Abstract
AbstractWe deal with the functional inequality $$\begin{aligned} f(x)f(y) - f(xy) \le f (x) + f (y) - f(x+y) \end{aligned}$$
f
(
x
)
f
(
y
)
-
f
(
x
y
)
≤
f
(
x
)
+
f
(
y
)
-
f
(
x
+
y
)
for $$f:{\mathbb {R}}\rightarrow {\mathbb {R}}$$
f
:
R
→
R
, which was introduced by Horst Alzer and Luis Salinas. We show that if f is a solution that is differentiable at 0 and $$f(0)=0$$
f
(
0
)
=
0
, then $$f=0$$
f
=
0
on $${\mathbb {R}}$$
R
or $$f(x) = x$$
f
(
x
)
=
x
for all $$x \in {\mathbb {R}}$$
x
∈
R
. Next, we prove that every solution f which satisfies some mild regularity and such that $$f(0)\ne 0$$
f
(
0
)
≠
0
is globally bounded.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computational Theory and Mathematics,Analysis
Reference3 articles.
1. Alzer, Horst, Salinas, Luis: On the functional inequality $$f(x)f(y) - f(xy) \le f (x) + f (y) - f(x+y)$$. Comp. Methods Funct. Theory 20, 623–627 (2020)
2. Young, Grace Chisholm: A note on derivates and differential coefficients. Acta Mathematica 37, 141–154 (1914)
3. Saks, Stanisław.: Theory of the integral, Second revised edition. With two addit. notes by S. Banach. G.E. Stechert & Co., New York (1937). Reprinted by Dover Publications, New York (1964)