Abstract
In this article we give q-analogs of the Opial inequality for q-decreasing functions. Using a closed form of the restricted q-integral (see Gauchman in Comput. Math. Appl. 47:281–300, 2004), we establish a new integral inequality of the q-Opial type.
Highlights
In 1960, Opial [10] established the following important integral inequality.Theorem 1.1 Let f ∈ C1[0, h], where f (0) = f (h) = 0 and f (t) > 0 for t ∈ (0, h)
2 Preliminaries Here we present necessary definitions and facts from the q-calculus
3 Results and discussions Our main results are contained in three theorems
Summary
1 Introduction In 1960, Opial [10] established the following important integral inequality. After interchanging the boundaries in the right-hand side integral, and replacing bqn with a, we find b b bp 1 – qn p Theorem 3.3 If f (x) and g(x) are absolutely continuous q-decreasing functions on (a, b) and f (bq0) = 0 and g(bq0) = 0, b a f (x)Dqg(x) + g(qx)Dqf (x)
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