Let $A$ be the family of functions $f(z)=z+a_2z^2+\cdots$ that are analytic in unit disk $D:\{z:|z|<1\}$ and $S$ is the subfamily of functions that are univalent in $D$. $R(a)$ is the subfamily of $A$ consisting of functions $f$ for which $\operatorname{Re}(f'(z)+zf''(z))>a$, $a<1$.
If $f\in R(a)$, then $\operatorname{Re}f'(z)>(1-a)(2\log2-1)+a$. How to obtain the infimum of the inequalities so that we can show $R(a) \subset S$ ?