We suggest that the reader reads first the entry on Bloch's constant. Let $\mathcal{F}$ be the set of all functions $f$ holomorphic on a region containing the closure of the disk $D=\{z\in\mathbb{C}:|z|<1\}$ and satisfying $f(0)=0$ and $f'(0)=1$ For each $f\in\mathcal{F}$ let $\lambda(f)$ be the supremum of all numbers $r$ such that there is a disk $S\subset D$ such that $f(S)$ contains a disk of radius $r$ (notice that here we don't require $f$ to be injective on $S$ .

Let $B$ be Bloch's constant. Then, clearly, $L\geq B$ The exact value of $L$ (as that of $B$ is not known but it has been shown that $0.5 \leq L \leq 0.56$ In particular, it is known that $L$ is strictly greater than $B$

Bibliography

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John B. Conway, Functions of One Complex Variable I, Second Edition, 1978, Springer-Verlag, New York.