Some inequalities connected with the exponential function

Włodzimierz Fechner

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Małgorzata Kuchta (1996)

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We derive and investigate integral inequalities of Opial type: $\int_{I} s | h h ̇ | d t \leq \int_{I} r h ̇ ² d t$ , where h ∈ H, I = (α,β) is any interval on the real line, H is a class of absolutely continuous functions h satisfying h(α) = 0 or h(β) = 0. Our method is a generalization of the method of [3]-[5]. Given the function r we determine the class of functions s for which quadratic integral inequalities of Opial type hold. Such classes have hitherto been described as the classes of solutions of a certain differential equation....

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We give some necessary and sufficient conditions for an n-1 times differentiable function to be a generalized convex function with respect to an unrestricted n-parameter family.