Let d(c) denote the Hausdorff dimension of the Julia set Jc
of the polynomial fc
(z) = z
2 +c. The function c ↦ d(c) is real-analytic on the interval (–3/4, 1/4), which is in the domain bounded by the main cardioid of the Mandelbrot set. We prove that the function d is convex close to 1/4 on the left side of it.