We consider the transformation \[ g(λ ) = ∫ φ (λ ,x)f(x)dμ (x)\], where f and φ) are non-negative Borel-measurable functions of nonnegative arguments and μ denotes the Lebesgue measure on [0,∞ ) or the counting measure on \ 0,1, ⋯ \. We show that under appropriate assumptions on φ, various geometric properties of f (such as star-shapedness and superadditivity) are inherited by g.
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Abdel‐Hameed et al. (1976) studied this question.