  The Upsilon invariant of a knot is a concordance invariant derived from knot
Floer homology theory. It is a piecewise linear continuous function defined on
the interval $[0,2]$. Borodzik and Hedden gave a question asking for which
knots the Upsilon invariant is a convex function. It is known that the Upsilon
invariant of any $L$-space knot, and a Floer thin knot after taking its mirror
image, if necessary, as well, is convex. Also, we can make infinitely many
knots whose Upsilon invariants are convex by the connected sum operation. In
this paper, we construct infinitely many mutually non-concordant hyperbolic
knots whose Upsilon invariants are convex. To calculate the full knot Floer
complex, we make use of a combinatorial method for $(1,1)$-knots.
