ON POLYNOMIAL EXPANSION OF MULTIVALENT FUNCTIONS

Coefficient bounds for mean p-valent functions, whose expansion in an ellipse has a Jacobi polynomial series, are given in this paper.

-I Let E {z cosh(s +i), 0 Let also r a+b be the sum of the seml-axls of E It where here and throughout this paper a, ) -l.
This expansion converges locally unlformly in Int(E ).
In [2] the author has given some coefficient bounds for o functlons mean p-valent and has an expansion in terms of Chebyshev polynomials in Int(Eo).Such polynomlals are generated by the speclal case a -I12 in Jacobl polynomlals.Other special cases of interest are the Legendre and the altraspherlcal polynomials generated by a-- 0 and a respectlvely if, p. 80-89].
In this paper we generalize results gven in [2] to functions of the form (I.I) and mean p-valent in Int(E ).In view of [2] we call f(z) mean p-valent in Int(g where 0 (R ( and n( f,Int(g )) denotes the number of roots of the equation f(z) w tn Interior E multtpltclty being take Into account.
We first recaii from [2]: TttEORgN Ao Let f(z) be mean p-valent in Int(E )o Then for z eosh(s+i:), exp(s) O where 0(t) and o(l) depend on a,b, and f only.PROOF OF THEOREM B. Using Schwarz's inequality we have Theorem B now follows in the same way as estimating inequality (14) of [2] by uslng [2, Lemmas 3 and 4].
Using (1.6) in (1.5)we tmmediately deduce (1.2).Now d|fferentlating (1.1) we see from equation (4.21.7) --(n+a++l)a P(a+l'B+l)(z) n= n n-I Again, as in the proof of (1.2), we deduce from this and [I, p. 245] for n > I, that B+lo(a+l,B+l)(z K(Cr+l,ff+l)/2z n where we have used the equation (z-l)a+l(z+l) n-I n-I which is deduced as in (1.6).This is equation (1.3) and the proof of the lemma is now complete.where 0(I) depends on a,b,a,B and f only.
This corollary follows upon setting p in Corollary 2.1.
REMARK.Using the formula (4.21.2) of [I] and the argument used in [2, Remark 2] we see by settlng z cosh s where II cos T + tanh s sin TI that o o f(E cosh s .F(2n+a+B+I) (__csh s )n((E 1/cosh s )n

i < 2
#, s tanh (b/a), a > b > O} be a fixed o o o ellipse whose focl are +/-l.
Szeg [I|, Theorem 9.1.11,see also p. 245) that a function f(z) which is regular in Int(E (this means the interior of E has an expansion of the form o r and < r < r we have o If(z)l 0(1) (l-r/r)-2p o where 0(I) depends on a,b and f only.

2 .
MAIN THEOREM.THEOREM 2.1.Let f(z).a P(a'B)(z) be mean p-valent In Int(E and n=O n n o M(r,f) C(1-r/r )-Y where C, > 0 and M(r,f) is as defined above.Then, I) and o(1) depend on a,b,a,,,f and f only.2. I. From (1.3) and Theorem B we deduce, using the bounds < (K(Cl'B+l)/h(Crl'l+l)(cosh I (r ')/sinhns) 1)/n)r and provided that 1-n/(n-1)r > 0, This completes the proof of Theorem 2.1.COROLLARY 2.1.Let f(z) a p(a,) (z) be mean p-valent in Int(E ).Then n=0 n n o as n (R)we have O(1)n 2p-I12 (p > 114) I) and o(I) depend on a,b,a,B,p and f only.In view of Theorem A, the proof of Corollary 2.1 follows by setting T 2p in Theorem 2.1.COROLLARY 2.2.Let f(z) a p(a,B) (z) be univalent in Int(E ).
Using this and Stlrllng's formula and letting r -we see that Theorem 2.1 and o Corollaries 2.1 and 2.2 correspond to analogous results for the unit disk (seeHayman  [61).