インテグラルキラー;489 said:Consider an analytic function $f$ and non-constant defined on a set $\mathcal U\subset\mathbb C$ open and connected. Prove that the real-valued functions $|f|,\,\text{Re}(z),\,\text{Im}(z)$ can't achieve local maximum.
This one looks hard, how to do it?
Yes, I know the principle, but it's supposed that I need an inequality for $f$ in order the principle to be applied. Is it true if I say that if $f$ is non-constant, then we have $|f(z)-w|\le\epsilon$ forall $\epsilon>0$ and $z,w\in\mathbb C$ ? But I don't think if helps me.AlexYoucis said:Well, the problem for $|f|$ is a pretty important theorem known as the Maximum Modulus Principle.
Ah okay so $\text{Re}(f(z))$ because $|e^{f(z)}|$ ain't bounded?For $\text{Re}(z)$ note that if $\text{Re}(f(z))$ acheived a maximum then so would $e^z$ since $|e^{f(z)}|=e^{\text{Re}(f(z))}$
But if you suppose that you have such a maximum you get such a desired inequality!インテグラルキラー;514 said:Yes, I know the principle, but it's supposed that I need an inequality for $f$ in order the principle to be applied. Is it true if I say that if $f$ is non-constant, then we have $|f(z)-w|\le\epsilon$ forall $\epsilon>0$ and $z,w\in\mathbb C$ ? But I don't think if helps me.
Ah okay so $\text{Re}(f(z))$ because $|e^{f(z)}|$ ain't bounded?
Thanks you but I don't see how to conclude, I'm supposed to prove that $|f|$ can't achieve a maximum. Okay so if that if such maximum exists, then it follows the inequality (why exactly?), but if the inequality doesn't hold, then $f$ is constant, which is a contradiction.AlexYoucis said:But if you suppose that you have such a maximum you get such a desired inequality!
Oh yes, so that concludes that $\text{Re}f(z)$ doesn't achieve its maximum.AlexYoucis said:No, because $|e^{f(z)}|$ can't reach a maximum neither can $\ln(|e^{f(z)}|)=\text{Re}(f(z))$.
A local maximum is a point on a graph where the function value is greater than or equal to all the nearby points, but not necessarily greater than all the points on the entire graph.
A global maximum is the highest point on the entire graph, while a local maximum is only the highest point within a specific interval or range.
A local maximum has a positive slope on both sides and a concave downward shape. It also has a higher function value than all the nearby points.
To identify a local maximum on a graph, you can look for the highest point on a specific interval or range. You can also calculate the derivative of the function and look for where it equals 0.
In real-world applications, a local maximum can represent the peak or highest value of a function within a specific range. This can be useful in optimizing processes or finding the maximum value of a variable.