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- Homework Statement
- please see below

- Relevant Equations
- please see below

**Problem:**

Find the set of all harmonic functions ##u(x,y,z)## that satisfy the following inequality in all of ##R^3##

$$|u(x,y,z)|\leq A+A(x^2+y^2+z^2)$$

where ##A## is a nonzero constant.

**Work:**

I removed the absolute value bars by re-writing the expression

$$-C-C(x^2+y^2+z^2)\leq u\leq C+C(x^2+y^2+z^2)$$

I am looking for a function ##u## that satisfies ##\Delta u = 0## in ##R^3##, because we can take ##A## as large as needed to satisfy the inequality for any ##u##

List of solutions: all harmonic functions, too many to list.

I suppose one obvious solution is obtained by setting ##\leq \Rightarrow =##

$$\pm u(x,y,z)=C+C(x^2+y^2+z^2)$$

I think the solution is supposed to be in abstract form to describe many types of functions. I don't understand why the term on the right side of the inequality is ##C+C(x^2+y^2+z^2)##. There must be a clue that I am oblivious to. Thank you in advance for any advice.