Internat. J. Math. & Math. Sci.
VOL. ii NO. 2 (1988) 251-258
251
ON A CLASS OF FUNCTIONS UNIFYING THE CLASSES OF
PAATERO, ROBERTSON AND OTHERS
S. BHARGAVA and S. NANJUNDA RAO
Department of Mathematics
University of Mysore
Manasagangothri
Mysore-570 006, India
(Received September 9, 1986 and in revised form November 26, 1986)
%(a
We study a class M
6,b,c) of analytic functions which unifies a
k
number of classes studied previously by Paatero, Robertson, Pinchuk, Moulis, Mocanu
ABSTRACT.
and others.
Thus our class includes convex and starlike functions of order 6,
,
spirallike functions of order 6 and functions for which zf’ is spirallike of order
functions of boundary rotation utmost k, a-convex functions etc. An integral
representation of Paatero and a variational principle of Robertson for the class
V
of functions of bounded boundary rotation, yield some representation theorems
k
and a variational principle for our class. A consequence of these basic theorems
is a theorem for this class
M (a,6,b,c)
which unifies some earlier results concerning
V() of Moulis and those concerning
the radii of convexity of functions in the class
the radii of starlikeness of functions in the classes U
k
of Pinchuk and
U2(8)
of
By applying an estimate of Moulis concerning functions in Vk(0)
we obtain an inequality in the class M (a,8,b,c) which will contain an estimate
k
Robertson etc.
for the Schwarzian deriative of functions in the class
Vk(8)
the estimate of Moulis for the Schwarzian of functions in
KEY WORDS AND PHRASES.
and in particular
V(0).
Convex functions, starlike functions, spirallike functions,
functions of bounded boundary rotation, a-convex functions.
1980 AMS SUBJECT CLASSIFICATION CODE. 30C45.
i.
INTRODUCTION.
Let N denote the set of all regular functions f on the unit disc E:
such that f(0)
0, f’(0)
Jf(z) Jf(z)
(a,b,c)
and let, for such a function f,
+ (1- a) [i- 1c +
a[1 + z f"(z)
b f’(z)
where a, b0 and c0 are complex numbers.
Let
l
M(a,8,b,c)
zl<
z f’(z)
c f
(
(1.1)
denote the class of
all functions f in N such that
2
0
where 0
8
IRe eijf
i,
/2
cosE
/2, z
d8
k
re
iO
(1.2)
(1-6) cos
0 & r
and k
2.
252
S. BHARGAVA AND S.N. RAO
This class unifies and generalizes various classes studied earlier by V.
Paatero [I-2], M.S. Robertson [3-4], E.J. Moulis [5-6], B. Pinchuk [7-9], K.S.
Padmanabhan and R. Parvatham [i0-II], P.N. Chichra [12], M.A. Nasr [13], G. Lakshma
Reddy [14] and P.T. Mocanu [15].
In particular, convex functions, convex functions
of order 8, starlike functions, starlike functions of order 8, X-splrallike functions,
functions
f
for which zf’ are X-spirallike, functions of bounded boundary rotations,
functions
f
for which
Re[e(l
zf’
zf"
+--) + (I-)---]
> 0
are all contained in the class
In this note, we study several properties of functions in the new class,
thereby giving a unified approach to proving and at the same time generalizing many of
-
the results of the earlier authors.
We use the following additional notations throughout our work.
3f (z) f (z) (,b,c)
f(0) defined by continuity,
with
Jf(z) -I
f"(z) + i- f’(z)
f ()
b T()
e
z
^2
(so that
Nf(z)() Nf(z)(,b,c,) f(z) Jf(z)
Nf(z)(l,l,l,) is the Schwarzian of f)
d(X,B)
d
X
X
f
is
N, let
(1.3)
z
(I 4)
complex)
secX (l-B) -I
iX
(1.5)
[(f,)/b (f/z)(l-e)/c]
(e,B, b,c)
Hf Hf
2.
e
(v:
If
d
(1.6)
SOME PRELIMINARY RESULTS.
The following Lemmas are immediate from the definitions (1.1)-(1.6) above and
will be used later.
LEMMA 2.1.
f(z)
If
(H)
(H)
3f (0)
d
z
+
A2z2 + A3z3 +
is in N, then
3f
(2.1)
u2A2’ J^’f (0)
u3A3
uA22_
(2 2
2
where
4e
u
PROOF.
2
u+uu2
-- - -
Nf(0)(v)
u
3
(A
+ I-
u
uA2),
3
2
2
I+ --c
(2.3)
u3
and
u
6a
3
+ 2 (I-)
c
3f (0)
Differentiating (1.6) logarithmically we have (2.1).
are respectively the constant term and the coefficient of
expansion of
(2.4)
0f(z)
power series for
z
and
3f(0)
in the power series
and to obtain this power series it is enough to substitute the
f
Thus (2.2) follows.
in (1.3).
Substituting (2.2) in (1.4) we
have (2.3).
LEMMA 2.2.
If
f(z)
z
+
A2z2 + A3z3 +
and
g(z)
z
+
Az 2 + Az 3 +
are in N, then
HfX(,B,b,c)
H
A’
g
(a’,B’,b’,c’)
(2.5)
if and only if
d
3f(a,b,c)
d’3 g (e’,b,’c’),
d’
d’(X’,8’).
(2.6)
CLASS OF ANALYTIC FUNCTIONS
and
f
Further, if
satisfy (2.5) or (2.6), then
g
see %(e
il
Re
8 cos %)
Jf
v
I-8
v’
and
6’
and
3
if’
8’ cos
J
$
(2.7)
l-B’
(2.8)
g
(A
uA)
d(u3A3
fiA)
3
d’u
3
’/d’,
/d
are complex numbers satisfying
du
6
%’(e
d’N (a’,b’,c’,’)
du2A 2 d’u2A2
where
see
Re
dNf(a,b,c,)
where
253
d’(u3A 3
u
A
2
(2.9)
fi’A ,2
(A
(2.10)
2
are complex numbers satisfying
a’u
Here u’, u
2’
PROOF.
and u
are as given by (2.4) with a,b,c replaced by a’,b’,c’ respectively.
3’
Equivalence of (2.5) and (2.6) are immediate from (2.1).
parts in (2.8) and using (1.3) we have (2.7).
and using (1.4).
On taking real
We get (2.8) on differentiating (2.6)
Again from (2.6) and (2.2) we have (2.9).
Finally (2.10) is
obtained on using (2.3) in (2.8).
LEMMA 2.3.
If
H
z+___a
where
f
and
g
are in
%’
(’ 8’ b’,c’)
g
al
N
and if
(I +
Constant
z) -2
(2.11)
Hf() (a,8,b,c)
< i, then
l+az
d
2 + d(l-lal 2) 3
l+z
+z 2
g
(2.12)
$
la12) 2 Nf()(,b,c,) d’Ng(z)(’,b’ ,CW,
(I+ az)4_
d(l-
+
2 ad(l
_
2 /d)
2
(I-
a(l + az)
/d
where
PROOF.
’/d’.
lal 2) a]f ()
+
a
d
az
Logarithmic differentiation of (2.11) yields (2.12).
(2 13)
Differentiating
(2.12) and subtracting from the resulting equation ’/d’ times the square of (2.12)
and simplifying we get (2.13).
Let B (k
2) denote the set of all real valued functions m(t) of bounded
k
variation on
[0,2] such that
2
2
f
dm(t)
2
and
0
f
Idm(t)
(2.14)
k.
0
The following is a result of Paatero [I] reformulated in terms of our notations.
LEMMA 2.4.
exists
A function g
m(t) in B k such that
is in
(i,0,i,I)
(=V k)
if and only if there
254
S. BHARGAVA AND S.N. RAO
H(l,O,l,l)
g
SOME BASIC RESULTS IN
log (l-ze -it) dm(t)
f
exp
(2.15)
0
X(a
b c)
k
The following representation theorem is a simple consequence of the definition of
3.
the class
M
X(
Bb c)
k
THEOREM 3.1.
Let (%,,B,b,c), (%’,’,8’,b’,c’) and
and if
g
g
f
If
be given.
is in
X’ (a’,’,b’,c’)
Hf(,B,b,c)
g
(e’,B’,b’,c’).
is in
PROOF.
k
is defined by
H
then
M
The hypothesis implies (2.7).
(2.7) and integrating with respect to
Taking absolute values on both sides of
between 0 and 2, we have the result
8
immediately.
The above theorem contains as special cases (i) Theorem 3 in [5] concerning the
V
(1,O,l,1), (ii) Theorem 5 in [lI] concerning the ciass Vk(g),
k
(iii) Lemmas 2, 3 and 4 in [6] concerning the class
and (iv) Lemmas 2,3,4,5,6
class
V(B,b)
and 9 in [14] concerning the classes
COROLLARY 3. i.
If
f
X(a
M
k
is in
,Id]f(0)
with equality for
f
_-<
V(13)
U(B).
and
B,b c)
then
(3.1)
k
satisfying
k
Hf%(a,B,b,c) (1-Z)k-+i
(l+z)
PROOF.
holds with
(I,0,I,I)
for which (2.6)
V
By Theorem 3.1, there exists a g in
k
%’
I. (3.1) now follows from (2.2) and (2.9) and on
b’
B’ 0,
’
IA2’I
using the Pick’s estimate [16] namely
side of (3.2) and
Theorem 3.1,
f
(3.2)
2
f
g(O)
0, then
k/2.
V
is in
(3.2) is in
by
given
g
k
g’
Now, if
(I,0,I,I)
<(,B,b,c).
is given by the right
[6] and therefore, by
That (3.1) is equality for this
follows easily.
The following representation theorem is a direct consequence of
Paatero’s Lemma
stated in Lemma 2.4 above.
THEOREM 3.2.
If
f
K(a,B,b,c)
is in
that
2
HfX(
m(t)
Conversely, given
PROOF.
If
f
f
exp
log(l-ze
-it
m(t)
in
Bk
such
(3 3)
dm(t)
0
B
k
and if
(,,b,c)
H(I
g
Then, by Theorem 3.1, g
B,b c)
in
is in
then there exists
0,I i)
is in
f
let us define
X
Hf(a,B,b,c).
(l,0,l,l).
when read with (3.4) yields (3.3).
is defined by
g
(3.3), then
f
is in
by
(3.4)
Hence by Lemma 2.4 we get (2.15) which
CLASS OF ANALYTIC FUNCTIONS
g
satisfy (3.4) for
f
implies that
B
m(t) in
Conversely, given
and
g
V
as in (3.3), we have by Lemma 2.4 that
f
and
k
in
255
By Theorem 3.1, (3.4)
given by (2.15).
k
(a,B,b,c).
is in
The above Theorem contains various integral representation theorems obtained
and the integral representation in Corollary 4 in
previously, for example, Theorem
[11], Theorem
M(a)
class
and 5 in [14].
[6], Theorems
in
Mo(a 0,1,1)
2
[15]
f’
(f/z)
is in the Mocanu
f
Further, if
then we have the representation
2
I-
log(l-ze -it) din(t) for
f
exp-
m(t) in
B
0
’
THEOREM 3.3.
(l,l,bl,Cl)
Hf
PROOF.
1,2,3
j
and
fj
Then
be given.
k
in
Mj
fl
is in
(aj,Bj,bj,cj)
and
(a
I’
Hf%2 (a2’ B2’b2’c2)
(k+2)/4
2
g ,b c
l’
3
Hf 3 (3’ B 3
(k-2)/4
b
3’ c 3
The Theorem follows immediately from Theorem 3.2 on using the representa-
tion (Brannan
k
(Xj,aj,gj,bj,cj),
2,3 such that
X,
B
Let
if and only if there are functions
2.
ml(t)
[17]) that
and
((k+2)/4)ml(t)
m(t)
m2(t)
((k-2)/4)m2(t)
where
m(t)
is in
B 2.
are in
Special cases of the above theorem are Theorem 4 in [5] and its corollary,
Corollary
in [Ii], Theorem 2 in
[6] and Theorem 2 in [14].
Theorem 6 in [14] is
also a special case of the above theorem on using the fact that the functions
and
zitS(z)
are together starlike of order
COROLLARY 3.2.
If
f
is in
(l-r)(k-2)/2
(k+2)/2
(1+r)
(a,,b,c),
iHf(
larg Hf(a,B,b
PROOF.
B
,b,c)
if
k sin
is real.
then
(l+r)(k-2)/2
(l-r)
c)
t
b,c)
(3.5)
(k+2)/2
-I r.
Theorem (3.3) gives that there are convex functions
Hf%(a
S(z)
(3.6)
g
and
h
such that
(g,)(k+2)/4
(h’)(k-2)/4
Using in this, well known distorsion and rotation theorems concerning convex functions
namely
(i+r)-2
larg
we immediately have
l’(z)l
’(z)
2
_-< (l_r) -2
sin-lr,
(3.5) and (3.6).
One can easily check that (3.5) becomes equality on taking the function defined by
(3.2) and then putting z
+r and z
-r respectively.
Inequalities (3.5) contain as special cases some of the known distorsion and
rotation theorems.
For example those found in [8], [9], [ii], and [14].
256
4.
S. BHARGAVA AND S.N. RAO
A VARIATIONAL FORMULA AND TWO MAIN INEQUALITIES.
(,B,b,c).
F(z)
Then
F(O)
is in M
k
H%F
0,
(z+a)/(l+z),
(z)
and
Izl
I.
<
Let
f
be in
defined by
(z)
Hf() (ct, B,b,c)
(a’ ,8 ,b ,c ’)
(4.1)
Hf(a)(a,B,b,c)
(l+az)
(a’,8’,b’,c’).
G(z)
g
By Theorem 3.1, there exists
PROOF.
Let
lal
Let
THEOREM 4.1.
<(I,0,I,I)
in
V
k
given by (3.4).
be defined by
G(0)
g((z))
G(z)
0,
(l-lal 2)
g(a)
g’(a)
(4.2)
Then by a variational principle of Robertson [4], G(z)
F(z)
again, there exists
V
is in
k.
By Theorem 3.1
(a’,g’,b’,c’) such that
in
,
,,
HF,z, (a’ ,8’ ,b’ ,c’)
HG,z.
G’. (z).
(1,0,1,1)
Using (4.2) and then (3.4) in this gives (4.1).
As special cases, the above Theorem contains Lemma 3 of [II], Theorem 6 of [5],
Lemma 5 of [6] and Lemma 7 of [14].
The following Theorem contains as special cases many earlier results concerning
radii of convexity and those concerning radii of starlikeness.
THEOREM 4.2.
f
If
Mk(=,8,b,c), then
is in
zf(z)- Izlmd-ll
[(I- Izl 2)
2
k
[zd-ll
(4.3)
and for any nonzero complex number
Re
when
r
Izl
a
2
>
(4.4)
0
2
air + a2r
kld-ll,
a
-I)
2 Re(d
2
a
-I
and
/a
4a
2a
IR=
k
PROOF.
r
< R, where
Q(r)
with
q(r)
Jf(z)(a’b/’c/)
2
2 Re(d
#
(4.6)
-I
2 Re(d
a
-I
2
We get (4.3) on choosing
using (3.1) and finally changing
(4.5)
g
to
Jf(z) (a,b/,c/)
-1
1.
(2.11) putting z
Now, we have from (1.3)
to satisfy
z.
+
zf(z)(a,b,c),
0 in (2.12),
# 0
and therefore,
Re
Jf(z)(a’b/’c/)
+ Re
zf(z)(a,b,c)
which yields the first part of (4.4) on using (4.3).
The second part of (4.4) follows
from the standard result for positivity of a quadratic form in r.
Lastly, it is easy
CLASS OF ANALYTIC FUNCTIONS
to check that the property
R
2 Re(d
-I)
257
is equivalent to
+
a
kld-ll
a
2
which is true.
REMARKS 4.2.
(i) The sharp radius of convexity of
f
V2(81
in
given by
Theorem 2 of [12] is a special case of (4.6).
The radius of convexity of
Vk(8,b)
I,
(l,,b,l)
given by (4.6) with
in Theorem 3 of [14].
(4.5) and
a
21d-ll
2
(r)
In fact, Q(r)
+ I, the
a ’r
2
alr
radius of positivity of
2
-al + /a +
4 a
f
in
is better than the one found
b
2
where
(r)
is as in
a
being
2
2a
which gives the result of Theorem 3 of
Q(r)
radius of positlvity of
convexity of
f
b
Vk(B)
in
given by (3.2) and
f
of starlikeness of
Putting
f
b.
-I
d
(O,,l,c)
R
z
is positive real.
provided
Uk(,c)
in
0
is given by
k
(=
b
O, b
2,
in (4.6).
c
O,
c.
(iv) We can
M(u 0,I,I)
2
(vl Inequality (4.3) contains
get the radius of "s-convexity" of functions in the Mocanu class
on putting
(iii) The radius
(4.6) with
further, we get an improvement over Theorem 7 of [14].
c
Clearly the
(ii) The Sharp radius of
found in Theorem 3 of [II] is given by (4.6) with
Jf(z)(,b/,c/)
Re
for
I,
(rl.
In fact, more generally, it is easy to see in (4.4) that
0.
and
[14] on putting
is bigger than that of
M()
I) Theorem 3 of [6].
In proving the inequality (4.7) below, we use an estimate of Moulls [5] for
,- 2/3,
I3-QA.I
(I,0,I,I)
Q >
V
k
more general class
f(z)
related to functions
z
+
A2z2 + A3z3 +
in the class
and thereby obtain a corresponding estimate for functions in the
(,8,b,c).
The inequality (4.7) is then applied to obtain the
estimate (4.9) which contains as special cases Moulis’s estimate [5] for the
Schwarzianderivative of functions in
LEMMA 4.1.
If
f
is in
INf(0)(,b,c,)
J(%,k,)
where
(Theorem 4 of [II]).
Vk(8)
(,8,b,c)
then
2
(,k,(l+(l-811,
6(1-81
(>01
(4.7)
is (in slightly different notation) the J-function of Moulis given
by
(I/3)cos %
{(3-21(k2cos%)/4
+ (k/21
](%,k,@)
’
have
cos}, k> 4/(3-21
(4.81
(I/3)cos % {(k-l) cos %+ (k/2)
PROOF.
By Theorem 3.1 we can find
b’
’
I,
Isin% I-
% and
8’
INf(o)(,b,c,)
0.
g
in
Isinl
}, k4/(3Q-2)
(I,0,I,I)
such that (2.5) holds with
Thus by (2.8) of Lemma 2.2 and 2.3 of Lemma 2.1, we
(I
6(1
)
8)
INg(o)(l,l,l,(l
liA3, ,A2’21
’
8))I
(213)(I+(I-8))
258
S. BHARGAVA AND S.N. RAO
[5] namely,
which gives (4.7) on using Moulis’s result
IA
where
J
fi’A
’21 (,k fi’)
fi’ > 2/3
2
is as in (4.8).
THEOREM 4.3.
,INf(z)(a,b,c,)l
0
where
PROOF.
If
f
<= 6(I-B)
is in
M(a,B,b,c),
then
(%,k,(2/3)(I+(I-B))),,
Iz12)
(I-
J
and
+
2[z/dll 1-2/dl
(4.9)
2
is as given by (4.8) above.
From (4.3) we easily have
I(I
Using this in (2.13) with
a21
la12) aJf(a)
z
0, %’
using Lemma 4.1 and finally replacing
d
%,
B’
a
by
lad-ll
B, a’
z
(k +
a, b’
we have
b,
c’
c
and then
(4.9).
REFERENCES
I.
PAATERO, V.
2.
PAATERO, V. Uber Gebiete von beschrankter Randdrehung, Ann. Acad. Sci. Fenn.
A37, (1933), 1-20.
3.
ROBERTSON, M.S.
374-408.
4.
ROBERTSON, M.S. Coefficients of Functions with Bounded Boundary Rotation, Canad.
J. Math. 21 (1969), 1477-1482.
5.
MOULIS, E.J.
6.
MOULIS, E.J. Generalization of the Robertson Functions, Pacific J. Maths. 81
(1979), 169-174.
PINCHUK, B. Functions with Bounded Boundary Rotation, Israel J. Math. I0 (1971),
7-16.
7.
Uber die konforme Abbildungen von Gebieten, deren Rander von
beschrankter Drehung Sind, Ann. Acad. Sci. Fenn. A33, (1931), 1-77.
On the Theory of Univalent Functions, Ann of Math.
3__7 (1936),
A Generalization of Univalent Functions with Bounded Boundary
Rotation, Trans. Amer. Math. Soc. 174 (1972), 369-381.
8.
PINCHUK, B. On Starlike and Convex Functions of Order a, Duke Math. J. 35 (1968),
721-734.
9.
PINCHUK, B.
I0.
Functions of Bounded Boundary Rotation, Trans Amer. Math. Soc.
(1969), 107-113.
PADMANABHAN, K.S. and PARVATHAM, R.
Indian J. Pure
11.
12.
Appl. Math.
13__8
On Functions with Bounded Boundary Rotation,
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