Positive Trigonometric Integrals Associated with Some Lommel Functions of the First Kind

We prove that integral(x)(0) t(mu-1/2) log t sin(x-t) dt = root x d/d mu s(mu,1/2) (x) > 0, for all mu >= 1/2 and x >= pi, where s(mu,1/2) (x) is the Lommel function of the first kind. This refines a result of Steinig published in 1972. As an application, we derive positive functional bounds for the functions s(mu,1/2) (x), mu >= 1/2 and x >= pi. Furthermore, we give estimates for the location of the zeros of the functions s(mu,1/2)(x), -3/2 < mu < 1/2, mu not equal - 1/2, x > 0.