Abstract
In the paper, the authors generalize Young’s integral inequality via Taylor’s theorems in terms of higher order derivatives and their norms, andapply newly-established integral inequalities to estimate several concrete definite integrals, including a definite integral of a function which plays an indispensable role in differential geometry and has a connection with the Lah numbers in combinatorics, the exponential integral, and the logarithmic integral.
Highlights
In 2009 and 2010, motivated by Theorem 1.2 and its proof in [4], among other things, Jakšetić and Pečarić extended and generalized Young’s integral inequality (1.1) and Hoorfar–Qi’s double inequality (1.2) as the following theorems
By virtue of Taylor’s theorems with different remainders, we establish some integral inequalities of Hoorfar–Qi’s type in terms of higher order derivatives and their norms, demonstrate that these newly-established integral inequalities generalize Young’s integral inequality (1.1), Hoorfar–Qi’s integral inequality (1.2), and Jakšetić–Pečarić’s integral inequalities (1.4), (1.5), and (1.6), and apply these integral inequalities to estimate several concrete definite integrals, including a definite integral of e–1/x which plays an indispensable role in differential geometry and has a connection with the Lah numbers in combinatorics, the exponential integral Ei(x), and the logarithmic integral li(x)
Applying these data to the double inequality (3.8), we arrive at [x – ln1/2(1 + x2)]n+2 (n + 1)!
Summary
In [9, Secton 2.7] and [10, Chapter XIV], many extensions, refinements, generalizations, and applications of Young’s integral inequality (1.1) in Theorem 1.1 were collected and reviewed. Lemma 2.4 ([10, Chapter IX] and [19]) Let f , g : [μ, ν] → R be integrable functions satisfying that they are both increasing or both decreasing. If φ is a concave function, inequality (2.6) is reversed These five lemmas are general knowledge in mathematics and they will continue to play important roles in this paper. Theorem 3.1 Let h(0) = 0 and h(x) be strictly increasing on [0, c] for c > 0, let h(n)(x) for n ≥ 0 be continuous on [0, c], let h(n+1)(x) be finite and strictly monotonic on (0, c), and let h–1 be the inverse function of h.
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