The following problem arose in asymptotic analysis of difference equations.

Numerical maximization suggests that for all nonzero complex numbers $a,b,c$ we have $$h\big(r(a,b,c),r(b,c,a),r(c,a,b)\big)\le2,\tag{1}$$ where $$r(a,b,c):=\Big|\frac{a b + a c - b c}{a^2}\Big|$$ and $h(\cdot,\cdot,\cdot)$ is the harmonic mean, with the equality in (1) iff $a,b,c$ are the vertices of an equilateral triangle centered at $0$.

Is this true?

**Remark 1:** If the above conjecture is true, then obviously it will also hold if the harmonic mean $h\big(r(a,b,c),r(b,c,a),r(c,a,b)\big)$ is replaced by $\min\big(r(a,b,c),r(b,c,a),r(c,a,b)\big)$.

**Remark 2:** For nonzero real $a,b,c$, it appears that the best upper bound $2$ on $h\big(r(a,b,c),r(b,c,a),r(c,a,b)\big)$ should be replaced by $3/2$, "attained in the limit" iff two of the three real numbers $a,b,c$ are equal to each other while the remaining one goes to $0$.

**Remark 3:** For nonzero real $a,b,c$, Mathematica says that the best upper bound on $\min\big(r(a,b,c),r(b,c,a),r(c,a,b)\big)$ is $1$.

A correct proof of the main, "harmonic-complex" conjecture or of the conjecture stated in Remark 2 or a "human" proof of the fact stated in Remark 3 would be enough for acceptance.