Thread: connected subharmonic function

Hi there
i need some help with this question

let D=D(0,1),let u be a continuous function on (\bar{D}) that is subharmonic on D,and let E={z \in \bar{D} :u(z)\leqslant 0}
Explain why C\E is connected,

and let u be subharmonic function on the strip U={z in C : -1< Im z < 1}
such that
lim sup u(z)\v(z)\leqslant 0 (w \in \partial \infty U)
where v(x+iy)=coshx cosy

show that u\leqslant 0

thanks

For the first use the maximum principle to see that there can be no bounded connected component, and since E is compact the complement has precisely one unbounded component.

The second one I don't understand, what does

lim sup u(z)\v(z)\leqslant 0 (w \in \partial \infty U)

mean