + b)) + P a−1 (2) (3.5) In the last step we used the property (3.2). For a → ∞ we can immediately observe that this is positive since the sum is a polynomial of degree 4 in a, which grows faster than a 2 (1 + ln (2a)). In fact, the expression (3.6) is positive for all a ≥ 15. For the remaining 5 ≤ a + b ≤ 28 we have checked with PARI/GP that P a,b (2) > 0.  Table 2. Values of P a,b (2) for a, b ∈ {1, 2, 3, 4}.
3.3. Proof of Theorem 1.4. Let n ≥ 5. The statement S (n) is true if for all partitions n = a + b with a, b ≥ 1 holds P a,b (x) > 0 for x > 1.8. Let 1 ≤ b ≤ a. Let n > N 0 and S(m) be true for all 5 ≤ m ≤ n − 1. We show that it is sufficient to put N 0 = 28. Note that S(n) is true for all 1 ≤ a, b ≤ 28 (compare Table 3). Let P a,b (x) = L + R with L and R defined as in (2.4) and (2.5), where we have to substitute p(n) by P n (x).
3.3.1. Left sum L. The left sum satisfies ln (a + b)) .

3.3.2.
Right sum R. We study R = R 1 + R 2 + R 3 . By the induction hypothesis we get R 2 > 0. R 1 associated to k = 1 leads to R 1 > b 2 a 2 P a−1 (x) P b (x).