Solution
= Solution
Let $z=tx+(1-t)y$. Apply the hypothesis with first argument $z$ and second argument successively $x,y$:
$$
f(z)-f(x)\le(1-t)\lambda(z)^T(y-x),
$$
$$
f(z)-f(y)\le t\lambda(z)^T(x-y).
$$
Multiply these inequalities by $t$ and $1-t$ respectively and add. The right sides cancel, leaving $f(z)\le tf(x)+(1-t)f(y)$.