Let C be a contour consisting of line segment from $-4$ to $0$ followed by the line segment from $0$ to $3i$. Without evaluating the integral, show that$$\bigg|\int_C(e^z-\bar{z})\, dz\bigg|\leq 28+4e^{-4}.$$
I started with $$\bigg|\int_C(e^z-\bar{z})\, dz\bigg|\leq\int_{z=-4}^0|e^z-\bar{z}|\, |dz|+\int_{z=0}^{3i} |e^z-\bar{z}|\, |dz|$$ $$\leq\int_{z=-4}^0|e^z|\, |dz|+\int_{z=-4}^0|\bar{z}|\, |dz|+\int_{z=0}^{3i} |e^z|\, |dz|+\int_{z=0}^{3i}|\bar{z}|\, |dz|$$ Please help how to go ahead.