443
Internat. J. Math. & Math. Sci.
VOL. 14 NO. 3 (1991) 443-446
ON POLYNOMIAL EXPANSION OF MULTIVALENT FUNCTIONS
M. M. ELHOSH
Department of Mathematics
Unlvers[ty of College of Wales
Aberys twyth, Wales
United Kingdom
(Received April 24, 1987)
ABSTRACT.
bounds
Coefficient
for
mean
p-valent
functions,
whose
expansion in an
ellipse has a Jacobi polynomial series, are given in this paper.
KEY WORDS AND PHRASES. Univalent and multivalued functions, orthogonal polynomials.
1980 AMS SUBJECT CLASSIFICATION CODES. 30C50, 30C55.
INTRODUCTION.
|.
<
-I
>
>
O} be a fixed
It
Let also r o
a+b be the sum of the seml-axls of E
o
is known (Szeg [I|, Theorem 9.1.11, see also p. 245) that a function f(z) which is
(this means the interior of E
has an expansion of the form
regular in Int(E
{z
Let E o
cosh(s +i),
o
0
i
2#,
s
tanh
o
(b/a),
a
b
ellipse whose focl are +/-l.
o
o
(a’ )(z)
(.i)
n--O anP n
This expansion converges locally
where here and throughout this paper a, ) -l.
Int(E
some coefficient bounds for
).
[2]
author
has
given
the
In
unlformly in
f(z)
o
functlons
Int(Eo).
in
p-valent
mean
and
has
an
expansion
in
terms
of
Chebyshev polynomials
-I12
Such polynomlals are generated by the speclal case a
in Jacobl
Other special cases of interest are the Legendre and the altraspherlcal
polynomlals.
0 and a
polynomials generated by a--
respectlvely if, p. 80-89].
In this paper we generalize results gven in [2] to functions of the form (I.I)
if
and mean p-valent in Int(E ). In view of [2] we call f(z) mean p-valent in Int(g
o
o
R 2
W(R,f)
/ / n(eib,f,lnt(E o ))pdd,
(I/w)
2
< pR
O0
where 0 (R (
and
w tn Interior E
f,Int(g )) denotes the number of roots of the equation f(z)
n(
multtpltclty being take Into account.
We first recaii from [2]:
TttEORgN
r and
<
Let f(z) be mean p-valent in Int(E )o
Ao
r
<
r o we have
If(z)l
0(1)
(l-r/r)-2p
o
where 0(I) depends on a,b and f only.
Then for z
eosh(s+i:), exp(s)
444
M.M. ELHOSH
Let f(z) be
THEOREM B.
where
>
c,$
<r <
r
O
,mean
p-valent
ma{If(z)l:z
0 and M(r,f)
a Int(E o
and
Int(E )}. Set z
o
M(r,f)
o
cosh(s+iT), exp(s)=r,
and
ll(r,f’)
Then as r
r
o
(I/2)
f If’(cosh(s+fT))llslnh(s+i1)Idt.
0
we have
(1-r/r)-Y
o
0(1)
-1}2 log(l/(l-r/r
(1-r/r)-1/2
O
0(1) (1-r/r)
0
o(1)
Ii(r,f’)
(y
0
))
> 1/2),
1/2)
(
(y
<
l/2)
where 0(t) and o(l) depend on a,b, and f only.
PROOF OF THEOREM B.
2w
,
II(r,f’)
Using Schwarz’s inequality we have
I/2)
0
I’(oh+OI21(oh(+))l 1-21slnh(s+iz) 2dz) 1/211/2
2
x
where 0
Z
<
2.
[(1/2)
f
0
If(cosh(s+l))[2-;kdT)I/2] I/2
Theorem B now follows in the same way as estimating inequality (14)
.
of [2] by uslng [2, Lemmas 3 and 4].
We now need a suitable coefficient formula.
(a,B) (z) be
Let f(z)
regular in Int(E O
anP
n
n=O
LEMMA I.I.
{z
E
a
(’8)
where K
n
8)=
(a,
2}.
Then for a fixed s so that 0
n( a’8)/h(a’B))(I/2vl) Ef f(z) dz,
(K
n
n
2n+a+8+Ir(n+a+l)r(n+B+l)/r(2n+a+8+2)
2a+8+1 r(n+ a+l
<
s
<
s
O
we have
(n) 0),
(1.2)
z
(K(a+l,B+l)/h(a+l,B+l))(l/2vi)
f
n
n-I
E
(n+a+B+l)an
h
n
<
cosh(s+iT), 0
and
f’(z_____)_
z
n
dz
(n) ])
(1.3)
and
r(n+ 8+I / 2n+ a+ 8+I r(n+l r(n+ a+ 8+I ).
We note here, using Stlrling’s formula from Titmarsh [3, p. 57], that
K
n
(a’8)/hn(a’8)
0(1)n
I/2/2n
(1.4)
% where 0(I) depends on a, 8 only.
as n
PROOF OF LEMHA.
a
where n
n
We have from [I, p. 245] that
{rih
a,
8)}-I f
E
(z-1)a
(a,
(z+l) 8Qn
8)(z)f(z)d z
(1.5)
0,1,2,
We now see from [1, Theorem 4.61.2], (see also Erdelyi, Magnus, Oberhettinger and
Trtcomt [4, p. 171], and Freud [5, p.44] that
(z-l)a(z+1)SQn(a’8)(z) (I/2)k=0 #1 -If (l-t)a(l+t)Stkpn(a’8)(t)dt
K
n
(a’B)/2z n+l
(1.6)
POLYNOMIAL EXPANSION OF MULTIVALENT FUNCTIONS
"a’B’(%
where K
445
In connection with this, see the argument used in
is as defined above.
[I, p.67].
the proof of formula (4.3.3) of
Using (1.6) in (1.5)we tmmediately deduce (1.2).
Now
d|fferentlating
(1.1) we see from equation (4.21.7) of [II that
--(n+a++l)a nP(a+l’B+l)(z)
n-I
f’(z)
n=
(1.2),
Again, as in the proof of
(n+ a+ B+1)a
{i
n
we deduce from this and
-I
h(a+l’B+l)}
n-I
(K a+l,
n-I
(z-l) a+l (z+l)
E
+I )/h a+l, B+I )(1/2i)
n-I
f
E
[I, p. 245]
for n
> I, that
B+I (a+l,+l) (z)f’ (z)dz
Qn-l
f’(z____)
z
n
dz
n
K(Cr+l,ff+l)/2z
n-I
where we have used the equation
(z-l)a+l(z+l) B+lo(a+l,B+l)(z
n-I
which is deduced as in (1.6).
This is equation (1.3) and the proof of the lemma is
now complete.
2.
MAIN THEOREM.
THEOREM 2.1.
Let f(z)
.
a
n=O
M(r,f)
n
)-Y
C(1-r/r
>
where C,
0
P(a’B)(z)
n n
be mean p-valent In Int(E
o
0 and M(r,f) is as defined above.
and
Then, as
we have
0(t)n
Y-1/2,
(y
--11
<
I/2),
(
0
o(I),
((
1/2),
<
1/2),
where 0(I) and o(1) depend on a,b,a,,,f and f only.
PROOF OF THEOREM 2. I.
From (1.3) and Theorem B we deduce, using the bounds
(K(Cl’B+l)/h(Crl’l+l)(cosh
n-1
n-I
<
<
a
O
’)/sinhns)
(a+l, +1)
(K(a+l’
B+I)/h n-I
)(2nil (r,f,)/rn( l-l/r)
n-I
0(1)n Y-1/2
(y
0()(Zogn)
(
o(1),
(y
((n-1)/n)r
where we have chosen r
I (r
and provided that 1-n/(n-1)r
>
>
1/2)
/z)
<
1/2),
0, This completes
the proof of Theorem 2.1.
COROLLARY 2.1.
as n
(R)we
Let f(z)
lanl
a
n= 0
have
r
-n
0
p(a,)
n n
(z) be mean p-valent in Int(E ).
o
O(1)n 2p-I12
(p
0(1) (log n)
(p
o(1),
(p
>
114)
1/4)
<
1/4),
Then
446
M.M. ELHOSH
where 0(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
COROLLARY 2.2.
Let f(z)
a
n--u
we have
n
2p in Theorem 2.1.
a
O(1)n
n
n
p(a,B)
(z) be univalent in Int(E ).
O
Then as
3/2r-n
o
where 0(I) depends on a,b,a,B and f only.
This corollary follows upon setting p
REMARK.
we see by settlng z
f(E cosh s o
.
nffi 0
cosh s
o
where
F(2n+a+B+I)
n r(n+ a+ I+
+ c l(-I/cosh
where
in Corollary 2.1.
Using the formula (4.21.2) of [I] and the argument used in
n--0
II
____o__s )n((E
an (__csh
cos
T
+
tanh s
o
sin
TI
[2, Remark 2]
that
1/cosh s )n
o
so)n-I+...+Cn/Coshnso}
n n
(a,B)()
n
(_ 1/cosh s o )n + c (- 1/cosh s o )n-l+.
+Cn/Coshns o
and
n
r(2n+a++l)a n coshns o /2nr(n+l)r(n+a+B+l).
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 (see Hayman
Using
[61).
REFERENCES
3.
SZEGO, G., Orthogonal polynomials, Amer. Math. Soc. Colleq. 23, (1939).
ELHOSH, M.M., On mean p-valent functions in an ellipse, Proc. Roy. Soc. Edlnb.
92A, (1982), I-II.
TITMARSH, E., The Theory of Functions, Oxford University Press (1939).
4.
ERDELYI, A., MAGNUS, W., OBERHETTINGER, F. and TRICOMI, F., H_her
I.
2.
Transcendental Functions,
5.
6.
II,
MacGraw-Hill (1953).
FREUD, G., .rthogonal Polynomials, Pergamon Press (1971).
HAYMAN, W.K., Coefficient problems for univalent functions and related function
classes, J. London Math. Soc. 40 (1965), 385-406.