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Let $(S_0,S_1,...)$ be a supermartingale relative to a nondecreasing sequenceof $\sigma$-algebras $H_{\le0},H_{\le1},...$, with $S_0\le0$ almost surely(a.s.) and differences $X_i:=S_i-S_{i-1}$. Suppose that $X_i\le d$ and $\mathsf{Var}(X_i|H_{\le i-1})\le \sigma_i^2$ a.s. for every $i=1,2,...$, where $d>0$and $\sigma_i>0$ are non-random constants. Let $T_n:=Z_1+...+Z_n$, where$Z_1,...,Z_n$ are i.i.d. r.v.'s each taking on only two values, one of which is$d$, and satisfying the conditions $\mathsf {E}Z_i=0$ and $\mathsf{Var}Z_i=\sigma ^2:=\frac{1}{n}(\sigma_1^2+...+\sigma_n^2)$. Then, based on acomparison inequality between generalized moments of $S_n$ and $T_n$ for a richclass of generalized moment functions, the tail comparison inequality $$\mathsf P(S_n\ge y) \le c \mathsf P^{\mathsf Lin,\mathsf L C}(T_n\gey+\tfrach2)\quad\forall y\in \mathbb R$$ is obtained, where$c:=e^2/2=3.694...$, $h:=d+\sigma ^2/d$, and the function $y\mapsto \mathsf{P}^{\mathsf {Lin},\mathsf {LC}}(T_n\ge y)$ is the least log-concave majorantof the linear interpolation of the tail function $y\mapsto \mathsf {P}(T_n\gey)$ over the lattice of all points of the form $nd+kh$ ($k\in \mathbb {Z}$). Anexplicit formula for $\mathsf {P}^{\mathsf {Lin},\mathsf {LC}}(T_n\gey+\tfrac{h}{2})$ is given. Another, similar bound is given under somewhatdifferent conditions. It is shown that these bounds improve significantly uponknown bounds.

Binomial upper bounds on generalized moments and tail probabilities of (super)martingales with differences bounded from above