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This question just came to my mind and I have no idea as to how to approach it. Let $z_1,z_2,\dots,z_n$ be $n$ complex numbers in the unit disc $|z| \leq 1.$ Consider a complex function on the unit disc wiith real values $$ f(z)=\sum_{i=1}^n \frac{|z-z_i|}{n}. $$ My questions:

• Does there exist a $z \in |z| \leq 1 $ so that $f(z)=|z|$?
• If not can we make $f(z)$ arbitrarily close to $|z|$ for some $ z \in |z| \leq 1 $?
• What about the maximum and the minimum value of $f(z)$?

Lots of thanks for any responces\hints\suggestions

This question just came to my mind and I have no idea as to how to approach it. Let $z_1,z_2,\dots,z_n$ be $n$ be any complex numbers in the unit disc $|z| \leq 1.$ Consider a complex function on the unit disc wiith real values $$ f(z)=\sum_{i=1}^n \frac{|z-z_i|}{n}. $$ My questions:

• Does there exist a $z \in |w| \leq 1 $ so that $f(z)=|z|$?
• If not can we make $f(z)$ arbitrarily close to $|z|$ for some $ z \in |w| \leq 1 $?
• What about the maximum and the minimum value of $f(z)$?

Lots of thanks for any responces\hints\suggestions

This question just came to my mind and I have no idea as to how to approach it. Let $z_1,z_2,\dots,z_n$ be $n$ complex numbers in the unit disc $|z| \leq 1.$ Consider a complex function on the unit disc wiith real values $$ f(z)=\sum_{i=1}^n \frac{|z-z_i|}{n}. $$ My questions:

• Does there exist a $z \in |z| \leq 1 $ so that $f(z)=|z|$?
• If not can we make $f(z)$ arbitrarily close to $|z|$ for some $ z\in\{w:|w|\leq 1\} $?
• What about the maximum and the minimum value of $f(z)$?

Lots of thanks for any responces\hints\suggestions

This question just came to my mind and I have no idea as to how to approach it. Let $z_1,z_2,\dots,z_n$ be $n$ complex numbers in the unit disc $|z| \leq 1.$ Consider a complex function on the unit disc wiith real values $$ f(z)=\sum_{i=1}^n \frac{|z-z_i|}{n}. $$ My questions:

• Does there exist a $z \in |z| \leq 1 $ so that $f(z)=|z|$?
• If not can we make $f(z)$ arbitrarily close to $|z|$ for some $ z \in |z| \leq 1 $?
• What about the maximum and the minimum value of $f(z)$?

Lots of thanks for any responces\hints\suggestions

This question just came to my mind and I have no idea as to how to approach it. Let $z_1,z_2,\dots,z_n$ be $n$ complex numbers in the unit disc $|z| \leq 1.$ Consider a complex function on the unit disc wiith real values $$ f(z)=\sum_{i=1}^n \frac{|z-z_i|}{n}. $$ My questions:

• Does there exist a $z \in |z| \leq 1 $ so that $f(z)=|z|$?
• If not can we make $f(z)$ arbitrarily close to $|z|$ for some $ z \in |z| \leq 1 $?
• What about the maximum and the minimum value of $f(z)$?

Lots of thanks for any responces\hints\suggestions

This question just came to my mind and I have no idea as to how to approach it. Let $z_1,z_2,\dots,z_n$ be $n$ complex numbers in the unit disc $|z| \leq 1.$ Consider a complex function on the unit disc wiith real values $$ f(z)=\sum_{i=1}^n \frac{|z-z_i|}{n}. $$ My questions:

• Does there exist a $z \in |z| \leq 1 $ so that $f(z)=|z|$?
• If not can we make $f(z)$ arbitrarily close to $|z|$ for some $ z\in\{w:|w|\leq 1\} $?
• What about the maximum and the minimum value of $f(z)$?

Lots of thanks for any responces\hints\suggestions