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?



Half spaces free of roots of a given polynomial
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>


