On a General Approach to Bessenrodt–Ono Type Inequalities and Log-Concavity Properties
Abstract In recent literature concerning integer partitions one can find many results related to both the Bessenrodt–Ono type inequalities and log-concavity properties. In this note, we offer some general approach to this type of problems. More precisely, we prove that under some mild conditions on...
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| Published in | Annals of combinatorics |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
21.05.2024
|
| Online Access | Get full text |
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| Summary: | Abstract
In recent literature concerning integer partitions one can find many results related to both the Bessenrodt–Ono type inequalities and log-concavity properties. In this note, we offer some general approach to this type of problems. More precisely, we prove that under some mild conditions on an increasing function
F
of at most exponential growth satisfying the condition
$$F(\mathbb {N})\subset \mathbb {R}_{+}$$
F
(
N
)
⊂
R
+
, we have
$$F(a)F(b)>F(a+b)$$
F
(
a
)
F
(
b
)
>
F
(
a
+
b
)
for sufficiently large positive integers
a
,
b
. Moreover, we show that if the sequence
$$(F(n))_{n\ge n_{0}}$$
(
F
(
n
)
)
n
≥
n
0
is log-concave and
$$\limsup _{n\rightarrow +\infty }F(n+n_{0})/F(n)<F(n_{0})$$
lim sup
n
→
+
∞
F
(
n
+
n
0
)
/
F
(
n
)
<
F
(
n
0
)
, then
F
satisfies the Bessenrodt–Ono type inequality. |
|---|---|
| ISSN: | 0218-0006 0219-3094 |
| DOI: | 10.1007/s00026-024-00700-7 |