I am working through Arfken, Weber, and Harris' s "Mathematical Methods for Physicists" $7^{th}$ edition. Example 12.7.1 is:
"Prove Jensen's theorem (that $\left| F(z) \right|^2$ can have no extremum in the interior of a region in which $F$ is analytic) by showing that the mean value of $|F|^2$ on a circle about any point $z_0$ is equal to $|F(z_0)|^2$. Explain why you can then conclude that there cannot be an extremum of $|F|$ at $z_0$."
I am utterly confused. I thought the only condition for analyticity is that $F$ s expressible as $F(z)=F(x+iy)$ and that its partial derivatives $\frac{\partial F}{\partial x}$ and $\frac{\partial F}{\partial y}$ exist and are continuous. this implies that $F$ is expressible as $F(x,y)=u(x,y)+iv(x,y)$, leading to the Cauchy-Riemann equations:
$$ \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} \ \ , \ \ \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}\to \frac{\partial ^2u}{\partial x^2}=-\frac{\partial ^2u}{\partial y^2} \ \ , \ \ \frac{\partial ^2v}{\partial x^2}=-\frac{\partial ^2v}{\partial y^2} $$
And so it is clear that $F$ has no extrenum since at any point $(x+iy)$, if $u(x_0,y)$ is concave up then $u(x,y_0)$ is concave down (similar for $v$) and so you can only be on a saddle and never on an extremum. Also, the Average of $F$ about any $z_0$:
$$ \frac{1 }{2 \pi }\int_0^{2 \pi } F\left(Re^{i\theta }+z_0\right) \, d\theta=\frac{1}{2 \pi }\int_0^{2 \pi } \left(F\left(z_0\right)+\frac{Re^{i\theta} }{1!}\left(\frac{dF}{dz}\right)_{z=z_0}+\frac{R^2 e^{2 \text{i$\theta $}}}{2!}\left(\frac{d^2F}{dz^2}\right){}_{z=z_0}+\text{...}\right) \, d\theta =F\left(z_0\right)+0+0\text{...} $$
is certainly equal to $F(z_0)$.
However, when we consider $|F|^2=u^2+v^2$, one can certainly think of an example where the value of $F$ at the saddle point is zero (example: $F(z)=z-iz=x+y-ix+iy \ \ , \ \ (x_0,y_0)=(0,0)$ ). The value of $u$ and $v$ either increases or decreases as you move away from $z_0$ and so $|F|^2$ only gets bigger (reminder, we assume, as in the given example, that $F(z_0)=0$). So the average on the circle also cannot be equal to zero. Considering the example, $|F|^2=2(x^2+y^2)$ clearly has a minimum at $(0,0)$ and the average of $|F|^2$ around a circle is not zero.
My question is why the book is correct / what am I missing. Also, when I search for Jensen's theorem, why do none of the three theorems I see resemble the one stated in the book?
The result you quoted looks like the maximum modulus theorem via the mean value theorem (for harmonic or analytic functions, not the one for differentiable functions on an interval), although it's also incorrect as stated. If $F$ is analytic and nonzero everywhere on a region $\Gamma$, then $|F|^2$ has no minimum in the interior of $\Gamma$. (I'm assuming "extremum" means "minimum or maximum.") The result for the minimum is just the maximum modulus theorem applied to $1/F$, and, like you said, $F(z) = z$ on the unit disk gives a counterexample if the nonzero condition is dropped.
I've always seen the third item on the page you linked to, the one giving an expression for $\log |f(0)|$, referred to as Jensen's theorem. (There's also a Jensen's inequality and variants, but those don't have much to do with this particular setting.)