Find the real part of function and determine whether it is convex or not. If we have $$g(z)=1+ \int_{0}^{1} \frac {e^{-i\delta}zu^\frac{\alpha}{\beta}}{1-zu} du$$ where $\alpha, \beta \in \mathbb{R},|\delta|\leq \frac {\pi}{2}, z $ is complex and $|z|<1$. How to find the real part of the function $g(z)$ and determine whether it is convex or not?
 A: I am assuming $z$ is real. First, we recall the fact, 

A function $f$ is convex on $(a,b)$ iff $f''(x)\geq 0$.

Using the change of variables $t=zu$ we have
$$ g(z)=1+  z e^{-i\delta}\int_{0}^{1}\frac {u^\frac{\alpha}{\beta}}{1-zu} du= 1+  z^{-\frac{\alpha}{\beta}} e^{-i\delta}\int_{0}^{1}\frac {t^\frac{\alpha}{\beta}}{1-t} du. $$
Now, the real part is 
$$ u(z)= 1+\cos(\delta)z^{-\frac{\alpha}{\beta}} \int_{0}^{1}\frac {t^\frac{\alpha}{\beta}}{1-t} du$$ 
which implies 
$$u''(z)= \frac{\alpha}{\beta}(\frac{\alpha}{\beta}+1)z^{-2-\frac{\alpha}{\beta}}\cos(\delta)\int_{0}^{1}\frac {t^\frac{\alpha}{\beta}}{1-t} du. $$
From the above, we have 
$$\int_{0}^{1}\frac {t^\frac{\alpha}{\beta}}{1-t} du \geq 0,\,\cos(\delta)\geq 0, \, \forall  \,|\delta| \leq  \pi/2. $$
Now, you just need to impose some conditions on $\alpha$ and $\beta$ if possible. So I leave it here for you to finish the task. 
A: Put ${\alpha\over\beta}=:\rho>-1$. Then
$${zu^\rho\over1-zu}=zu^\rho\sum_{k=0}^\infty z^k u^k\qquad(|z u|<1)\ .$$
It follows that
$$\int_0^1 {zu^\rho\over1-zu}\ du=\sum_{k=1}^\infty {z^k\over\rho+k}\qquad(|z|<1)$$
and therefore
$$g(z)=1+e^{-i\delta}\sum_{k=1}^\infty {z^k\over\rho+k}\qquad(|z|<1)\ .$$
As $g$ is an analytic function in $D:\> |z|<1\>$ its real part $u$ is a harmonic function of $x$ and $y$. Now nonconstant harmonic functions cannot be convex, nor concave. This can be seen as follows: In two variables the convexity criterion is that the Hessian should be positive semidefinite. This requires
$$u_{xx}\geq0,\quad u_{xx}u_{yy}-u_{xy}^2\geq0\ .$$
Since for a harmonic $u$ we have $u_{yy}\equiv-u_{xx}$ it follows that the Hessian is indefinite apart from a set of isolated points.
