The reverse inequality for $(C-S)^*$ on nonincreasing sequences]{The sharp reverse inequality for the transpose of the Ces\`aro operator minus the shift on nonincreasing sequences

Let $Cx(n)=(x(1)+\dots+x(n))/n$ be the Ces\`aro means of a sequence $x$ and $Sx(n)=x(n+1)$ its left shift. Ben Said, Boza and Soria determined the best constants in reverse inequalities for $C-S$ and for its transpose $(C-S)^*$ on the cones of arbitrary, nonnegative, and nonnegative nonincreasing sequences in $\ell^p$ for $1<p<\infty$, except the best constant $K(p)$ in $K(p)\|x\|_p\le\|(C-S)^*x\|_p$ for nonnegative nonincreasing $x$ and $2<p<\infty$. They conjectured that $K(p)^p=C_p=\int_0^1|\ln t+1|^p\,dt$, the best constant in the continuous inequality $\|(H^*-I)f\|_p^p\ge C_p\|f\|_p^p$ for nonnegative nonincreasing $f$, where $H^*f(x)=\int_x^\infty f(t)\,dt/t$. We prove that $K(p)^p=\min(2,C_p)$ for every $p\ge2$. The ratio $\|(C-S)^*x\|_p^p/\|x\|_p^p$ equals $2$ at the first unit vector and tends to $C_p$ along the indicators of $\{1,\dots,N\}$ as $N\to\infty$, which gives the upper bound $K(p)^p\le\min(2,C_p)$. Because $C_p$ increases in $p$, the conjecture holds for $2<p\le p^*$, where $p^*\approx2.8274$ solves $C_p=2$, and fails for every $p>p^*$. A telescoping argument reduces the lower bound $K(p)^p\ge\min(2,C_p)$ to two inequalities on the image of each indicator under $(C-S)^*$. The first, between two sums over its entries, follows from a comparison, independent of $p$, between its positive and negative entries. The second compares the $p$-th powers of the entries with the integral defining $C_p$. Both are proved by estimates on harmonic numbers for large $N$ and by computer for small $N$. The telescoping argument replaces a reduction of lower bounds to indicators due to Bennett, which needs a matrix with nonnegative entries, whereas the matrix of $(C-S)^*$ has a negative entry in every row after the first.