A Hermite–Hadamard Inequality for Superquadratic Functions on Euclidean Balls

Let d≥1, let Kd=[0,∞)d, and let B(A,R)⊂Kd be a closed Euclidean ball. For a continuous superquadratic function φ:Kd→R, we establish a corrected Hermite–Hadamard estimate that relates the integral mean of φ over the ball to its value at the center and its integral mean over the boundary sphere. The correction is given explicitly by a dimension-dependent radial functional. For arbitrary superquadratic functions the result is a corrected estimate; when φ≥0, the correction is nonnegative and the upper estimate genuinely refines the corresponding convex Hermite–Hadamard bound. Equality is attained for φ(x)=∥x∥2. A one-dimensional specialization on [A−R,A+R]⊂[0,∞) is also obtained. As applications, we show that the modified Bessel function Iν, ν≥2, is superquadratic and derive corresponding integral inequalities, including an explicit hyperbolic-function example.