I nternat. J. Math. & Math. S ci.
Vol. 3 No. 3 (1980) 483-489
483
THE RADIUS OF CONVEXITY OF CERTAIN ANALYTIC FUNCTIONS II
J.S. RATTI
Department of Mathematics
University of South Florida
Tampa, Florida
33620
(Received August 7, 1979)
ABSTRACT.
2], MacGregor found the radius of convexity of the functions
In
z+a2z2+a3z3+
f(z)
f’ (z)
analytic and univalent such that
i
< i.
This
paper generalized MacGregor’s theorem, by considering another univalent function
z+b2z2+b 3z3+...
g(z)
such that
f’ (z)
IgT(z)
i
< i for
Izl
< i.
Several theorems are
proved with sharp results for the radius of convexity of the subfamilies of functions associated with the cases:
Izl
<
I, Re{g’(z)}
> a
g(z) is starlike for
Izl
< i,
g(z) is convex for
(a=0, 1/2).
KEY WORDS AND PHRASES. Univalent, analytic, starlike, convex, radius of starlikeness and radius of convexly.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i.
INTROUDCTION.
Throughout we suppose that f(z)
g(z)
30A32, 30A36, 30A42.
z
+
b2z
2
+
z+a2z2+..,
is analytic for
is analytic and univalent for
solved the following problem:
zl
< i.
zl
< and
In [4] the author
what is the radius of convexity of the family of
484
J.S. RATTI
functions f(z) which satisfy
f’
Re(g, (z))>
(z)
Izl<l
0 for
The problem was solved
g(z) is starlike for
also for each of the subfamilies associated with the cases:
Izl
Re{g’ (z)} > a (a
< i,
e(0 __<
Izl
< i) for
i
)
0,
for
Izl
g(z) is convex of order
< i,
< i.
f’(z)
In this paper we consider functions f(z) which satisfy
ii<
gF(z)
The radius of convexity of this family of functions is determined.
i for
Izl<
Also, we find
the radius of convexity for the subfamilies associated with each of the cases:
g(z) is starlike for
The case g(z)
2.
Izl
g(z) is convex for
< i,
Re{g’(z)}
e (=-
1
0,).
[2J
z has already been proved by MacGregor
[4]
i.
The function h(z) is analytic for
Izl
,zl
<
I and satisfies
1 and Re{h(z)} > a
(0 <_
< i) for
+ (2e l)z(z)
i + z(z)
where
(z) is analytic and satisfies
1
h(z)
< i,
The following lemmas will be used in the proofs of our theorems.
LEMMA
h(0)
Izl
l+(z)l_<
for
LEMMA
Izl
<
If #(z) is analytic for
2.
< 1 if and only if
Izl
<
l#(z)
I and
_<
i for
Izl
< i,
then
2
l< i i- l#.(.z.)l2
Izl
z’ (z) + #.(z) I<_
1 + z,(z)
1
’(z)
(i)
(ii)
Part (i) of Lemma 2 is well-known
[3]
I
and part (ii) follows easily, by first
applying triangular inequalities and then using part (i).
LEMMA
for
z]
If h(z)
3.
< i, then
i
+
ClZ +
i
>
Re{h(z)} --i+
This is a well-known result due to C.
THEOREM
3.
g(z)
z
+
b2z
2
+
i.
Suppose
zl
is analytic for
< 1 and
Re{h(z)}
> 0
Iz
z
Carathe’odory.
f(z)-z + a,z
2
+
is analytic and univalent for
is analytic for
zl
<
I.
zl
<
I and
RADIUS OF CONVEXITY OF ANALYTIC FUNCTIONS
If
f’ (.Z.)
-ii
Ig,(z
< i
Izl
for
Izl
< i, then f(z)maps
485
1/5 onto a convex domain.
<
The result is sharp.
therefore
h(0)
f’ (z)
g’ (z)
Let h(z)
PROOF
g’ (z)
0 and
zl
0 for
lh(z)
< I.
Izl
< 1 for
The function g(z) is univalent for
I
<
<
I
I,
Thus by Schwarz’s lemma we have
< i.
z(z)
h(z)
zl
The function h(z) is analytic for
Izl
l(z)
with
< I.
Therefore
g’(z)(l + z(z)).
f’(z)
Taking the logarithmic derivative we obtain
z’(z.). + #(z)
g"(z)
g’(z) +
f"(z)
f’ (z)
i
+ z(z)
Using lemma 2(ll) we get
Re
{zf"f,(z)(Z) + 1} _> Re{ zgWlg,(z)(z) +
Since g(z)is univalent, we have
Re
{zg"(z)
g’(z)
zl
i}-
[i1
+ i} _>
Izi (2JzJ
+
i
(3.1)
zl
4)
,,izl 2
Using this estimate in (3.1) we obtain
f"_ + I}
z(z)
(z)
Re
Th+/-s
=
> 0
for
zl
ep=ss+/-o +/-s o+/-tv +/-f
zl
< r
>
i
+
/5.
<
izl 2
S= th =ondt+/-o
is necessary and sufficient for
zl
convex domain, we conclude that f(z) maps
<
f(z) to map
zl
zf" (z)
R-e,Cz
+
< r onto a
1/5 onto a convex domain.
To
show that the estimate obtained in the theorem is sharp, we consider the function
2
z
(I + z)
f(z) such that f’ (z)
with g(z)
z)3
(1
(1
z)2
This function f(z) satisfies the hypotheses of the theorem and a short computation shows that
THEOREM
2.
zf"(z)
f’ (z)
+
Let f(z)
1
1
+ 5z
I- z
z
2
This expression vanishes at z
+ a2 z2 +...
be analytic for
< 1
and
1/5.
486
J.S. RATTI
g(z)
for
+
z
zl
b2 z2
+
< i, then
zl
f(z) maps
<
zl
If
i.
1/5 onto a convex domain.
<
Since g(z) is starlike for
PROOF:
Izl
be analytic and starlike for
f’(z)
ii
Ig(z)
< 1
The result is sharp.
implies g(z) is univalent there,
< 1
the proof of this theorem follows from that of theorem i.
THEOREM
g(z)
for
+
z
zl
b2z
2
+
< i, then
PROOF.
+0
g (z)
Suppose f(z)
3.
a2z2 +
+
z
zl
zl
zl
g’ (z) + I
The function
zg"(z)
+
i
I.
ClZ +
< 1
f’ (z)
If
Izl
< 1
is regular for
zl
> 0 for
and
11
gV (z)
< 1
The result is sharp.
I it is univalent there.
<
+ I}
Re{--g,(z)
and
< i
<
i/3 onto a convex domain.
<
Since g(z) is convex for
for
zl
is analytic and convex for
f(z) maps
zl
is analytic for
Therefore
< i and has positive
real part, therefore by lemma 3,
Re
’’(z)
{zg’(z)
+}->
l.z
i
z
Using this estimate in (3.1) we get
Re{Zf"(z)
f"(z)
+ I}
>
Izl
ii +
zl
This last expression is positive for
a convex domain.
<
Izl
Izl)
i- 3
Izl
i/3.
Izl 2
i
Thus f(z) maps
<
z
I/3 onto
To see that the estimate obtained is sharp, we consider f(z)
l+z
f’(z)
such that
I
z
(i- z)
with
2
Thus f(z) satisfies the
I- z
zf"(z)
f’ (z)
However
hypotheses of the theorem.
z
g(z)
i
+ I
+ 3z
z
1
which vanishes at
2
1/3.
z
4.
THEOREM
g(z)
z
(z)
If’gi(z)
+
b2 z2
I
<
PROOF.
+
i
convex domain.
f(z)
Suppose
+
z
is analytic and
for
lz]
< i, then
a2z2 +
is analytic for
Re g’(z) > 0
f(z)
maps
Izl
for
<
zl
(/TF-
< i.
3)/4
zl
< i and
If
onto a
The result is sharp.
Since Re
g’ (z)
> 0
for
Izl
< i, it follows from lemma i, with e
0
487
RADIUS OF CONVEXITY OF ANALYTIC FUNCTIONS
that
i
g’(z)
1
z(z)
+ z(z) where
[(z)
1.
Taking the logarithmic derivative of this expression we get
g" (z) .__2 (z (z) + (z))
2
g’(z)
i
(z)
z2
Using lemma 2 (ii) and simplifying we get
Ig"(Z)g,(z)
2[
<
Izl
l]2
2
Thus
Re{ zg"(z)
i}
g,(z) +
2
Izl 2
Izl
7!
1
> 1-
1
Using this estimate in (3.1) we get
Re{Zf"(z)f’(z)
+ I}>
2[zl
1
Izl 2
for Izl
1-
This last expression is positive
31z
izl
<(/17
3)/4.
To show that the
2
estimate obtained is sharp we consider f(z) such that f’ (z)-
g(z)
z- 2 log(l- z).
2
21zl
2
(I +
i-
-_)
with
This f(z) satisfies the hypotheses of the theorem.
However
zf"(z) + i
f ()
g(z)
z
(z)
if’gi(z)
where r
+
11
0
5.
Suppose f(z)
b2z2 +
< i for
z
I
+
<
3z- 2z
2
z
a2 z2 +
is analytic for
Izl
+
(3- / 17)/4.
This last expression vanishes at z
THEOREM
i
zl
Re{g’(z)}
< i and
i, then f(z)maps
is the smallest positive root of
zl
is analytic for
Izl
4-4r
<
13r
r
2
0
>
<
I and
1/2 for
zl
<
I. If
onto a convex domain,
2r
3
r
4
0.
The result
is sharp.
PROOF.
Since
Re{g’ (z)) > 1/2 for
zl
< i, we have by lemma
I with
1/2,
J.S. PATTI
488
Z’(z)
i
Thus
+ z(z)
g’ (z)
1
+ z(z)
From (3.1) we get
-
1} > Re{ zg"(z) + 1
Re. zf’’-z)
g’ (z)
f’(z) +
zg"(z)
i
-Re{
g,(z) +
zl
Re{
-z21 + (z)z(z) z (z)
Re{ zf"(z.)
i}
f, (z) +
Therefore,
21z[)
/1..
is positive if
( + z(z))
Izl)
(
This will be true if
Re{Llzl(1-
{(3lzl
z+(z))
lz[)z2+
+ (i-
I)
t/z@(z)]*)
,(z)}]
0,
(asterisks denote the conjugate of a complex number)
Re{Izl
(i-
Re{[(3lz
Izl21,(z)l 2)
[(31zl
Izl)z 2
-i)+ (i-
i)
/
(i-Izl)z2(h
(z)] ix + z,(z)] * }<
(z)] [I + z(h(z)]
*}
>
o,
Iz121,(z)12).
Izl(1-
By lemma 2, it is easily seen that this last inequality will be true if
31zl
i
Izl)Izl(
+ (I
--I(z)
x- Izl
2
<
Izl(l
Izll(z) l).
This inequality is equivalent to showing
r
where
Let
Izl
p(x)
+
r (0 < r < I)
r
+ 3r 2 +
r2(l
3r
2
+
and
r
2(I
+ r)x- r 2 x 2
I(z)
+ r)x
x
< i,
(0 < x < i).
2 2
r x
We see that p(x) attains its maximum value q(r) at x= l+r
2
2
2
q(r)
r + 3r +
(I + r)
consequently
2.
Since r
+ 3r 2 +
2
r
q-- (I
+ r) 2 ,< 1
positive root of the equation
holds for all r < r, where
2
2
r
2
i.
r + 3r + ,.---(i + r)
rn
is the smallest
RADIUS OF CONVEXITY OF ANALYTIC FUNCTIONS
489
This simplifies to
4
4r- 13r
2
2r
3
r
4
(3.2)
0
To show that the estimate obtained above is sharp, we let
I
g’(z)
z+b
(z)
where
+ z(z)
i
i
b
i + bz
2
where r
+ r0
l(z)
< 1
for
Izl
g’(z) > 1/2 for
< i, we have Re
is the smallest
(i- z)g’(z).
positive root of (3.2); and we select f(z) so that f’(z)
Since
0
Izl
Thus f(z)
< i.
satisfies the hypotheses of the theorem, and
zf"(z) + 1
;" (z)
Setting
z
r
0
i
+ (2b
i
and b
2
2z3
2
2
4b
5b
2)z + (2b
l)z
2
z) (i + bx)(l + 2bz + z
(I
+ r0
bz
4
we see that the numerator of the above
expression is
(i
+
r
0)
(4
4r
0
13r 2r03
r
4
0)
which vanishes.
Theorems 4 and 5 give the radius of convexity for the class of functions f(z)
associated with g(z) such that Re
g’ (z)
> s
our method seems to give only estimates for
determination of r
c
when e
r
0 and
1/2.
For
+ 0, 1/2
the radius of convexity and
is still open.
REFERENCES
i.
W. K. Hayman, Multlvalent Functions,Cambridge University Press, Cambridge,
1958.
2.
T. H. MacGregor, A Class of Univalent Functions, Proc. Am. Math. Soc. 15
(1964), 311-317.
3.
Z. Nehari,
4.
J. S. Ratti, The Radius of Convexity of Certain Analytic Functions, J.
of Pure and App. Math. 1 (1970), 30-36.
.Co.nformal
Mapping, McGraw Hill, New York, 1952.