MathDB

Consider functions

f

satisfying the following four conditions: (1)

f

is real-valued and defined for all real numbers. (2) For any two real numbers

x

and

y

we have

f(xy)=f(x)f(y)

. (3) For any two real numbers

x

and

y

we have

f(x+y) \le 2(f(x)+f(y))

. (4) We have

f(2)=4

.
Prove that: a) There is a function

f

with

f(3)=9

satisfying the four conditions. b) For any function

f

satisfying the four conditions, we have

f(3) \le 9

Problem Statement