An elementary inequality for three complex numbers

Question

The following problem arose in asymptotic analysis of difference equations.

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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?

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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 $1$.

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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.

Answer 1

I will prove the original inequality.

First, performing the change of variables $x=1/a$, etc., and inverting the harmonic mean, we need $$ \sum \left|\frac{yz}{x(y+z-x)}\right|\geq \frac32. $$ Next, denoting $p=y+z-x$, etc., we transform the inequality to $$ \sum\left|\frac{(p+q)(p+r)}{p(q+r)}\right|\geq 3, $$ or $$ \sum\left|2+\frac{(p-q)(p-r)}{p(q+r)}\right|\geq 3. $$ Now we perform the last change $$ u= \frac{(p-q)(p-r)}{p(q+r)}, \quad\text{etc.,} $$ and notice that $$ \sum\frac 1u=\sum \frac{p(q+r)}{(p-q)(p-r)} =-1 $$ (which is just a three-variable identity). Thus, it suffices to show that $$ \sum |2+u|\geq 3 \quad \text{whenever}\quad \sum\frac1u=-1. $$

Denote $|2+u|,|2+v|,|2+w|$ by $r_{1,2,3}$, respectively.

Assume first that $r_i\leq 2$ for all $i$. The inverse image of $|2+z|=r$ (for $r\leq 2$) is a circle whose rightmost point is $-1/(r+2)$, hence, say, $$ \Re \frac1u\leq - \frac1{r_1+2}. \qquad(*) $$ Adding up three such inequalities, we obtain $$ -1\leq-\sum\frac1{r_i+2} \leq-\frac 9{\sum(r_i+2)}, $$ whence $\sum(r_i+2)\geq 9$, as desired.

Assume now that, say, $r_1>2$ (but $r_1<\sum r_i<3$, arguing indirectly). Then $\frac1u$ lies outside the disk having diameter $[-1/5,1]$ (as $|2+u|<3$). On the other hand, we have $r_2+r_3<1$, so $\frac1v, \frac1w$ lie in the open disks having diameters $$ \left[-\frac 1{2-r_i},-\frac1{2+r_i}\right]. $$ Since $$ \frac1{2-r_2}+\frac1{2-r_3}\leq \frac32 \quad\text{and}\quad \frac1{2+r_2}+\frac1{2+r_3}\geq \frac45, $$ point $\frac1u =-1-\frac1v-\frac1w$ lies in the open disk having diameter $$ \left[-1+\frac45,-1+\frac32\right] = \left[-\frac15, \frac12\right]. $$ which contradicts the previous region indicated for $\frac1u $.

Equality arises only for $u=v=w=-3$, which seems to be rolled back easily.