  Let $\BS_1,...,\BS_n$ be independent identically distributed random variables
each having the standardized Bernoulli distribution with parameter $p\in(0,1)$.
Let $m_*(p):=(1+p+2p^2)/(2\sqrt{p-p^2}+4p^2)$ if $0<p\le 1/2$ and $m_*(p):=1$
if $1/2\le p<1$. Let $m\ge m_*(p)$. Let $f$ be such a function that $f$ and
$f''$ are nondecreasing and convex. Then it is proved that for all nonnegative
numbers $c_1,...,c_n$ one has the inequality $$\E f(c_1\BS_1+...+c_n\BS_n)\le\E
f(s^{(m)}\cdot(\BS_1+...+\BS_n)),$$ where $s^{(m)}:=(\frac1n \sum_{i=1}^n
c_i^{2m})^\frac1{2m}$. The lower bound $m_*(p)$ on $m$ is exact for each
$p\in(0,1)$. Moreover, $\E f(c_1\BS_1+...+c_n\BS_n)$ is Schur-concave in
$(c_1^{2m},...,c_n^{2m})$. A number of related results are presented, including
ones for the ``symmetric'' case.
  A number of corollaries are obtained, including upper bounds on generalized
moments and tail probabilities of (super)martingales with differences of
bounded asymmetry, and also upper bounds on the maximal function of such
(super)martingales. It is shown that these results may be important in certain
statistical applications.
