Aequat. Math. c  Springer Basel 2014 DOI 10.1007/s00010-013-0252-4 Aequationes Mathematicae General Opial type inequality Ana Barbir, Kristina Kruli´ c Himmelreich and Josip Peˇ cari´ c Abstract. In this paper our goal is to give a general Opial type inequality. We consider two functions, convex and concave and prove a new general inequality on a measure space (Ω, Σ,μ). The obtained inequalities are not direct generalizations of the Opial inequality but are of Opial type because the integrals contain a function and its integral representation. We apply our result to numerous symmetric functions and obtain new results that involve Green’s functions, Lidstone series and the Hermite’s interpolating polynomials. Mathematics Subject Classifications (2000). Primary 26D15; Secondary 26A51. Keywords. Opial inequality, Green function, Jensen inequality, Lidstone polynomial, Hermite interpolating polynomial. 1. Introduction In 1960, Opial [6] proved the following inequality: Let f ∈ C 1 [0,h] be such that f (0) = f (h) = 0 and f (x) > 0 for x ∈ (0,h). Then h  0 |f (x)f ′ (x)|dx ≤ h 4 h  0 [f ′ (x)] 2 dx, (1.1) where h/4 is the best possible. This inequality has been generalized and extended in several directions (for more details see e.g. [1, 3]). Kruli´ c et al. [4] (see also [5, Chapter II, p. 15]) studied measure spaces (Ω 1 , Σ 1 ,μ 1 ), (Ω 2 , Σ 2 ,μ 2 ), and the general integral operator A k defined by A k f (x)= 1 K(x)  Ω2 k(x, y)f (y) dμ(y),x ∈ Ω 1 , (1.2) where f :Ω 2 -→ R is a measurable function, k :Ω 1 × Ω 2 → R is measurable and non-negative, and