# A complex log inequality

Standard

Let $z$ be a complex number. The power series expansion of $\text{log}(1+z)$ about $z_0=0$ is given by

$\displaystyle{\text{log}(1+z) = \sum_{n=1}^\infty (-1)^{n-1}\frac{z^n}{n} = z-\frac{z^2}{2} + \ldots}$

which has radius of convergence 1. If $|z|<1$ then

$\displaystyle{\left|1-\frac{\log(1+z)}{z}\right|\leq \frac{|z|}{2(1-|z|)}}$

If we further assume $|z|<1/2$ then

$\displaystyle{\left|1-\frac{\log(1+z)}{z}\right|\leq \frac{1}{2}}$

This gives,

$\displaystyle{\frac{|z|}{2}\leq |\log(1+z)|\leq \frac{3|z|}{2}, \quad |z|<\frac{1}{2}}$

I just wanted to see how this inequality will appear graphically, so here are the plots made using SageMath 7.5.1 (by fixing the real part of z to zero and varying the imaginary part till 1/2)

i = CDF.0; p1 = plot(lambda t: abs(log(1+t*i)), 0, 0.5, rgbcolor=(0.8,0,0),legend_label=’ $|log(1+z)|$’, thickness=2); p2 = plot(lambda t: abs((t*i)/2), 0, 0.5, rgbcolor=(0,0.8,0), legend_label=’$|z|/2$’, thickness=2); p3 = plot(lambda t: abs(3*(t*i)/2), 0, 0.5, rgbcolor=(0,0,0.8), legend_label=’ $3|z|/2$’, thickness=2); p1+p2+p3

I tried to get a graph where this inequlaity fails (i.e. the plots intersect), but failed to do so.

# Real vs Complex Plane

Standard

Real plane is denoted by $\mathbb{R}^2$ and is commonly referred to as  Cartesian plane. When we talk about $\mathbb{R}^2$ we mean that $\mathbb{R}^2$ is a vector space over $\mathbb{R}$. But when you view $\mathbb{R}^2$ as Cartesian plane, then it’s not technically a vector space but rather an affine space, on which a vector space acts by translations, i.e. there is no canonical choice of where the origin should go in the space, because it can be translated anywhere.

Cartesian Plane (345Kai at the English language Wikipedia [Public domain, GFDL or CC-BY-SA-3.0], via Wikimedia Commons)

On the other hand, complex plane is denoted by $\mathbb{C}$ and is commonly referred to as Argand plane. But when we talk about $\mathbb{C}$, we mean that $\mathbb{R}^2$ is a field (by exploiting the tuple structure of elements) since there is only way to explicitly define the field structure on the set $\mathbb{R}^2$ and that’s how we view $\mathbb{C}$ as a field (if you allow axiom of choice, there are more possibilities; see this Math.SE discussion).

Argand Plane (Shiva Sitaraman at Quora)

So, when we want to bother about the vector space structure of $\mathbb{R}^2$ we refer to Cartesian plane and when we want to bother about the field structure of $\mathbb{R}^2$ we refer to Argand plane. An immediate consequence of the above difference in real and complex plane is seen when we study multivariable analysis and complex analysis, where we consider vector space structure and field structure, respectively (see this Math.SE discussion for details). Hence the definition of differentiation of a function defined on $\mathbb{C}$ is a special case of definition of differentiation of a function defined on $\mathbb{R}^2$.