On the log-concavity of the fractional integral of the sine function

We prove that the function Fλ(x):=∫0x(x−t)λsintdt is logarithmically concave on (0,∞) if and only if λ≥2. As a consequence, a Turán type inequality for certain Lommel functions of the first kind is obtained. Furthermore, some monotonicity properties of functions involving the remainders of the Taylor series expansion of the functions sinx and cosx are given.