Use the exp_cone set documented at http://cvxr.com/cvx/doc/funcref.html#sets . That documentation does not tell you that you need to implement
y > 0 , y*exp(x/y) <= z
as
{x,y,z} == exponential(1)
In your case, the roles of x and y are reversed from the above. So you could do
cvx_begin
variables x y z
minimize(z)
subject to
{y,x,z} == exponential(1)
0 <= y <= 1
cvx_end
I recommend you read CVXQUAD: How to use CVXQUAD's Pade Approximant instead of CVX's unreliable Successive Approximation for GP mode, log, exp, entr, rel_entr, kl_div, log_det, det_rootn, exponential cone. CVXQUAD's Quantum (Matrix) Entropy & Matrix Log related functions . The code I provided above will invoke CVXQUAD’s Pade Approximation, presuming that CXVQUAD and its exponential.m replacement are installed per the instructions.
That said, this is a rather boring problem, exactly as stated, because the optimal solution has x = y = z = 0. So presumably, you ultimately want to solve a more interesting problem building on this construct.