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I thank Loic Teyssier and Emil Jerabek who helped me to revise the two previous version

This question is motivated by the following fact in complex variable:(I learned this fact from the book of Ahlfors, Complex Analysis)

Fact: If all roots of a complex polynomial $p(z)$ lie in a half plane then all roots of its derivative $p'(z)$ lie in the same half plane.

this implies that

If all roots of a polynomial $p(z)$ is contained in a convex set $K$ then all roots of $p'(z)$ is contained in $K$, too.

This means that The algebra $A$ of polynomials satisfies the following property:

Property $P$:

$A$ is an algebra of entire functions which is closed under derivation and for every $f\in A$ and every convex set $K$ with $Z(f)\subseteq K $ we have $Z(f') \subseteq K$.

I have a question on this fact:

Question: Is there an algebra $A$ of entire functions with the above property $P$ but $A$ is not equal to the polynomial algebra?

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    $\begingroup$ A stronger, classical fact gives the answer to 1). If $C$ is the convex hull of the roots of $p$, then all roots of $p'$ also lie in $C$. $\endgroup$
    – Loïc Teyssier
    Commented Jun 16, 2016 at 13:32
  • $\begingroup$ @LoicTeyssier Thank you very much Loic for your comment. Is not it a consequence of the initial one(about half space).According to your comment one csn replace the half plane by any arbitrary convex set.thanks again for your interesting comment. $\endgroup$
    – Ali Taghavi
    Commented Jun 16, 2016 at 13:42
  • $\begingroup$ @Loic Is there an algebra of entire functions which is closed under differentiation and satisfy the property under this post: $\endgroup$
    – Ali Taghavi
    Commented Jun 16, 2016 at 14:17
  • $\begingroup$ That is for every convex set K containing all roots of a typical f in the algebra, K must contain all roots of f'? $\endgroup$
    – Ali Taghavi
    Commented Jun 16, 2016 at 14:20
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    $\begingroup$ Every closed convex set is an intersection of half-planes, so the statements for half-planes and arbitrary convex sets are equivalent. $\endgroup$
    – Emil Jeřábek
    Commented Jun 16, 2016 at 14:26

1 Answer 1

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The property that the zeros of the derivative of a polynomial $P$ lie in the convex hull of the zeros of $P$ is usually called the Gauss-Lucas theorem.

About question 2), the algebra of entire functions of order less than 1 satisfy Property P : it is closed under derivation and satisfies the property concerning the zeros, see Corollary (3.1) p.835 in

M. Marden, On the zeros of the derivative of an entire function. Amer. Math. Monthly 75 1968 829–839,

or Theorem (4.2) in

M. Marden, On the derivative of an entire function. Proc. Amer. Math. Soc. 19 1968 1045–1051, http://www.ams.org/journals/proc/1968-019-05/S0002-9939-1968-0231996-0/S0002-9939-1968-0231996-0.pdf

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  • $\begingroup$ @use111 Thank you for your perfect answer. $\endgroup$
    – Ali Taghavi
    Commented Jun 19, 2016 at 10:25

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