The positivity of some Cesàro mean is proven for Jacobi series
Σ
a
n
P
n
(
α
,
β
)
(
x
)
,
α
,
β
≧
−
1
2
\Sigma {a_n}P_n^{(\alpha ,\beta )}(x),\alpha ,\beta \geqq - \tfrac {1}{2}
. This has applications to the mean convergence of Lagrange interpolation at the zeros of Jacobi polynomials. The positivity of the
(
C
,
α
+
β
+
2
)
(C,\alpha + \beta + 2)
means is conjectured and proven for some
(
α
,
β
)
(\alpha ,\beta )
. One consequence of this conjecture would be the complete monotonicity of
x
−
c
(
x
2
+
1
)
−
c
,
c
≧
1
{x^{ - c}}{({x^2} + 1)^{ - c}},c \geqq 1
.