55
Internat. J. Math. & Math. Sci.
(1986) 55-64
Vol. 9 No
ON CERTAIN CLASSES OF p-VALENT FUNCTIONS
M. K. AOUF
Department of Mathematics
Faculty of Science
University of Mansoura
Mansoura, Egypt
(Received February 20, 1985)
V(a,b,p)(k
Let
ABSTRACT.
2, b#o
f(z)
denote the class of functions
Re{e il
2
lim sup
u
r+l
z
U
having (p-l) critical points in
a
is any complex number, o
p
n
a z
+ n
p+l n
p and
Ill
{z:
Izl
U
analytic in
<
/2)
I}
and satisfying
zf"(z)
f’(z)
p-e
cosl}
p)]
[p+(l+
de
k coal.
f(z) which are p-valent convex of
o< e < p, with bounded boundary rotation and those p-valent functions f(z)
e < p.
zf’(z)/p is l-spirallike of order e, o
In this paper we generalize both those functions
e,
order
for which
KEY WORDS AND PHRASES.
p-VaZp.nt, starZ.ike, convex, spinallike functions, funotions
with bounded boundary rotation.
AMS SUBJECT CLASSIFICATION CODE.
I.
INTRODUCTION.
Mk(k _>
Let
variation on
Let
z
p
A
+
P
denote the lower class of real-valued functions
[o,2]
where
n p+l
2)
a z
n
p
n
which satisfy the conditions
f2ndm(T)
0
2
m(t) of bounded
/2n idm(t)
0
and
is a positive integer, denote the class of functions
V(a,b,p)(k _> 2,
/2) if there exists
f 2 Re
> o
0
b
#
o
Izl
< i}.
For
f
g
A
p
is any complex number, o
we say that
_<
a < p
and
f"(re
f’(reiO)
d=2p (I-6<r<I)
(1.1)
and
lira sup
r
ri2
v
Re
e
il
[p+
(I+
p_
zf"(z)
f’(z) -p)]
acosl}
d
_<
k.
f(z)
such that
re
I+
{z:
U
which are analytic in
belongs to the class
Ill
30A32, 30A36.
kcos.
(1.2)
f
M.K. AOUF
56
ondition (I.I) implies that
U.
has (p-l) critical points in
f
It is noticed that, by giving specific values to
k,a,b,p
A
and
(a,b,p)
in V
k
we obtain the following important subclasses studied by various authors in earlier
papers:
(1)
V
(2)
V
I(o
l,p)
k
Vk(P).
is the class of p-valent functions investigated by Silvia
[i].
k(o,l,l)
V k, is the class of bounded boundary rotation introduced by
V (a
k
Lwner [2] and Paatero [3, 4]
f(z) e A
tions
studied by Padmanabhan and Parvatham
f(z)
class of functions
Vk(a)
V
k(o,b,l)
V
k%(a,b,l)= V kk(a,b),
(3)
Vk(b),
V
a, o < a < p,
zf’(z)/p
f(z) e
is
Ap
(o
k
I,I)
f(z)
A
E
is the
Vk
[7],
V
x(a
k
investigated by Moulis [8],
[9] and
f(z) e A
introduced by Nasr
is the class of functions
f(z)
investigated by Lakshma[D,
A
C(p), is the class of p-valent convex functions considered by
o
C(a,p), is the class of p-valent convex functions of order
V2(a,I,P)
C
A(p)
%-spirallike in
U,
V
V
is the class of functions
2(o,l,p)
[12],
Goodman
[5],
I, is the class of func-
investigated by Moulis [6] and Silvia
A
I, is the class of functions
a
o
o < a <
Vk(a)
(o
2
for which
l,p)
is the class of functions
V2(a,l, p)
zf’ (z)/p is
f(z) e A
P
for which
C (a,p), is the class of functions
%-spirallike of order
a, o < a < p,
o
V 2(o,b,p)
C(b,p), is the class of p-valent functions satisfying
+
Re [p
C(b,p)
The class
V
introduced by Aouf
Cl(a,b,p)
2%(a,b,p)
Re e
Also from
(4)
o
Vk(a
V
il
[p +
l(a,b,p)
k
f(z)
f’(z)
(i+
Vk(a, p)
f(z)
and finally for k=2 any function
zf"(z)
f() -p)]
> a cosl
z
U.
we can obtain the following important subclasses:
Vk(a,p),
l,p)
[13]
> o,z e U.
if and only if
lim sup
r+ I_
i.e.,
zf"(z) -p)]
f,(z
(I+
/02
is the class of p-valent functions
Re {i+
p
zf"Iz)}-a
f’ z)
a
f(z)
d8 < k
if and only if
pzp-l{exp
-(p-a) I
0
2
log(l-ze
-it
dm(t)}
m(t) e Mk
A
p
satisfying
CERTAIN CLASSES OF p-VALENT FUNCTIONS
Vk-A.a( ,l,p) VkO(a,cos
(5)
f(z) e A
functions
f(z)
i.e.,
f,
V
g
(z)
pz
(
p-I
-(p-)cos e
o
Vk(b,p),
sup
r/l
i.e.,
f(z)
Vk(b,p)
f’ (z)
VkO(a
(7)
pz
cosX
dO < k cosX
P
exp
lira
is the class of p-valent
if and only if
Vk(O,b, p)
(6)
eiX(l+ zf"(z))}-a
f’(z)
Re
f02
p)
k
-,
<
satisfying
P
sup
lim
r+l
IXl
V (a,p),
,p)
e
57
-i% / 2
-it
log(l-ze
0
dm(t)}, re(t)
f(z) e A
is the class of p-valent functions
/2 IR e
0
{p +
(I+
zf"(z)
f’(z)
p))
dO
M
k.
P
satisfying
kp,
if and only if
p-I
b,p)
f02
exp {-pb
Vk(
b p)
2
Re
log(l-ze -it) dm(t)}, m(t) e Mk.
is the class of p-valent functions
f(z)
Ap
saris-
fying
lim
r+l
i.e.,
f(z) e
f’(z)
p-I
(I+
o
exp
zf"(z)
f’(z)
P)}
a
p
Vk(O.,b,p
pz
{p+
sup
-[d0
< k,
if and only if
{-(p-a)b rj2 log(l-ze
dm(t)}
0
m(t) e
_.
We state below some lemmas that are needed in our investigation.
LEMMA
(I)
(2)
[0]. h(z)
h’(z)
h’(z)
V
X(,b)
if and only if
k
e-iX f2
0
exp{-(l-a)b cosX
[f’l(Z)](l-a)b cosXe
b
cosXe
-iX
log(l-ze
-it
[f2(z)]
(4)
h’(z)
[f(z)]
(5)
h’(z)
[f4(z)]
f4
(6)
h’(z)
[f(z)]
(l-a)cs% e
(7)
there exists two normalized starlike functions
f2 Vk()"
f3 e Vk
V (a)
k+2
f’ (z)
4
z
s2(z)
z
k
fl Vk
h’(z)
sl(z)
L.
-iX
(3)
(l-a)b
dm(t)}, re(t)
k-2
4
f5 Vk(b)
(1-a)b cosX e
-iX
s
l(z)
and
s
2(z)
such that
58
M.K. AOUF
LEMMA 2[I.
h(z)
If
V(a,b),
belongs to
then
H(z)
is defined by
.z+a
h’l-az)
h,(a)(l+z)2(l-)b
H’(z)
H(o)
Iz;
o,
lal
i,
cosXe -iX
3[I. Suppose h(z)
LEMMA
k
]b2[ _< 3(I-
[b[
a)
V
is also in the class
+
z
b2z
2
+
k
+
b3z
belongs to
V (a,b)
then
cosX
and
2
[b[
Ib3] _< (l-a)
X[l-(l-a)
cos
k
]b[cosX -]
These bounds are sharp, with equality for the function
(I
(1+z)
defined by
cose -iX
)b
-/
2
X
Vk(P),
g(z) e
LEMMA 4[lj.
p >_ I,
if and only if
g’(z)
pzp-l[f 3 (z) ]P
for
f3 Vk
some
REPRESENTATION FORMULAS FOR THE CLASS
2.
VXk(a,b)
2
(l-z)
h’(z)
h(z) e
pzp-l[h ’(z)]
f’(z)
h e V
for some
z
+
k
X(a,b,p)
k
if and only if
pz
n-I
E
n a z
n
n=p+l
p-I +
(2.1)
(,b).
pzp-l[h ’(z)]
f’(z)
Let
PROOF.
h(z)
Vka,b, p)
fz)
LEMMA 5.
V
n=E2 bnZ
n
e V
X
k(a,b),
pz
p-I
l
+ n--p+l
n a z
n
n-1
for
z e U
By direct computation, we obtain
12
0
j.2=
iX
Re {e [p
+
(i
+ zf"(z)
f’(z)
P)
-a cos
Re
{eiX[l"
+
zh" (z)
i)
b
0
’
X}
IdO
p-a
cosX
Id0
l-a
and the result follows from (1.2).
An immediate consequence of Lemmas 5 and
THEOREM i.
(I)
f’(z)
f(z) e V
pz
p-I
X(a,b,p)
k
is
if and only if
exp {-(p-a)b cosX e
-iX
12
0
log(l-ze
-it
.)dm(t)}, m(t)e
M
k
CERTAIN CLASSES OF p-VALENT FUNCTIONS
cos
-iX
(2)
f’(z)
PzP-l[fl’(z)](P-a)b
(3)
f’(z)
)’ ](l_Pe)
pzp_l[f2(z
(4)
f, (z)
PzP-I
(5)
f (z)
b
pzp-I [f4(z) (I-P) f4 Vk(a)
(6)
f’ (z)
PzP-I
(7)
there exists two normalized starlike functions
f5’(z)]
fl
b cosk e
k
-i%
f3 Vk
(p-a)cos
e
-i
f5 Vk(b)
(p-a) b cos%e
k+2
V
E
f2 E Vk(a)
(p-a)b
f3 (z)
e
59
s
(z)
and
s2(z)
such that
-i
sl(z)
f’ (z)
-q--
p_
pz
k-2
4
s2(z)
Z
(4) and Lemma 4 is
Also an immediate consequence of Theorem
THEOREM 2.
V(a,b,p)
f(z)
pz
3.
if and only if
g
p-t
COEFFICIENT ESTIMATES FOR THE CLASS
THEOREM 3.
f(z)
If
z
p
V(a,b,p).
+n__Zp+l anZ
[ap+ll _< ,[P(P-a)]p+l k lbl
lap+2[ ! P(P-a)[b[p+2
Vk(p)
E
x
n
e V (a,b,p), then
k
3.1)
cosX
2
cosX {[I
(l-a)
k
+
Ib[cosX -]
(p-l)
[b[
k2
cosX -}
(3.2)
These bounds are sharp with equality for
k
f(z)
pz
p-I
-iK
(p-a)b cos, e
[(I-z) k
(l+z)
2
(3 3)
+
PROOF.
By Lemma 5, there exists an
f’(z)
pz
p-I
+
n__Zp+l
n
anZ
n-I
pz
h(z)
z
p-I [I +
n__Z2
n=E2 bn zn
+
n
V
k(a,b)
bn zn-l]
such that
(3.4)
Expanding the right side of (3.4), we obtain
f’(z)
pz
p-1
p-a b zp
p-a
p-a p-1
+ 2p (l_a)
+ p{3 (l_a)
2
b3 + 2 (l_a)(l_a) b 2 2}
Equating coefficents from (3.5) and (3.4), we have
zp+l +
(3.5)
M.K. AOUF
60
p-a
(p+l)ap+
2P(l_a) b2
(p+2)ap+ 2
p-a
p{3 (l-a)
b
3
p-a
+ 2 (l-a)
(l_al)b 2 2}
Thus the result follows from lemma 3.
4.
X(a,b,p)
SHARP RADIUS OF CONVEXITY OF THE CLASS V k
LEMMA 6.
Vk(a,b,p),
f e
If
p
and
aP-lz p-I
f,
F’a (z)
F (o)
o,
p)__
Vk(a,b,
is in
satisfying
z+a
(i--)
2(p-a)b-ra cos
f’(a)(z+a)P-1(l’--z)
a
Fa
then the transformation
for all
e
lal
(4.1)
z e U
-i
p+l
I.
<
The proof of this lemma follows by using lemmas 5 and 2.
LEMMA 7.
zf"(z)
f’(z)
of
For
Izl
< r
and
f
V(a,b,p),
ranging over
the domain of values
is the disc with center
([2(p-a)Re{b
cos %e
-i%
r2+(p-l)
p+l]
2(p-a)Im{b
l-r 2
p(p-a)klb
and radius
cos Re
-jR
r2
l-r 2
cos% r
l-r 2
PROOF.
f(z)
For
f e V
for
lal
Whenever
z
%(a
k
p
np+
+
b,p)
< I.
a z
n
n
e
Vk (a,b
F (z)
let
z
a
p
p),
f" (z)-P (P- I)
lim
zP-2
zp-I
z/ o
n
E
A z
+ n=p+l
n
e
Vk (a
b,p)
p(p-l)a p+!
be given by Lemma 6
By direct calculation we have
p[2(p-a)b cos
f" (a)
p(l-lal2)T(a)
p(p+l)Ap+
e-i-p+l]lal2+p(p-1)
(4.2)
a
Combining (3.1) and (4.2), we obtain
lfvf"(a)
(a)
[2(p-a)b cos
a(1_ a
e
-i_p+ ]lal2+(p-l)
From (4.3) it follows that for
[zf"(z)
(z)
[2(.p-a)b cos%
<
e)
p(p-a)klbcos
(1_ a
Iz
(4.3)
12
r < I,
e-i-p+l]r2+(p-l)
<
(i_r2)
p(p-a)klblcosr
(4.4)
(i_r2)
and the proof is completed.
THEOREM 4.
r
The sharp radius of convexity of the class
2{(p-a)k[b[cos + [(p-a)2k2[bl 2 cos2
PROOF.
From (4.4), we have
4(2(I-
V
X(a,b,p)
k
is
)Re{bcos le-l }-D] -I
a
(4.5)
61
CERTAIN CLASSES OF p-VALENT FUNCTIONS
I(i+
e-)b
1+(2(1-
zf" (z)
f’
cos e
P
(z))-P(
-i
-I) r 2
p(p-a)klb Icos%
r
l-r 2
l-r 2
which implies that
zf"z)
f’ [z)
Re{l+
l-(p-a)klblcos r+(2(l--)
p
cos
Re{b
e
-i%
-I) r2
l-r 2
zf"(z)}
f’(z)
Re{l+
Therefore
f (z)
P[
> o
Izl
for
r
is given by (4.5).
o
The function
defined by
X
pzp-I
f.(z)
(l-ez)
z
r+p
e
r,
(p-)b cos%
e
-il
(4.6)
k
(l-cz)
when
k
2
-+
2
e
r-pei%
c
and
b
l_rpeikbb-
l+rpei/
shows that the bound in (4.5) is sharp.
5.
V(,b,p).
DISTORTION AND ROTATION THEOREMS FOR THE CLASS
In [ii] Lakshma used variational method to solve the extremal problems for the
class
V(e,b).
He proved the following:
LEMMA 8111],
#
Let
be a given point in
o
U
and let
H(h’(),(h"()
analytic in a neighbourhood of each point
H(Xl,X 2 Xn+
h(n)($),), h e
Then the functional
J(h’)
(or minimum) in
attains its maximum
M
h’(z)
where
M_< n,
lelm--inlel
A
(l-a)b
%(,b)
k
-iX
I1 levi
h(z) e V
If
(k_
I)A
k+
I)A
’(l-ez)2
(l-ez
cos
e
(a,b),
k
_<
larg
h’(z)
If
-
N
j-1 (1-ejz) -ej(l-a)b
M
jl J -<
I,
k
cos% e
-i
N
k
j1 J -< +I"
1,
then we have
lh’(z)
_<
k
((l-z)
lelmaxle
I)A
k
(1_ez)( + 1)A
-i
Bounds are sharp with properly selected
LEMMA i0[II].
only for a function of the form
(1-%z)J
N_< n,
LEMMA 9[ii].
where
V
8 (l-e)b cos%e
jH=1
h(n)(),)
Re H(h’(),h"($)
h(z) c
_< (l-a)k
e
and
e
Vk(,b), then for Izl
blcos arc sin iz;.
with
M
N
in (5.1).
r < I, we obtain
be
V
k
62
M.K. AOUF
proof of the following two theorems follows from Lemma 5 and the above bounds.
The
THEOREM 5.
f e V
If
l(a,b,p),
k
((l-z)
(p-a)b cos% e
we obtain
k
f’ (z)
_<
k
B
r <
lelm=axll
pzp-I
f (z)
pzp-l[h’(z)
by (5.1) with properly selected
and
with
If
6.
e
V(a,b,p),
f
(p-l)0
larg f’(z) !
k
(l_ez)( + I)B
-il
Bounds are sharp with equality for
THEOREM 6.
I)B
(3(l-ez)
I)B
(l_ez)( + I)B
where
Izl
then for
k
Izl
then for
Iblcos
V(a,b).
+ (p-a)k
HARDY CLASSES FOR THE CLASS
In order to obtain the
M
l_a
h(z)
where
is defined
I.
N
r < I, we obtain
arc sin
Izl.
V
classes for the class
k(a,b)-
we will use the
k
following lemmas.
IF
LEMMA ii[ 7].
LEMMA 12114].
e V
f3
f’
If
1
Then for
k.
H(o
e
<
I)
Izl
r
f3,(rei0 )I
larg
f e H I-
then
! k cosA arc
where, for
=I,
H
sin r.
is the
class of bounded functions
LEMMA13[15]. If
LEMMA
n
for
f(z)
1411
f(z) e
If
e V
f3
2
<
H(o
l
k
e
n=o
for all
a z
n
then
n
a
2
U
o(n
n
and
cos21(k+2)
2
if
Furthermore
(k+2)
[cos2l (k+2)-2]
Z
f(z)
and
e H
f3
then
cos2l
[fo’ (z)]CSl
I)
<
f3
f3 HN
is not of the form
-il
where
fo
is given by
k
(l-ze _ito
f’(z)
O
(m(t)
e
(+I)
2n
-it
)dm(t)}
exp {/
-log(l-ze
e(f3)>
6
H
cs2l
THEOREM 7.
U <
6
6(f3)>
and
o
such that
o
(2+6)
f3(z)
(6.1)
O
a probability measure on [o,2]), then there exists
(2+c)
(k+2)
h e V
If
and
l(a,b)
k
2
(I-a)Re{ b} cos21(k+2)
f3
e H
[cs2%
Re{ b}
and
o
h e H
(k+2) -2]
then
for
for
h’
n
H
<
).
2
cos2
(k+2)
for all
2
[(I-a)Re{ b}cos2l(k+2)-2]
CERTAIN CLASSES OF p-VALENT FUNCTIONS
cos2%
where
h’
if
Furthermore,
63
is not of the form
h’(z)
(l-a)Re{ b} (k+2)
[f3’(z)](l-a)b
e(h)
exists
h’ e H
for
l-)b cos%e
[f (Z)
is given by
O
(6.1)
then there
such that
o
2
E+
f (z)
where
(h)
and
o
-i%
and
(l-a) Re{ b}cos2% (k+2)
+
h e H
[(1-a)Re{ b}cos2%(k+2)-2]
2
cos
(1-)Re{ b}(k+2)
PROOF.
If
h e
k(R,b),
then it follows from Lemma I(4) that
[f3’(z)](l-)b
h’ (z)
f3 Vk
Then
lh,(z)i
If3(z)l
(l-a)Re{ b}
exp{-(l-a)Im{b}arg
By Lemma 11, the exponential factor is bounded.
12 and 14.
f3(z)}.
Thus the result follows from Lemmas
From Theorem 7 and Lemma 13, we obtain a growth estimate for the Taylor’s
coefficients of
COROLLARY I.
h e V
If
(a
k
h(z)
b)
z
n bn zn
+
e V
2
then
k(a,b)
and
cos2%
2
(l-a)Re{ b}(k+2)
[(l-)Re{ b}cosZ(k+2)-4]
b
Acknowledgment.
n
o
2
(n
The author is thankful to Professor Dr. S. M. Shah for reading
the manuscript and for helpful suggestions.
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