A GENERALIZATION OF AN INEQUALITY OF ZYGMUND

The well known Bernstein Inequallty states that if D is a disk 
centered at the origin with radius R and if p(z) is a polynomial of 
degree n, then maxz∈D|p′(z)|≤nRmaxz∈D|p(z)| with equality iff p(z)=AZn. 
However it is true that we have the following better inequallty: 
maxz∈D|p′(z)|≤nRmaxz∈D|Rep(z)| 
with equality iff p(z)=AZn.


INTROICTION.
The classical result of Bernstein as it appears in [2] is Bernstein Inequality If D is a Euclidean disk and P is a polynomial of degree n over C then n (1) liP D tr(D) IIPlID where llfllD sup If(z)l and tr(D) is the transfinite diameter of D (which D is the disk's radius in this case).
This result was generalized to various directions.The following theorem appears in [I] Let 0 s k s and let E be a closed k-quasldlsk, then THEOREm.For any polynomial P of degree n we have and P(Zl)-P(Z2) Zl-Z2 Let # be the class of all analytic functions f(z) akzk in k=O Izl < such that 0 < If(z)l < A problem posed by Krzyz [4]  The proof of this uses the following generalization of (1): Let S(0,1} {z C Iz[ < I} and let p be any polynomial of degree n over C then llp' liD(0, I) s nRe PlID(0, 1) This follows from an inequality of Zygmund [7] TI{EORE.
For any polynomial p of degree n and for any s p < D(0,1) by using the same ideas as in Zygmund's proof applied to g is a quite general mapping D(O,I) E. pog 2. RESULTS.
THEOI 1.Let g be a complex valued function of Suppose that {arg g(eiX)10 s x s 2} [0,2=/n] and that dx then for any non-negative, non-decreasing convex function for any and for any polynomial P of degree n over C we have O(n-lllm{elg(eie)p'(g(eie))}l]de max(X(IRe{p(elg(ele))}II de} 0 equality occurs in (7) iff p(z) Az We remark that the consequences of Theorem hold true even if the condition (arg g(eiX)lo x 2} 2 [0,2/n] is dropped.
We will indicate at the end of Section 4 how to prove that.
As a consequence we derive an analogous theorem to (I), The last corollary can be seen directly, but, it shows that we cannot drop "max" on the right hand of the above inequalities since it is easy to find a while lira ,,llgl]p g such that [[Re g]Ip We denote g(e ix) R(x)e i#(x), R(x) Ig(eiX)l #(x) arg g(e ix) where the coefficients a,b are such that S(x,@(x)) S(g(eiX)) (x,@(x)) (g(eiX)) (15) As in Zygmund we denote the modified Dirichlet kernel and it's conjugate kernel by D (u), D (u) respectiveiy.
A PROOF OF TIIEOBEM 1.

THEOREN 3 .
If E is a simply connected domain such that G D(0,1) E is a Riemann mapping normalized by G(G(reie))IP de P max IRe{P(eiG(re riG'(O)This last inequality is not sharp.