Radii of convexity and close-to-convexity of certain integral representations

Strict upper bounds are determined for $|s(z)|$, $|\mathrm{Re}\,s(z)|$, and $|\mathrm{Im}\,s(z)|$ in the class of functions $s(z)=a_nz^n+a_{n+1}z^{n+1}+\dots$ ($n\geqslant1$) regular in $|z|<1$ and satisfying the condition
$$ |u(\theta_1)-u(\theta_2)|\leqslant K|\theta_1-\theta_2|, $$
where $u(\theta)=\mathrm{Re}\,s(e^{i\theta})$, $K>0$, and $\theta_1$ and $\theta_2$ are arbitrary real numbers. These bounds are used in the determination of radii of convexity and close-to-convexity of certain integral representations.