Let $(x,y,z)$ be the sides of a triangle inscribed in a unit circle and $a+bi$ be a complex number with positive real and imaginary parts.
Question: What is the minimum $a_0$ such that for all $a > a_0$ and $b > 0$,
$$ \max\left(\left|x^{a+bi} + y^{a+bi} - z^{a+bi}\right|\right) \le 2^{a+1}? $$
Experimental evidence show that $a_0$ is about $2.196$. For example if $a = 2.196$, I found that for $b = 1$ and
$$ x = 1.9993004636979215, y = 1.9992973006020347, z = 0.10586804423264035 $$
the value of the above expression is $9.167956548880614$ and this is already greater than $2^{a+1}$. But if we increase $a$ slightly to $a = 2.1967$, I could not find any $b$ and $(x,y,z)$ for which the above expression is greater than $2^{a+1}$. So $a_0$ must to close to this value.
Motivation: For real exponents, in this question it was proved that the probability $$P\left(x^a + y^a \ge z^a\right) = \frac{1}{a^2}$$ and trivially the maximum value of $x^a+y^a-z^a$ is $2^{a+1}$. The maxima happens in the case of a degenerate triangle when $x = y = 2$ and $z=0$. So non-degenerate triangle, the maxima is $< 2^{a+1}$. So it logical to ask does this maxima hold for complex exponents. I observed experimentally the maxima holds for complex exponents only if the real part does not exceed a critical value $a_0 \approx 2.196$. It is interesting that:
$a_0$ is critical value below which both real exponents and complex exponents give the same maxima.
