Inequalities involving partial derivatives

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Some sharp integral inequalities involving partial derivatives

* Correspondence: chjzhao@163. com Department of Mathematics, China Jiliang University, Hangzhou 310018, P. R. China Full list of author information is available at the end of the article Abstract The main purpose of the present article is to establish some new sharp integral inequalities in 2n independent variables. Our results in special cases yield some of the recent results on Pachpatter, A...

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Sharp Integral Inequalities Involving High-Order Partial Derivatives

Inequalities involving functions of n independent variables, their partial derivatives, integrals play a fundamental role in establishing the existence and uniqueness of initial and boundary value problems for ordinary and partial differential equations as well as difference equations 1–10 . Especially, in view of wider applications, inequalities due to Agarwal, Opial, Pachpatte, Wirtinger, Poi...

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In this note we study the upper bound of the integral f {tW(x))2w(x)dx Jo where t(x) is a trigonometric polynomial with real coefficients such that \\t\\ao < 1 and w(x) is a nonnegative function defined on [0, n]. When w{x) = sin; x , where j is a positive integer, we obtain the exact upper bound for the above integral.

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Inequalities for the Derivatives * †

The following question is studied and answered: Is it possible to stably approximate f if one knows: 1) f δ ∈ L ∞ (R) such that f − f δ < δ, and 2) f ∈ C ∞ (R), f + f ≤ c? Here f := sup x∈R |f (x)| and c > 0 is a given constant. By a stable approximation one means L δ f δ − f ≤ η(δ) → 0 as δ → 0. By L δ f δ one denotes an estimate of f. The basic result of this paper is the inequality for L δ f...

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Inequalities Involving Generalized Bessel Functions

Let up denote the normalized, generalized Bessel function of order p which depends on two parameters b and c and let λp(x) = up(x), x ≥ 0. It is proven that under some conditions imposed on p, b, and c the Askey inequality holds true for the function λp , i.e., that λp(x) +λp(y) ≤ 1 +λp(z), where x, y ≥ 0 and z = x + y. The lower and upper bounds for the function λp are also established.

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ژورنال

عنوان ژورنال: Journal of Mathematical Analysis and Applications

سال: 1982

ISSN: 0022-247X

DOI: 10.1016/0022-247x(82)90121-4