Ngô Quốc Anh

January 17, 2010

New Inequalities of Ostrowski-like Type Involving n knots and L^p-norm of m-th derivative

Filed under: Nghiên Cứu Khoa Học — Tags: — Ngô Quốc Anh @ 22:58

Let me introduce my recent result with V.N. Huy published in Applied Mathematics Letters last year.

I want to start by recalling the following results due to Nenad Ujevic:

Let I \subset \mathbb R be an open interval such that [a,b] \subset I and let f : I \to \mathbb R be a twice differentiable function such that f'' is bounded and integrable. Then we have

\displaystyle\begin{gathered}\Bigg|\int\limits_a^b {f\left( x \right)dx}- \frac{{b - a}}{2}\Bigg(f\left( {\frac{{a + b}}{2} - \left( {2 - \sqrt 3 } \right)\left( {b - a} \right)} \right) \hfill \\ \qquad\qquad+ f\left( {\frac{{a + b}}{2} + \left( {2 - \sqrt 3 } \right)\left( {b - a} \right)} \right)\Bigg)\Bigg| \leqq \frac{{7 - 4\sqrt 3 }}{8}{\left\| {f''} \right\|_\infty }{\left( {b - a} \right)^3}. \hfill \\ \end{gathered}

And

Let I \subset \mathbb R be an open interval such that [a,b] \subset I and let f : I \to \mathbb R be a twice differentiable function such that f'' \in L^2(a,b). Then we have

\displaystyle\begin{gathered}\Bigg|\int\limits_a^b {f\left( x \right)dx}- \frac{{b - a}}{2}\Bigg(f\left( {\frac{{a + b}}{2} - \frac{{3 - \sqrt 6 }}{2}\left( {b - a} \right)} \right) \hfill \\ \qquad\qquad+ f\left( {\frac{{a + b}}{2} + \frac{{3 - \sqrt 6 }}{2}\left( {b - a} \right)} \right)\Bigg)\Bigg| \leqq \sqrt {\frac{{49}}{{80}} - \frac{1}{4}\sqrt 6 } {\left\| {f''} \right\|_2}{\left( {b - a} \right)^{\frac{5}{2}}}. \hfill \\ \end{gathered}

In the above mentioned results, constants \frac{{7 - 4\sqrt 3 }}{8} in the first and \sqrt {\frac{{49}}{{80}} - \frac{1}{4}\sqrt 6 } in the second result are sharp in sense that these cannot be replaced by smaller ones. This leads us to strengthen them by enlarging the number of knots (2 knots in both results) and replacing the norms \| \cdot \|_\infty in the first and \| \cdot \|_2 in the second.

Before stating our main result, let us introduce the following notation

\displaystyle I\left( f \right) =\int\limits _a^b {f\left( x \right)dx } .

Let 1 \leqq m, n<\infty and 1 \leqq p \leqq \infty. For each i = \overline{1,n}, we assume 0 < x_i < 1 such that

\displaystyle\left\{ \begin{gathered}{x_1} + {x_2} +\cdots+ {x_n} = \frac{n}{2}, \hfill \\ \cdots\hfill \\x_1^j + x_2^j +\cdots+ x_n^j = \frac{n}{{j + 1}}, \hfill \\ \cdots\hfill \\x_1^{m - 1} + x_2^{m - 1} +\cdots+ x_n^{m - 1} = \frac{n}{m}. \hfill \\ \end{gathered}\right.

Put

\displaystyle Q\left( {f,n,m,x_1 ,...,x_n } \right) = \frac{{b - a}}{n}\sum\limits_{i = 1}^n {f\left( {a + x_i \left( {b - a} \right)} \right)} .

We are in a position to state our main result.

Let I \subset \mathbb R be an open interval such that [a,b] \subset I and let f : I \to \mathbb R be a m-th differentiable function such that f^{(m)} \in L^p(a,b). Then we have

\displaystyle\left| {I\left( f \right) - Q\left( {f,n,m,{x_1},...,{x_n}} \right)} \right| \leqq \frac{1}{{m!}}\left( {{{\left( {\frac{1}{{mq + 1}}} \right)}^{\frac{1}{q}}} + {{\left( {\frac{1}{{\left( {m - 1} \right)q + 1}}} \right)}^{\frac{1}{q}}}} \right){\left\| {{f^{\left( m \right)}}} \right\|_p}{\left( {b - a} \right)^{m + \frac{1}{q}}}.

As can be seen the above result is not sharp. It will be very interesting if we can derive a sharp estimate. Note that, the results due to Nenad Ujevic can be rewritten as the following

\displaystyle\left| {I\left( f \right) - Q\left( {f,2,2,\frac{1}{2} - \left( {2 - \sqrt 3 } \right),\frac{1}{2} + \left( {2 - \sqrt 3 } \right)} \right)} \right| \leqq \frac{{7 - 4\sqrt 3 }}{8}{\left\| {f''} \right\|_\infty }{\left( {b - a} \right)^3}

and

\displaystyle\left| {I\left( f \right) - Q\left( {f,2,2,\frac{1}{2} - \frac{{3 - \sqrt 6 }}{2},\frac{1}{2} + \frac{{3 - \sqrt 6 }}{2}} \right)} \right| \leqq \sqrt {\frac{{49}}{{80}} - \frac{1}{4}\sqrt 6 } {\left\| {f''} \right\|_2}{\left( {b - a} \right)^{\frac{5}{2}}}.

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