Convexity of relative entropy in its first argument ($D(\lambda q+(1-\lambda )r\Vert p)\le \lambda D(q\Vert p)+(1-\lambda )D(r\Vert p)$)

For a fixed probability mass function $p$ on a finite alphabet, relative entropy is a convex function of its first probability mass function. For $p_i>0$, the coordinate function $f_i(t)=t{\log }_2(t/p_i)$ has $f_i^{\prime \prime }(t)=1/(t\ln 2)>0$ on $(0,\infty )$ and extends continuously with $f_i(0)=0$. Applying scalar convexity coordinate by coordinate proves the inequality. If $p_i=0$, finite relative entropy requires $q_i=r_i=0$ for every positive-weight constituent; otherwise the right side is infinite. Thus the assertion also holds as an extended-valued convexity statement. On the probability simplex supported where $p>0$, the function is strictly convex: two distinct probability mass functions give a strict inequality for $0<\lambda <1$.