2 problems

Let $a$ and $b$ be integers with $0\leq k\leq a<b$. The codicil conjecture. The following equivalent-looking coefficientwise inequalities hold: … or … Here $\binom{n}{k}_q$ denotes…

Let $a$ and $b$ be integers with $0<a<b$, and let $\beta\geq1$. The $\beta$-conjecture. One has … where the inequality is coefficientwise for polynomials in $q$. This is the three-…