Internat. J. Math. & Math. $ci.
Vol. I0 No. 2 (1987) 259-266
259
ON COEFFICIENT BOUNDS OF A CERTAIN CLASS
p-VALENT ,-SPIRAL FUNCTIONS OF ORDER O
M.K. AOUF
Department of Mathematics, Faculty of Science
University of Mansoura
Mansoura, Egypt
and
Department of Mathematics, Faculty of Science
University of Qatar
P.O. Box 2713
Doha
Qatar
(Received April II, 1986)
ABSTRACT.
Let
(A,B,p,a)(II
S
f(z)
class of functions
satisfy for
e
w(z)
re
z
il
sec%
z
a z
n
n
and
Izl
{z:
U
analytic in
a < p), denote the
o
<
I}, which
U
zfz)
ip tan
f(z)
l
S (A,B,p,a)
[
+
U
p + [pB+(A-B) (p-a) ]w(z)
+ Bw(z)
w(o)= o
with
an
paper we obtain the bounds of
class
-I & A < B &
<
n=p+l
iO
is analytic in
p
and
lw(z)
and we maximize
Izl
for
lap+2
z
a2p+l
U.
In this
over the
for complex values of
KEY WORDS AND PHRASES. p-Valent, analytic, bounds, -spirallike functions of order
19BOAMS SUBJECT CLASSIFICATION CODE. 30A32
1.
INTRODUCTION.
A (p a fixed integer greater than zero) denote the class of functions
P
p
< I}. We use
{z:
to denote
z +
which are analytic in U
f(z)
k=p+l
the class of bounded analytic functions w(z) in U satisfies the conditions
Let
akzk
w(o)--o
and
Izl
lw(z) l-<-Izl
for
z e U.
Also let
e (p a) (with p
denote the class of functions with positive real part of order
P(z)
p
+
[
k=l
a
a postive integer)
that have the form
ckzk
U and satisfy the conditions P(o)
p and Re{P(z)} > a
(o --< a < p) in U. The class P(p,a) was introduced by Patil and Thakare [I].
It was shown in [I] that the function P
P(p,a) if and only if
which are analytic in
(I.I)
M.K. AOUF
260
p-(p-2a)w(z)
+ w(z)
P(z)
Ill
For
<
and
Re
for
U
z
e
and
zfz)
5
zz)
i
$ (p,a)
Sl(p,e)
analytic in
U
.
A < B
P(A,B)(-I
Let
S%(p,a)
i)
was introduced by Patil and Thakare [I].
It
p e P(p,a)
if and only if there exists a function
(1.4)
denote the class of functions
Pl(Z)
l+Aw(z)
I+Bw(z)
are p-valent l-spiral-
P(z) + ip sin
and such that
PI(Z)
P(A,B)
n
Z
n=l
if and only if
(1.5)
U
z
w
+
Pl(Z)
P(A,B) was introduced by Janowski [2].
-I =< A < B
and o =< a < p, denote by
The class
For
P2(z)
tions
cos
f(z)
(1.3)
We say the functions in
such that
S (p,a) denote
I
> a cos
f
let
which satisfy
P
The class
like of order a
was shown in [i] that
e
A
a < p.
o
(1.2)
a fixed integer greater than zero
p
f(z)
the class of functions
i
w
of form (I.I) which satisfy that
(p-a)P l(z) + a,
P2(z)
Pl(Z)
Using (1.5) in (1.6), one can show that
p
P2(z)
f(z)
A
P
zf)
i
P2(z)
E
P(A,B,p,a) if and only if
(1.7)
w
e < p)
A < B
and
, P2(z)
P(A,B,p,a)
o
denote the class
which satify
P2(z)+ip
cos
f(z)
-I
<
(1.6)
P(A,B)
E
+ [pB+(A-B) (p-a) ]w(z)
+ Bw(z)
(A,B,p,s)(ll
Also let
of functions
e
P(A,B,p,a) the class of funcP2(z) P(A,B,p,a) if and only if
sin
(1.8)
Using (1.7) in (1.8) one can easily show that:
f(z)
(i) e
(ii)
il
sec
z’(z)
f(z)
s l (A,B,p,=)
zfz)
ip
f(z)
if and only if
(p-s) ]w(z)
tan: p + [pB+(A-B)
+ Bw(z)
p + [pB+(A-B)(p-a)cos
+ Bw(z)
e
]w(z)
w
w
.
.
(1.9)
(l.iO)
We shall need the following lemma in our investigation:
LEMMA i_[.3]. Let
w(z)
k=l
Ib 2
b 2
Equality is attained for
max
bkzk
,
if
v
is any complex number, then
(l.il)
{I,
w(z)
#.2
and
w(z)
z.
COEFFICIENT BOUNDS OF A CERTAIN CLASS OF p-VALENT FUNCTIONS
2.
LEMMA 2.
and
-I
If integers
A < B
<--
and
p
(j+l) 2
j=o
cos2%
.
BIt
2
+
I) sec 2
[k 2(B 2
2
k-I
II
+ 2kB(B-A)(p-e)]
(B-A) (p-a)cos
We prove the lemma by induction on
ql
{(B-A)2(p-a)2 +
m.
For
m=q-l.
Next suppose that the result is true for
[k2(B 2
,
e
-il
2
+ Bj[
}.
(2.1)
(j+l) 2
j=o
I)
the lemma is obvious.
m=l
We have
sec2l
k=l
2
k-I
+ (B-A)2(p-a) 2 + 2kB(B-A)(p-a)]
x
(B-A) (p-e)cos
][
e
+Bi
}:
(j+l) 2
j=o
cos2
q2
z2
k=l
+ (B-A)2(p-e)
cos2
q2
<
m=l
(B_A) 2 (p_e) 2
m2
PROOF.
I1
< p,
o
i, then
l(B-A)(p-a)cos e -ix +
m-i
are greater than zero;
m
261
Sk(A,B,p,a).
COEFFICIENT ESTIMATES FOR THE CLASS
q-2
[ [k2(B 2
{(B-A)2(p-a) 2 +
sec2%
I)
k=l
k-I
+ (B-A)2(p-a) 2 + 2kB(B-A)(p-a)]
H
[(B-A)(p-e)cos %e -i%
+
Bil
+
B[
2
(j+l) 2
j=o
[(q-l)2(B2-1)sec2l + (B-A)e(p-e) 2 + 2(q-l)B(B-A)(p-e)]x
q-2
(B-A) (p-a)cos
j=o
-il
(B-A) (p-a)cos e
+
2
(j+l)
e
-i%
2
(j+l) 2
j=o
q-2
,
2
B]
(q-I)2B2+(B-A)2(p-s)2cos2I + 2(q-l) B(B-A)(p-e)cos2I
q2
II
(B-A) (p-a)cos
,
+
e
(j+l)
j=o
Showing that the result is valid for
THEOREM I.
If
f(z)
z
p
+
m=q.
[
k=p+l
n-(p+l)
a
n
k=o
B
akzk
This proves the lemma.
S%(A,B,p,e),
l(B-A)(p-e)cos le -il +
k+l
Bk
then
(2.2)
M.K. AOUF
262
for
p+l
n
and these bounds are sharp for all admissible
A,B, and a and for each
n.
As
PROOF.
S%(A,B,p,),
f
zfz)
sec
e
from (1.9), we have
p+[pB+(A-B) (p-s) ]w(z)
-ip tan
f(z)
w e
l+Bw(z)
.
This may be written as
{Bei
+ [-pB+(B-A)(p-e)-ipB
sec lsf(z)
(p+ip tan l)f(z)
sec
e
tan
If(z)} w(z)
lzfz)
Hence
Be
iI
{pz p +
sec
[.
(p+k)ap+kZ
k=l
zp
[-pB+(B-A) (p-s)-ipB tan
(p+ip tan)
z
p
[
+
see
p zp +
[.
(p+k)a
+
ap+kzP+k}
+
(P+k)
k;1
p+k
w(z)
ap+kzP+k}
p+kzP+k}
or
Be
il
+
sec l+ [-pB+(B-A)(p-) -ip B tan I
(p+k) Be
il
sec A
(p+ip tan I
+ [-pB+(B-A) (p-s)
[
secl) +
p e
ip B tan ]
ap+kZ
p+ip tanl-(p+k)e
i
1
k
w(z)
sec }
k=l
ap+kZ
k
which may be written as
[
(p+k) Be
sec l+ [-pB+(B-A)(p-s)
ip B tan l]
k=o
il
sec
[p+ip tan I- (p+k)e
where
a
P
=I
and
w(z)
[
k=o
Equating coefficients of
m-I
[.
bk+
z
m
z
]ap+kZ
iR
k
(2.3)
on both sides of (2.3), we obtain
{(p+k)Be il sec l+ [-pB + (B-A)(p-a)
(p+m)e
w(z)
k+l
k=o
p+ip tan I
1
ap+kZk
sec X} a
p+m
ip B tanl
]}
ap+k bin_k
COEFFICIENT BOUNDS OF A CERTAIN CLASS OF p-VALENT FUNCTIONS
ap+m
which shows that
on right-hand side depends only on
ap ap+
ap+(m_l)
of left-hand side.
m-1
E
[
.
k=o
Hence we can write
(p+k)B e
+
sec
m
[p+ip tan- (p+k)e
[-pB+(B-A)(p-a)
i%
sec]ap+k
k=o
for
z
re
m-I
2
.
f
.
2m-i
2
m
f
2
+
Akzk
+ [-pB+(B-A)(p-a)
sec
lap+kl
ip B tan%]
{(p+k)Be i%
sec+[-pB+(B-A)(p-a)-ip’B’tan]}ap+kr
{p+ip tan%-(p+k)e il
sec%}ap+kr
Ip+ip
tan
l-(p+k)e
i%
2
lap+kl
secl
Ip+ip tan-(p+k)e il
k i0k
k=o
k=m+l
2
r
me +
[.
I lap+kl
2
r
r
k iSk
e
2
r
2
r
2k
k
e
i0k 2
lw( re i8 )I
AkrkeiSkl2
IAkl
k--m+l
2
sec
[.
+
e
2
ap+k
tanl]
-ipB
k--o
m
Setting
k=+l
[-pB + (B-A)(p-a)
k=o
m
[
+
2
sec
(p+k)Beil
k=o
>_L
k
k w(z)
and a proper choice of
=> 0).
i0
2, then
0 < r < i, 0 --< 0
l(p+k)Be i%
m-I
2
>__
z
]} a p+k z
ip B tan
Ak(k
m=1,2,3
Let
263
dO
2
"d8
dO
2k
2k
(2.4)
in (2.4), the inequality (2.4) may be written as
r
m-I
l(p+k)Be i
sec
+
[-pB+(B-A)(p-a) -ip B tan
i
secl
]I
2
k=o
p+ip tanX
(p+k)e
i
see
P + ip tan X-(p+m)e
2
2
lap+kl
2
I lap+ml
2
(2.5)
Simplification of (2.5) leads to
2
ap+m
Replacing
lanl
2
-<
cos21
m2
(n-p) 2
{k2 (B2-1)sec2 +(B-A) (p-a) (B-A) (p-a)+2kB
k=o
p+m by n
cos2
m-I
12
(2.6)
in (2.6), we are led to
n-(+l) {kZ(B2-1)sec2 +(B-A)(p-a)[(B-A)(p-a)+2kB]}
k--o
ap+k
]ap+k 12,
2.7)
264
M.K. AOUF
where
p+l.
n
For
n=p+l, (2.7) reduces to
lap+112
lap+ 11-
(B-A)2(p_a)2cos2 %
9
or
(-) (p-a)cos
x
(2.8)
which is equivalent to (2.2).
To establish (2.2) for
n, n
Fix
n > p+l, we will apply induction argument.
p+2, and suppose (2.2) holds for
24
fan 12 __< cos
(n-p)
k
1,2
n-(p+l).
Then
(B-A) 2 (p-a) 2 +
2
n- p+l)
{k2(B2-1)sec2% +
(B-A) (p-a) (B-A) (p-a) + 2kB]
k--I
k-i
2
(B-A) (p-a)cos e -i% +
(j+l) 2
H
j=o
Bj]
Thus from (2.7), (2.9) and lemma 2 with
[an
2
n-(p+l)
H
<
(B-A)(p-)cos
e
-i
+
(2.9)
m--n-p, we obtain
2
B’I
(j+l) 2
j--o
This completes the proof of (2.2).
This proof is based on a technique found in
Clunie [4].
For sharpness of (2.2) consider
z
f(z)
p
[]
() (p-a)cose -i
(I-BE)
l, B
# 0
Remarks on Theorem I:
(I)
B=I
Setting
and
A=-I
in Theorem
I, we get the result of Patil and
Thakare [I].
(2)
(3)
Setting
B=I, A=-I
and
p=l
in Theorem I, we get the result of Libera
Setting
B=I, A=-I, p=l
and
a=0
in Theorem I, we get the result of
B=I, A=-I, p=l
and
%=0
in Theorem I, we get the result of
[5].
Zamorski [6].
(4)
Setting
Robertson [7] and Schild [8].
THEOREM 2.
f(z)=z p +
If
k=p+l
number, then
a2 p+l <= (B-A)(p-a)
cosmax
2
lap+2-
This inequality is sharp for each
PROOF.
e
il
As
sec
f
E
SA(A,B,p,a),
zf’(z)
f(z
-ip tan
.
akzk
i,
E
S%(A,B,p,a)
and
(B-A)(p-a)(2-l)cos
is any complex
-eil}
(2.10)
from (1.9) we have
p+[pB+(A-B) (p-a) ]w(z)
l+Bw(z)
(2.11)
.
265
COEFFICIENT BOUNDS OF A CERTAIN CLASS OF p-VALENT FUNCTIONS
where
w(z)
bk zk
k=l
c
.
Rewriting the form (2.11) as
p-e
w(z)
ik
zf’(z) + ip
f(z)
sac k
Beiksec kzf’(.z)
f(z)
+ [-pB+(B-A)(p-a)-ip tank
seck [pf (z)-zf’(z)
e
Bei
secl-(zf (z)) +[-Bp e
-e
il
, ap+
.
.
k
k=l
-e
iX
z
k
i
sac k
k
ap+k
(B-A) (p-a)+ <
x {2 a
ap+l
sac k
(B-A) (p-a)
z
k
z
iX
e
sec X
(B-A) (p-)
bl
e
2
ik
z2
k
k
+ (B-A) (p-a)
and
ap+kZ
Beiksec } ap+k]Z
2
z
k
see
-((B-A)(P-a)+Beikse.Cl)a
p+l
(B-A) (p-a)
p+2
and then comparing coefficients of
b
z
k=
k=l
-e
+(B-A)(p-a)]f(z)
Beil
+
k=l
(B-A) (p-a) +
lseck
i
kap+k
sec i
k;1
(B-A)(p-a)[l +
tan k
x
z2
+ ...]
on both sides, we have
ap+l
sack
2 (B-A) (p-a)
(B-A) 2 (p-a) 2
ap+ 2
(B-A)
(p-a)+eiksecA
a2
p+1
]"
Thus
(B-A) (p-a)
ap+l
e
i)t
seek
bl
and
(B-A) (p-a)
ap+2
2 e
i
sec
k
b2
(p-a)+eiksecl, a2
+ (B-A)
2 (B-A) (p-a)
p+l
Hence
ap+2
a2
p+l
,(B-A) (p-a)
2etXsec k
(B-A, (p-a)
2eiksec k
b
2
b
+
2
(p-a)+eiksecl
+ (B-A)
2 (B-A) (p-a)
(p-a)+eikseck
(B-A)
2 (B-A) (p-a)
]a2p+l
P
(B-A) 2 (p-a) 2
e
2
lsec2
bl 2
(2.12)
M.K. AOUF
266
Thus taking modulus of both sides of (2.12), we are led to
a2p+ll
lap+2
(B-A) (p-e.) cos
Ib 2
(B-A)(p-e)2
Ib2
2
{e
sec
}
2(B-A)(p-e)ll
b121
e sec I
l-il(B-A)(p-e)(2-l)
bl I.
e
sec I
2
(2.13)
in (2.13), we get
Using lemma
lap+2
cos
(.p-e)+ei%secl
(B-A)
2(B-A)(p-n)
a2p+ll (B-A)2(P-n)
coslmax {I,
l(B-A)(p-n)(2-l)
cos l-e
I}
and since (I.ii) is sharp, then (2.10) is also sharp.
Remark on Theorem 2.
Setting (i) B=I and A=-I,
(iii) B=I, A=-I, p=l and e=0,
(li) B=I, A=-I and p=l,
(iv) B=I, A=-I and =0, in Theorem 2, we get the
results of Patil and Thakare [I].
In conclusion, I would like to thank Prof. Dr. D. K. Thomas for his
ACKNOWLEDGEMENT.
kind encouragement and helpful guidance in preparing this paper.
REFERENCES
I.
2.
3.
4.
5.
6.
7.
PATIL, D.A. and THAKARE, N.K. On Coefficients Bound of p-Valent h-Spiral Functions
of Order e, Indian J. Pure appl. Math. 10(7) (1979). 842-853.
JANOWSKI, W. Some Extremal Problems for Certain Families of Analytic Functions,
Ann. Polon. Math. 28(1973), 297-326.
KEOGH, F.R. and MERKES, E.P. A Coefficient Inequality for Certain Classes of
Analytic Functions, Proc. Amer. Math. Soc. 20(1969), 8-12.
CLUNIE, J. On Meromorphic Schlicht Functions, J. London Math. Soc. 34(1959),
215-216.
LIBERA, R.J. Univalent n-Spiral Function, Canad. J. Math. 19(1967), 449-456.
ZAMORSKI, J. About the Extremal Spiral Schlicht Functions, Ann. Polon. Math. 9
(1962), 265-273.
ROBERTSON, M.S. On the Theory of Univalent Functions, Ann. of Math. 37(1936),
374-408.
8.
SCHILD, A.
On Starlike Functions of Order e, Amer. J. Math. 87(i) (1965), 65-70.