Inequality: $$-\frac{22}{24}h^2 ||f^{(4)}||_{\infty}\leq 0$$

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In summary, the inequality states that the sum of the squares of the differences of two successive real valued functions is smaller than the maximum of the function.
  • #1
mathmari
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Hey! :eek:

Does the following inequality hold?

$$|-\frac{5}{24}h^2f^{(4)}(x_1)+\frac{64}{24}h^2f^{(4)}(x_2)-\frac{81}{24}h^2f^{(4)}(x_3)| \\ \leq |-\frac{5}{24}h^2+\frac{64}{24}h^2-\frac{81}{24}h^2|\max_x |f^{(4)}(x)| \\ = |-\frac{22}{24}h^2| ||f^{(4)}||_{\infty}= \frac{22}{24}h^2 ||f^{(4)}||_{\infty}$$

where $h>0$
 
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  • #2
mathmari said:
Hey! :eek:

Does the following inequality hold?

$$|-\frac{5}{24}h^2f^{(4)}(x_1)+\frac{64}{24}h^2f^{(4)}(x_2)-\frac{81}{24}h^2f^{(4)}(x_3)| \\ \leq |-\frac{5}{24}h^2+\frac{64}{24}h^2-\frac{81}{24}h^2|\max_x |f^{(4)}(x)| \\ = |-\frac{22}{24}h^2| ||f^{(4)}||_{\infty}= \frac{22}{24}h^2 ||f^{(4)}||_{\infty}$$

where $h>0$
No, because $f^{(4)}(x_2)$ might have a different sign from $f^{(4)}(x_1)$ and $f^{(4)}(x_3)$. Unless you know that $f^{(4)}(x_1)$, $f^{(4)}(x_2)$ and $f^{(4)}(x_3)$ all have the same sign, the best you can say is that $$\left|-\frac{5}{24}h^2f^{(4)}(x_1)+\frac{64}{24}h^2f^{(4)}(x_2)-\frac{81}{24}h^2f^{(4)}(x_3)\right| \leqslant \left(\frac5{24} + \frac{64}{24} + \frac{81}{24}\right)h^2\|f^{(4)}\|_{\infty} = \frac{150}{24}h^2\|f^{(4)}\|_{\infty}.$$
 
  • #3
I used the Taylor expansion three times, once for $f(x+h)$, once for $f(x+2h)$ and once for $f(x+3h)$ and the $f(x_1), f(x_2), f(x_3)$ of the above expression are the remainders of each expansion. Do we know if they have all the same sign?
 
  • #4
I want to show that $$|\delta_{h,r}f(x)- f''(x)| \leq \frac{11}{12} h^2 ||f^{(4)}||_{\infty}$$ where $$\delta_{h,r}f(x)=\frac{1}{h^2} (2f(x)-5f(x+h)+4f(x+2h)-f(x+3h))$$

I applied the Taylor expanson at $f(x+h)$, $f(x+2h)$ and $f(x+3h)$ and $f(x_1), f(x_2), f(x_3)$ are the corresponding remainder of each expansion.

$$f(x+h)=f(x)+h f'(x)+\frac{h^2}{2} f''(x)+\frac{h^3}{6} f'''(x)+\frac{h^4}{24} f^{(4)}(x_1), x_1 \in (x,x+h)$$

$$f(x+2h)=f(x)+2hf'(x)+2h^2 f''(x)+\frac{4}{3} h^3 f'''(x)+\frac{16}{24}f^{(4)}(x_2), x_2 \in (x,x+2h)$$

$$f(x+3h)=f(x)+3hf'(x)+\frac{9}{2}h^2 f''(x)+\frac{27}{6}h^3 f'''(x)+\frac{81}{24} h^4 f^{(4)}(x_3), x_3 \in (x,x+3h)$$Substituting at $$\delta_{h,r}f(x)=\frac{1}{h^2} (2f(x)-5f(x+h)+4f(x+2h)-f(x+3h))$$ we get
$$\delta_{h,r}=f''(x)+\frac{h^2}{24}(64 f^{(4)}(x_2)-5f^{(4)}(x_1)-81 f^{(4)}(x_3))$$

Since $$|-\frac{5}{24}h^2f^{(4)}(x_1)+\frac{64}{24}h^2f^{(4)}(x_2)-\frac{81}{24}h^2f^{(4)}(x_3)| \\ \leq |-\frac{5}{24}h^2+\frac{64}{24}h^2-\frac{81}{24}h^2|\max_x |f^{(4)}(x)| \\ = |-\frac{22}{24}h^2| ||f^{(4)}||_{\infty}= \frac{22}{24}h^2 ||f^{(4)}||_{\infty}$$
we have

$$|\delta_{h,r}f(x)- f''(x)| \leq \frac{11}{12} h^2 ||f^{(4)}||_{\infty}$$

($h>0$)

Is this correct? Is there somewhere a mistake?
 

FAQ: Inequality: $$-\frac{22}{24}h^2 ||f^{(4)}||_{\infty}\leq 0$$

What is the meaning of "inequality" in this equation?

The term "inequality" refers to a mathematical statement that compares two quantities or expressions and indicates that one is greater than, less than, or not equal to the other.

What does $$-\frac{22}{24}h^2$$ represent in this equation?

This expression represents a negative value that is dependent on the variable h. It is multiplied by the norm of the fourth derivative of the function f, denoted by ||f^(4)||∞, to create the left side of the inequality.

How is the value of ||f^(4)||∞ determined?

The norm of the fourth derivative of a function is calculated by finding the maximum absolute value of the function's fourth derivative over its entire domain. This value is denoted by ||f^(4)||∞, where ∞ represents infinity.

What does it mean for the inequality to be less than or equal to 0?

If the left side of the inequality is less than or equal to 0, it indicates that the function's fourth derivative is either negative or equal to 0 over its entire domain. This implies that the function is either decreasing or constant, respectively.

How can this inequality be applied in a scientific context?

This inequality can be used in scientific research to evaluate the behavior of a function and its derivatives. It can also be applied in fields such as economics, physics, and engineering to analyze various systems and phenomena.

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