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Zbl 1115.26010
Pachpatte, B.G.
New Ostrowski and Grüss type inequalities. (English)
[J] An. Ştiinţ. Univ. Al. I. Cuza Iaşi, Ser. Nouă, Mat. 51, No. 2, 377-386 (2005). ISSN 1221-8421; ISSN 0041-9109/e

The author presents elementary proofs of the following extensions of the Ostrowski and Grüss inequalities for triples of continuous functions $f$, $g$, $h$ on the interval $[a,b]$ with bounded derivatives $f'$, $g'$, $h'$: $$\multline\left\vert f(x)g(x)h(x)-\frac1{3(b-a)}\left[g(x)h(x)\int_a^bf(y)dy+h(x)f(x )\int_a^bg(y)dy+f(x)g(x)\int_a^bh(y)dy\right]\right\vert \\ \le\frac13[\vert g(x)\vert\vert h(x)\vert\Vert f'\Vert_\infty+\vert h(x)\vert\vert f(x)\vert\Vert g'\Vert_\infty+\vert f(x)\vert\vert g(x)\vert\Vert h'\Vert_\infty]A(x)\endmultline$$ and $$\multline\left\vert\frac1{b-a}\int_a^b f(x)g(x)h(x)dx-\frac13\left[\left(\frac1{b-a}\int_a^b g(x)h(x)dx\right)\right. \left(\frac1{b-a}\int_a^b f(x)dx\right) \right. +\\ \left.\left.\left(\frac1{b-a}\int_a^b h(x)f(x)dx\right)\left(\frac1{b-a}\int_a^b g(x)dx\right)+ \left(\frac1{b-a}\int_a^b f(x)g(x)dx\right)\left(\frac1{b-a}\int_a^b h(x)dx\right)\right]\right\vert\\ \le\frac1{3(b-a)}\int_a^b[\vert g(x)\vert\vert h(x)\vert\Vert f'\Vert_\infty+\vert h(x)\vert\vert f(x)\vert\Vert g'\Vert_\infty+\vert f(x)\vert\vert g(x)\vert\Vert h'\Vert_\infty]\vert A(x)dx,\endmultline$$ where $A(x)=[1/4+(x-(a+b)/2)^2/(b-a)^2](b-a)$. In addition, discrete analogues of the inequalities are proven.
[Jiri Rakosnik (Praha)]

MSC 2000:: *26D10 Inequalities involving derivatives, diff. and integral operators
26D15 Inequalities for sums, series and integrals of real functions

Keywords: Ostrowki inequality; discrete analogue; elementary proof

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