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

Let d(ɛ) and $\mathcal{D}\left(\delta \right)$ denote the Hausdorff dimension of the Julia sets of the polynomials p_ɛ(z) = z² + 1/4 + ɛ and f_δ(z) = (1 + δ)z + z² respectively. In this paper we will study the directional derivative of the functions d(ɛ) and $\mathcal{D}\left(\delta \right)$ along directions landing at the parameter 0, which corresponds to 1/4 in the case of family z² + c. We will consider all directions, except the one $\varepsilon \in {\mathbb{R}}^{+}$ (or two imaginary directions in the δ 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°) around ${\mathbb{R}}^{+}$.