1. No category

Faber polynomial coe¢ cients of classes of meromorphic bi-starlike
functions
Jay M. Jahangiri1 and Samaneh G. Hamidi2
1 Department of Mathematical Sciences, Kent State University, Burton, Ohio 44021-9500, U.S.A.
2 Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia.
Correspondence should be addressed to Jay M. Jahangiri: [email protected]
Applying the Faber polynomial coe¢ cient expansions to certain classes of meromorphic bi-starlike functions,
we demonstrate the unpredictability of their early coe¢ cients and also obtain general coe¢ cient estimates for
such functions subject to a given gap series condition. Our results improve some of the coe¢ cient bounds
published earlier.
Let
be the family of functions g of the form
(1)
g(z) =
1
X
1
+ b0 +
bn z n ;
z
n=1
that are univalent in the punctured unit disk D := fz : 0 < jzj < 1g :
For the real constants A and B (0 B 1; B A < B) let (A; B) consist of functions g 2
so that
zg 0 (z)
1 + A'(z)
=
g(z)
1 + B'(z)
P1
n
where '(z) = n=1 cn z is a Schwarz function, that is, '(z) is analytic in the open unit disk jzj < 1
and j'(z)j jzj < 1: Note that jcn j 1 (Duren [8]) and the functions in (A; B) are meromorphic
starlike in the punctured unit disk D (e.g. see Clunie [7] and Karunakaran [12]). It has been proved
by Libera and Livingston [15] and Karunakaran [12] that jbn j Bn+1A for g 2 [A; B]:
The coe¢ cients of h = g 1 , the inverse map of g; are given by the Faber polynomial expansion (e.g.
see Airault and Bouali [3] or Airault and Ren [4, p. 349])
h(w) = g
1
(w) =
1
1 X
1
+
Bn wn =
w
w
X1
K n (b0 ; b1 ; : : : ; bn )wn ;
n n+1
b0
n=0
n 1
where w 2 D,
n
Kn+1
(b0 ; b1 ; : : : ; bn ) = nbn0
+
n(n
1
1)bn0
b1 + n(n
1)(n 2)(n
3!
2
3)
1
b2 + n(n
2
bn0
4
1)(n
(b4 + 3b1 b2 ) +
2)bn0
X
bn0
3
(b3 + b21 )
j
Vj
j 5
and Vj is a homogeneous polynomial of degree j in the variables b1 ; b2 ;
; bn .
In 1923, Lowner [16] proved that the inverse of the Koebe function k(z) = z=(1 z)2 provides the
best upper bounds for the coe¢ cients of the inverses of analytic univalent functions. Although the
estimates for the coe¢ cients of the inverses of analytic univalent functions have been obtained in a
surprisingly straightforward way (e.g. see [13, page 104]), but the case turns out to be a challenge when
the bi-univalency condition is imposed on these functions. A function is said to be bi-univalent in a
1
Jahangiri and Hamidi
2
given domain if both the function and its inverse are univalent there. By the same token, a function is
said to be bi-starlike in a given domain if both the function and its inverse are starlike there. Finding
bounds for the coe¢ cients of classes of bi-univalent functions dates back to 1967 (see Lewin [14]).
The interest on the bounds for the coe¢ cients of subclasses of bi-univalent functions picked up by
the publications [6], [17], [9], [1] and [10] where the estimates for the …rst two coe¢ cients of certain
classes of bi-univalent functions were provided. Not much is known about the higher coe¢ cients of
subclasses bi-univalent functions as Ali, Lee, Ravichandran and Supramaniam [1] also declared …nding
the bounds for jan j; n 4 an open problem. In this paper, we use the Faber polynomial expansions
of the functions g and h = g 1 in (A; B) to obtain bounds for their general coe¢ cients jan j as well
as providing estimates for the early coe¢ cients of these types of functions.
We shall need the following well-known two lemmas, the …rst of which can be found in [11] (also
see Duren [8]).
P
1
n
Lemma 1. Let p(z) = 1 + 1
n=1 pn z be so that Re(p(z)) > 0 for jzj < 1: If
2 then
p2 + p21
(2)
2 + jp1 j2 :
Consequently, we have the following lemma, which we shall provide a short proof for the sake of
completeness.
Lemma 2. Consider the Schwarz function '(z) =
then
c2 + c21
(3)
P1
n=1 cn z
1+(
n
where j'(z)j < 1 for jzj < 1: If
0
1)jc1 j2 :
Proof. Write
p(z) = [1 + '(z)]=[1 '(z)]
where p(z) = 1 + n=1 pn
is so that Re(p(z)) > 0 for jzj < 1. Comparing the corresponding
coe¢ cients of powers of z in p(z) = [1 + '(z)]=[1 '(z)] shows that p1 = 2c1 and p2 = 2(c2 + c21 ).
By substituting for p1 = 2c1 and p2 = 2(c2 + c21 ) in (2) we obtain
P1
zn
2(c2 + c21 ) + (2c1 )2
or
2 + j2c1 j2
c2 + (1 + 2 )c21
1 + 2 jc1 j2 :
Now (3) follows upon substation of = 1 + 2
0 in the above inequality.
In the following theorem we shall observe the unpredictability of the early coe¢ cients of the functions
g and its inverse map h = g 1 in [A; B] as well as providing an estimate for the general coe¢ cients
of such functions.
Theorem 1. For 0 B
[A; B]. Then
8 B A
< p2B A ;
(i): jb0 j
:
B A;
8 B A 1 A
< 2
2(B
(ii): jb1 j
: B A
2 ;
(iii): b1
(iv): jbn j
1 and
B
A < B let the function g and its inverse map h = g
if 2B
A
1;
otherwise:
2B
2
A) jb0 j ;
B A
2B A 2
b
;
2(B A) 0
2
B A
; if bk = 0 for 0 k
n+1
if 0
A
otherwise:
n 1:
1
2B;
1
be in
Meromorphic bi-starlike functions
3
Proof. Consider the function g 2
given by (1).Therefore (see [3] and [4].)
zg 0 (z)
=
g(z)
(4)
1
X
1
Fn+1 (b0 ; b1 ; b2 ;
; bn )z n+1
n=0
where Fn+1 (b0 ; b1 ; b2 ;
; bn ) is a Faber polynomial of degree n + 1. We note that F1 = b0 , F2 =
b20 2b1 , F3 = b30 + 3b1 b0 3b2 , F4 = b40 4b20 b1 + 4b0 b2 + 2b21 4b3 and F5 = b50 + 5b30 b1 5b20 b2
5b0 b21 + 5b1 b2 + 5b0 b3 5b4 : In general (Bouali [5], p.52)
Fn+1 (b0 ; b1 ;
X
; bn ) =
; in+1 ) bi01 bi12
A(i1 ; i2 ;
binn+1
i1 +2i2 + +(n+1)in+1 =n+1
where
A(i1 ; i2 ;
Similarly, for the inverse map h = g
wh0 (w)
=
h(w)
(5)
where Fn+1 (B0 ; B1 ; B2 ;
Fn+1 =
+(n+2)in+1 (i1
; in+1 ) := ( 1)(n+1)+2i1 +
1
1
+ i2 +
+ in+1 1)!(n + 1)
:
(i1 !)(i2 !) (in+1 !)
we have
1
X
Fn+1 (B0 ; B1 ; B2 ;
; Bn )wn+1 ;
n=1
; Bn ) is a Faber polynomial of degree n + 1 given by
n(n (n + 1))! n
n(n (n + 1))!
B0
B n 2 B1
n!(n 2n)!
(n 2)!(n (2n 1))! 0
n(n (n + 1))!
B n 3 B2
(n 3)!(n (2n 2))! 0
X n
n(n (n + 1))!
n (2n 3) 2
B0n 4 B3 +
B1
B0
(n 4)!(n (2n 3))!
2
j
Kj ;
j 5
Kj is a homogeneous polynomial of degree j in the variables B1 ; B2 ;
(6)
B0 =
b0 ;
and
Bn =
; Bn
1
and
1 n
K
(b0 ; b1 ; : : : ; bn ):
n n+1
Since, both g and its inverse map hP= g 1 are in [A; B], P
by the de…nition of subordination, there
1
n and (w) =
n
exist two Schwarz functions '(z) = 1
c
z
n=1 n
n=1 dn w so that
(7)
zg 0 (z)
=
g(z)
1 + A'(z)
=
1 + B'(z)
1+
wh0 (w)
=
h(w)
1 + A (w)
=
1 + B (w)
1+
1
X
(A
B)Kn 1 (c1 ; c2 ;
; cn ; B)z n
B)Kn 1 (d1 ; d2 ;
; dn ; B)wn :
n=1
and
(8)
1
X
(A
n=1
In general (see Airault [2] or Airault and Bouali [3]), the coe¢ cients Knp (k1 ; k2 ;
; kn ; B) are given
Jahangiri and Hamidi
4
by
Knp (k1 ; k2 ;
; kn ; B) =
(p
+
(p
+
(p
+
(p
X
+
p!
p!
k1n B n 1 +
k n 2 k2 B n 2
n)!n!
(p n + 1)!(n 2)! 1
p!
k n 3 k3 B n 3
n + 2)!(n 3)! 1
p!
p n+3 2
k n 4 k4 B n 4 +
k3 B
n + 3)!(n 4)! 1
2
p!
k n 5 k5 B n 5 + (p n + 4)k3 k4 B
n + 4)!(n 5)! 1
k1n
j
Xj ;
j 6
where Xj is a homogeneous polynomial of degree j in the variables k2 ; k3 ;
Comparing the corresponding coe¢ cients of (4) and (7) implies
(9)
Fn+1 (b0 ; b1 ; b2 ;
; bn ) = (A
1
B)Kn+1
(c1 ; c2 ;
; kn .
; cn+1 ; B):
Similarly, comparing the corresponding coe¢ cients of (5) and (8) gives
(10)
Fn+1 (B0 ; B1 ; B2 ;
1
B)Kn+1
(d1 ; d2 ;
; Bn ) = (A
; dn+1 ; B):
Substituting n = 0, n = 1, and n = 2 in equations (6), (9) and (10), respectively, yield
b0 = (A B)c1 ;
b0 = (A B)d1 ;
(11)
and
2b1 b20 = (B
2b1 + b20 = (A
(12)
A)(c21 B
B)(d21 B
c2 );
d2 ):
Taking the absolute values of either of the equation in (11), we obtain jb0 j
B A: Obviously,
from the equations (11) we note that c1 = d1 . Solving the equations in (12) for b20 and then adding
them gives
2b20 = (B A) c2 + d2 Bc21 Bd21 :
Now in light of equations (11), we conclude that
2b20 = (B
A) c2 + d2
2B
b2 :
(B A)2 0
Once again, solving for b20 and taking square root of both sides we obtain
s
(B A)2 (c2 + d2 )
B A
p
jb0 j
:
2(2B A)
2B A
Now the …rst part of Theorem 1 follows since for 2B A > 1 it is easy to see that
B A
p
< B A:
2B A
Adding the equations in (12) and using the fact that c1 =
4b1 = (B
A) (d2
c2 )
(d21
d1 we obtain
c21 )B = (B
Dividing by 4 and taking the absolute values of both sides yield
B A
jb1 j
:
2
A)(d2
c2 ):
Meromorphic bi-starlike functions
5
On the other hand, from the second equations in (11) and (12) we obtain
2b1 = (B
A)(d2 + Ad21
2Bd21 ):
Taking the absolute values of both sides and applying Lemma 2 it follows
1
jb1 j
(B A) d2 + Ad21 + 2Bjd1 j2
2
1
(B A) 1 + (A 1)jd1 j2 + 2Bjd1 j2
2
B A 1 A 2B
=
jb0 j2 :
2
2(B A)
This concludes the second part of Theorem 1 since for 0 < A < 1 2B we have
B A 1 A 2B
B A
jb0 j2 <
:
2
2(B A)
2
Substituting the equations (11) in the equations (12) we obtain
8
B
< 2b1 b2 = (B A)
b2 c2 ;
0
(B A)2 0
(13)
B
: 2b1 + b2 = (A B)
b2 d 2 :
0
(A B 2 0
Following a simple algebraic manipulation we obtain the coe¢ cient body
b1
Finally, for bk = 0; 0
k
n
(14)
2B
2(B
A 2
b
A) 0
B
A
2
:
1 the equation (9) yields
(n + 1)bn = (A
B)cn+1 :
Solving for bn and taking the absolute values of both sides we obtain
B A
jbn j
:
n+1
Remark 1. The estimate jb0 j
Theorem 2(i)).
pB A
2B A
given by Theorem 1(i) is better than that given in ([10],
Remark 2. In ( [12], Theorem 1) the bound jbn j Bn+1A was declared to be sharp for the coe¢ cients
B A
1 A 2B
2
pB A and jb1 j
of the function g 2 [A; B]. The coe¢ cient estimates jb0 j
2
2(B A) jb0 j
2B A
given by our Theorem 1 show that the coe¢ cient bound jbn j Bn+1A is not sharp for the meromorphic
bi-starlike functions, that is, if both g and its inverse map g 1 are in [A; B]. Finding sharp coe¢ cient
bound for meromorphic bi-starlike functions remains an open problem.
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