I ntnat. J. Mah. & Math. Sci.
Vol. 5 No. 2 (1982) 281-288
281
THE SECOND COEFFICIENT OF A FUNCTION
WITH ALL DERIVATIVES UNIVALENT
A. SATHAYE and S.M. SHAH
Department of Mathematics
University of Kentucky
Lexington, Kentucky 40506 U.S.A.
(Received July 7, 1981 and in revised form December 9, 1981)
ABSTRACT.
We consider the second coefficient of a class of functions, univalent
and normalized, and with all derivatives univalent in the unit disk D, and improve
on a known result.
It is also shown that this bound is in a sense best possible.
KEY WORDS AND PHRASES.
Typily real
funns
Univalent
functions of exponential type.
Pry 30C50, Secondary 30D15.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i.
funns Ee
INTRODUCT ION.
Let S be the family of functions of the form
f(z)
ak zk
+
z
k=2
which are analytic and univalent in the unit disk D:
E, then each
family of S such that if f
f(n)
(1.1)
zl
< i.
is univalent in
Let E be the sub-
D, n
0,1,2,
Set
sup
{la21
f
E}.
(1.2)
Then it is known that f must be an entire function of exponential type not exceed-
ing 2 [2,3] and [1,2]
1.5910 <
<
1.7208.
(1.3)
We construct, in Section 2, a function f belonging to E for which
a2
f"(0) > 3.18781
2
2
1.593905.
(1.4)
A. SATHAYE AND S.M. SHAH
282
E+ defined
Consider now a subfamily of E,
if
E+
f(k)(0)
E,
f
as follows:
> 0, k
1,2
}.
Set
E+}.
sup {a 2, f
It follows from an unpublished result of Ted Suffridge that
+ < 1.62031.
(1.5)
Note that the function f defined below by (2.2) belongs to
for
+
2.
MAIN RESULT.
and so a lower bound
(1.4).
is given by
In the following we shall write
{(T + )e
where T is a positive constant.
THEOREM.
E+,
-
+ (T-
)e}/2
Let
(z)
exp(-(l + z))}/2 + Az
{exp((l + z))
2
--+
+ (AT + )
Bz
3
(2.1)
where A, B, T are nonnegative constants, and let
f(z)
f(z;A,B,T)
(i)
If T
3.18781018, A
{(z)
0.03666, B
(0)}/’ (0).
(2.2)
0 then f defined by (2.2) belongs to
E and
2a
(ii)
If T
T > 3.18781.
f"(0)
2
(2.3)
3.18782 then f defined by (2.2) does not belong to E for any
choice of nonnegative constants A,B.
PROOF.
We require the following:
DEFINITION.
Then F(z) is called typically-real if
Let f(z) be regular in D.
and only if
Im f(z)
0<>Im z
0.
The following theorem is due to S.Y. Trimble.
THEOREM A [4].
a
n
are real.
Let f(z)
z
+
En=2
a z
n
n
be regular in D and suppose that all
Then a necessary and sufficient condition that f e E is that each
283
FUNCTION WITH ALL DERIVATIVES UNIVALENT
f(k)
(k
is typically real in D.
0,i,2
PROOF OF (i).
sinh
It is easily seen that (0)
> 0, @’(0)
cosh
+A>0;
and
f(z)
+ T
z
z
2
+
+
e
e- )+B
+
/
e
+
z
is entire and of exponential type n and has all positive coefficients.
writing z
x
+
Further,
iy, we have
sin
zy
+ x)
sinh (i
+ (AT + ) + 6Bx.
Y
Write
(Im @’(z))
2(x
Im z
Iz;
i"
Then
2(x)
@2
This function
T, A and B.
sin
/i
x2
#’ (z)
sinh n(l
+
6B < AT
by our choice of
Further,
is typically real.
"(z)
and since B
-< x -< i, since 0
is positive for -i
Hence
z sinh (i + x) + (AT + ) + 6Bx.
+ z) + AT +
+ 6Bz,
0
Im "(z)
This shows that
"(z)
sinh
is typically real.
y cosh (i-+ x).
(k)(z),
k
3,4
is
+ A + (AT + )x + B(4x2
I).
(2.4)
Similarly each
typically real.
Consider now
Im
)
{cosh (i + x)}
Note that
l(X)
F(x)
/i- x2
z/1 x2
sin
> 0 for 0 < x < i.
{cosh (i
x)}
l(-X)"
Write now F(x)
/i/i x 2
sinh
x2
Then
+ A- (AT + )x + B(4x2 -i). (2.5)
Write
i x2)/n i-
C(x)
cosh (i- x),
H(x)
(sinh
G(x)
A- (AT + )x,
S(x)
sinh z(l
x).
x2)
A. SATHAYE AND S.M. SHAH
284
Then since B
0,
C(x)H(x) + G(x)
F(x)
{-
F’(x)
sinh (i-
x)}H(x) + C(x)H’(x) + G’(x)
and
H(x) --x
2
i+
n=2
Now B
n
A
(AT + )
0 and for 0 < x < 0.046 <
2
A- (AT + )x > 0.
G(x)
[0.92,1] we have
Further for x
min C(x)
rain
C(1)
i
H(0.92) > 0.765
H(x)
and
G(1) > -0. 752.
mln G(x)
Thus F(x)
[0.92,1] and also for x
0 for x
E
[0,0.04].
Differentiating log H(x), we have
H’ (x)
H(x)
2
1
+ 2x
n--2n
(x)
2
1
where
(2x3)/{(n 2
e(x)
l)(n
2
1
+
x2)}
n--2
Write u(x)
(2/x) + (3x/2).
-S(x)H(x) + C(x)u(x)H(x)
F’(x)
Now 0 < g(x) < (0.27)x
u,
3
3
(x) < 0.27x and
Let x
I
< x < x
I
Tl(Xl,X2)
-T
T2(Xl,X2)
T
<
2, T I
If T
PROPOSITION 2.
so if F(x
I)
i-0, H(0)
> O, F(x
0, H(x) > 0 and
Denote by subscript 2
where x I, x belong to this interval.
2
2
< x < x
i as x
1 and so
and positive.
the value of these functions at x
Then for x
Further in this interval,
Now u(1) > 0 and so u(x) > 0, C(1)
0 and so S(x) > 0, H(x)
x3+
(AT + )
e(x)C(x)H(x)
in the interval (0.040,1).
C+, S, H’ > 0 and so H+.
C(x) > 0, S(1)
Then
2)
1
I
2
2
and by subscript i at x
u2C2H I H2(S 1 +
UlCIH 2 HI(nS2)
F’(x)
<
> 0 or T
T
2
2.
I.
Ci(0.27x23))
Let
(AT + ),
(AT + )
Hence we have
< 0 then F(x) is monotone in this interval and
> 0 then F(x) is positive.
285
FUNCTION WITH ALL DERIVATIVES UNIVALENT
COROLLARY 3.
Suppose that the quantities
{T.}l
and
{F(xi)}
are machine calcu-
lated with at least 6-digit accuracy and we have:
(i)
(ii)
-> i0 -6 or T < -i0 -6
2
-6
F(x 2) > 10
F(xI) > 10
T
1
-6,
[Xl,X2].
then F(x) is positive in
By machine calculations we have:
COROLLARY 4.
(i)
For
T
F(x2)
(ii)
0.0133178
0.0081411
For
[Xl,X2]-T
(2.7)
[0.847885,0.918943]
F(x
0.0373463,
I
F(x2)
(iii)
F(Xl)=
-0.0410121,
2
(2.6)
[0.040000,0.054844]
[Xl,X2]
I)
0.0354489
0.1001402
For
T
F(Xl)
0.8434650,
1
F(x2)
(2.8)
[0.918943,0.990000]
[Xl,X2]
0.1001402.
0.2242320
Hence F(x) is positive in the intervals (2.6), (2.7) and (2.8).
[XI,X2] say.
Xl). By the
enough to show that F(x) > 0 in [0.054844,0.847885]
< 0, T > 0. Write
2
I
F(xI) +
Theorem, F(xo)
when T
M
M(Xl,X2)
F’()(Xo Xl)’
F(Xo
for every
Xo
in
[Xl,X2].
PROPOSITION 5.
then F(x) > 0 in
COROLLARY 6.
Let T
->
F(Xl)
+
>
F(Xl)
+ TI
-T
(x 2
I
where x
I
<
Thus it is now
Xo
< x
2
We now consider
Mean-Value
(Xl,Xo).
and
Hence
T1 (Xo Xl)
(x2
Xl)
F(Xl)
M
Hence we have
1
< 0 and F(x I) > 0, F(x2) > 0.
If F(x
I)
>
M(Xl,X2)
[Xl,X2].
Suppose that
{F(xi)
at leat 6-digit accuracy and we have
and
M(Xl,X2)
are machine calculated with
286
A. SATHAYE AND S.M. SHAH
(i)
-6, F(x2)
> 10 -6
-6
M(Xl,X2) + i0
F(x) is positive in [Xl,X2].
PROOF THAT F(x) > 0 IN [Xl,X2].
(ii)
then
> 10
F(Xl)
F(xI)
>
This was carried out by running the following
simple program and repeatedly applying Corollary 6, as indicated.
(1)
Set x
(2)
Apply test of Corollary 6.
Xl,
I
X
x
2
2.
Is the hypothesis satisfied?
If not, then
go to (5).
(3)
F(x) is now positive in
If X
is reasonable.
2
> I0
x
[Xl,X2]
and x
-6
is too large.
If x
2
x
I
by (x
Replace x
2
If it is, then stop.
I.
<
2
If it is, then
is as close to X
2
as
I
by x
2
and x
2
by X
Go
2.
(2).
[xl,x2]
x
2
then continue.
Start with the next interval by replacing x
to
(5)
x
2
10-6?
Is X
2
We have proved F(x) > 0 in
stop.
(4)
[Xl,X2].
I
+
x2)/2.
Is (x
2
x
10-67
I)
This method cannot be reliably continued beyond
a i0 -6
This program terminated by x
then go to (2).
X..
reaching
2
In fact the program was run from
0.04 to 0.99 by incorporating the test of Corollary 4, as well as another mean value
test comparing
F(x2)
and T
2
(x
x
2
I)
when T
2
> 0.
The numbers presented here
were generated by the program and are not intrinsically significant.
The table for
the relevant calculations is available upon request.
0 in -i < x -< i and
This completes the proof of the statement that F(x)
hence that of (i).
(ii)
We require the following:
L4MA.
Consider a family of functions of two parameters (a,b) defined by
Suppose that there exist a
o
Xl,
x
2
g(x
I)
(2.9)
satisfying
(xi,ao,0)
f(xl)
bg(x) + h(x).
af(x)
#(x,a,b)
> 0
< 0,
i
(2.10)
1,2
f(x2) g(x2)
< 0
(2.11)
287
FUNCTION WITH ALL DERIVATIVES UNIVALENT
and
f(xl)/g(xl)
_<
f(x2)/g(x2)
or
(x2,a,b)
(2.12)
Then for all a, b -> 0 we must have either
(xl,a,b)
(2.13)
< 0
In this lemma and proof of (ii), f is different from f in (2.2).
REMARK.
PROOF.
< 0
We will show that the resulting system of in-
Assume the contrary.
Set e
equalities has no solution.
a
a
B
o
b.
Then we get from (2.10) and
our assumption,
0 <
(xi,a,b)
f(xi)
(xi,ao,0)
g(xi),
i
1,2.
(2.14)
Also by our assumption
(2.15)
From (2.15), (2.14) and (2.11) we get e
2) and (2.11) give f(x2) > 0, that is
(2.14) (with i
wise
Now we claim that
0.
# 0; for other< 0 leading to a
contradiction.
Thus
(with i
B
> 0 and from (2.14) (with i
i) and (2.11) we get
> 0.
Now (2.14)
1,2) and (2.11) yield
f(x 1)
g(xI)
in contradiction to (2.12).
PROOF OF (ii).
a
A,
h(x)
B
f(x
>
g(x
2
2)
The lemma is proved.
Let, in the lemma,
C(x)H(x)
(x,a,b)
>
b
f(x)
B,
x
+ a(l
rE)
Tx,
g(x)
i-
b(l
4x
i- 4x
2,
2)
T
3.18782
x
C(x)H(x)
Then from (2.5) and (2.9) we see that
(x)
Let x
I
(x,a,b)
0. 7033334,
0.1160000, x2
ao
(x,A,B)
0.0366100.
F(x)
Then the following are correct
up to 6 decimals at least.
(x I)
-0.0000062,
f(xI)
1
Tx
I
(x 2)
0.6302128,
-0.0000037
f(x2)
-1.2421002
A. SATHAYE AND S.M. SHAH
288
g(x
I)
f(x
I)
g
4x21
I
0.9461760,
f(x
0. 66606 30,
(I)
g (x
2
2)
g(x2)
-0.9787114
1.2691179
Hence the conditions of the lemma are satisfied.
or
(x2)
F(x2)
This means that
(x I)
F(x I) < 0
< 0 for all choices of A, B > 0 and consequently the function f
defined by (2.2) cannot be in the class E, if T
3.18782, for any choices of A -> 0
and B >0.
REMARK.
The choices of T and A, in part (1) of the theorem are not intrin-
sically significant.
and given interval
Another computer program was set up which, for any given T,A
[Xl,X2]
could quickly chec the values of the function at sev-
eral points in the interval and calculate relevant quantities in
actual
the lemma.
The
T,A,B were then found simply by trial and error by taking some value of T
and eliminating possible choices of A,B by applying the lemma.
all A,B got eliminated, then a lower T was chosen.
If for a given T
Finally, all calculations were
rechecked as before for the chosen T,A,B.
The fact that the computer used for this process
(Radio Shack TRS 80) was
interactive and totally dedicated was very vital in the success of this "trial and
error method.
ACKNOWLEDGEMENT.
The authors are grateful to Professors J.D. Buckholtz and S.Y.
Trimble for many helpful discussions, and to the referee for minor corrections and
some suggestions.
REFERENCES
i.
LACHANCE, M.
2.
SHAH, S.M. and TRIMBLE, S.Y. Univalent functions with univalent derivatives,
Bull. Amer. Math. Soc. 75 (1969), 15 3-15 7.
3.
SHAH, S.M. and TRIMBLE, S.Y. Analytic functions with univalent derivatives,
Indian J. of Math. 20 (1978), 1-35.
4.
TRIMBLE, S.Y.
Remark of functions with all derivatives univalent, Inter. J. of
Math. and Math. Sciences 3 (1980), 193-196.
An extension of close-to-convex functions with an application to
entire functions with univalent derivatives, to appear.