In this paper, we present a method that allows one to obtain a number of sharp inequalities for expectations of functions of infinite-degree
U-statistics. Using the approach, we prove, in particular, the following result: Let
D be the class of functions
f :
R+→
R+ such that the function
f(
x+
z)−
f(
x) is concave in
xR+ for all
zR+. Then the following estimate holds: for all
fD and all
U-statistics ∑
1i1<<ilnYi1,…,il(
Xi1,…,
Xil) with nonnegative kernels
Yi1,…,il :
Rl→
R+, 1
ikn;
ir≠
is,
r≠
s;
k,
r,
s=1,…,
l;
l=0,…,
m, in independent r.v.'s
X1,…,
Xn. Similar inequality holds for sums of decoupled
U-statistics. The class
D is quite wide and includes all nonnegative twice differentiable functions
f such that the function
f″(
x) is nonincreasing in
x>0, and, in particular, the power functions
f(
x)=
xt, 1<
t2; the power functions multiplied by logarithm
f(
x)= (
x+
x0)
t ln(
x+
x0), 1<
t<2,
x0max(e
(3t2−6t+2)/(t(t−1)(2−t)),1); and the entropy-type functions
f(
x)=(
x+
x0)ln(
x+
x0),
x01. As an application of the results, we determine the best constants in Burkholder–Rosenthal-type inequalities for sums of
U-statistics and prove new decoupling inequalities for those objects. The results obtained in the paper are, to our knowledge, the first known results on the best constants in sharp moment estimates for
U-statistics of a general type.
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