On the directional derivative of the Hausdorff dimension of quadratic polynomial Julia sets at 1/4

Let d(epsilon) and D(delta) denote the Hausdorff dimension of the Julia sets of the polynomials p(epsilon)(z) = z(2) + 1/4 + epsilon and f(delta)(z) = (1 + delta) z + z(2) respectively. In this paper we will study the directional derivative of the functions d(epsilon) and D(delta) along directions landing at the parameter 0, which corresponds to 1/4 in the case of family z(2) + c. We will consider all directions, except the one epsilon is an element of R+ (or two imaginary directions in the delta parametrization) which is outside the Mandelbrot set and is related to the parabolic implosion phenomenon. We prove that for directions in the closed left half-plane the derivative of d is negative. Computer calculations show that it is negative except a cone (with opening angle approximately 150 degrees) around R+.