Radius of convexity of partial sums of odd functions in the close-to-convex family

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We consider the class of all analytic and locally univalent functions f of the form f(z)=z+∑ₙ₌₂∞ a₂ₙ₋₁ z²ⁿ⁻¹, |z|<1, satisfying the condition Re(1+zf′′(z)f′ (z))>-1/2. We show that every section s₂ₙ₋₁(z)=z+∑ₖ₌₂ⁿa₂ₖ₋₁z²ᵏ⁻¹, of f, is convex in the disk |z|<√2/3. We also prove that the radius √2/3 is best possible, i.e. the number √2/3 cannot be replaced by a larger one.