A note on certain inequalities for univalent [627589]

A note on certain inequalities for univalent
functions
Roxana S endrut iu
Faculty of Environmental Protection, University of Oradea,
Str. B-dul Gen. Magheru, No.26, 410048 Oradea, Romania
[anonimizat]
Abstract
The concept of uniformly starlike functions and uniformly convex
functions were rst introduced in [3] by A. W. Goodman and then
studied by various authors. In this paper we use a parabolic region to
prove certain inequalities for uniformly univalent functions in the open
unit disk U.
Key words : univalent functions, S al agean operator, uniformly star-
like, uniformly convex, uniformly close-to-convex.
2000 AMS subject classications : 97U99.
1 Introduction
Denote byUthe unit disc of the complex plane:
U=fz2C:jzj<1g:
LetH(U) be the space of holomorphic functions in Uand
A=ff2H(U) :f(z) =z+a2z2+


; z2Ug:
Let
S=ff2A:f univalent in U g
be the class of holomorphic and univalent functions from the open unit disc
U.
Forf2A; n2N[f0g;letInfbe the S al agean dierential operator (see
[6]) dened as In:A!A
I0f(z) =f(z)
I1f(z) =zf0(z)




In+1f(z) =z[Inf(z)]0(z2U):
1

2 Preliminary results
Denition 2.1 [3] A function f2Sis said to be in SP(), the class of
uniformly starlike functions of order , with2[0;1], if it satises the
condition
Refzf0(z)
f(z)gjzf0(z)
f(z)1j: (1)
Replacingfbyzf0(z) we obtain
Denition 2.2 [3] A function f2Sis said to be in the subclass UCV ()
of uniformly convex functions of order , if it satises the condition
Ref1 +zf00(z)
f0(z)gjzf00(z)
f0(z)j: (2)
The concept of uniformly starlike functions and uniformly convex functions
were rst introduced in [3] by A. W. Goodman and then studied by various
authors.
We set

=fu+{v;u>p
(u1)2+v2
with
q(z) =zf0(z)
f(z)
or
q(z) = 1 +zf00(z)
f0(z)
and consider the functions which map Uonto the parabolic domain
such
thatq(z)2
:By the properties of the domain
;we have
Re (q(z))>Re (Q(z))>1 +
2; (3)
where
Q(z) = 1 +2(1)
2(log(1 +pz
1pz))2:
Furthermore, from [5] we have the following denition
Denition 2.3 [5] A function f2Sis said to be in the subclass UCC ()
of uniformly close-to-convex functions of order , if it satises the inequality
Refzf0(z)
g(z)gjzf0(z)
g(z)1j (4)
for someg(z)2SP():
2

Remark 2.1 A function h(z)is uniformly convex in Uif and only if zh0(z)
is uniformly starlike in U(see, for details, [1], [2], [5]).
In order to prove the main results we use the following lemma:
Lemma 2.1 [5] (Jack's Lemma) Let the function w(z)be (non-constant)
analytic in Uwithw(0) = 0 . Ifjw(z)jattains its maximum value on the
circlejzj=r<1at a pointz0;then
z0w0(z0) =cw(z0);
cis real and c1:
3 Main results
Theorem 3.1 Letf2A; n2N[f0g:If the dierential operator Inf
satises the following inequality:
Re (In+2f(z)
In+1f(z)1
In+1f(z)
Inf(z)1)<5
3; (5)
thenInf(z)is uniformly starlike in U.
Proof. We denew(z) by
In+1f(z)
Inf(z)1 =1
2w(z);(z2U): (6)
Thenw(z) is analytic in Uandw(0) = 0:Furthermore, by logarithmically
dierentiating (6), we nd that
In+2f(z)
In+1f(z)1 =1
2w(z) +zw0(z)
2 +w(z);(z2U);
which, in view of (5), readily yields
In+2f(z)
In+1f(z)1
In+1f(z)
Inf(z)1= 1 +zw0(z)
1
2w(z)(2 +w(z));(z2U): (7)
Suppose now that there exists a point z02Usuch that
maxjw(z)j:jzjjz0j=jw(z0)j= 1;(w(z0)6= 1);
3

and, letw(z0) =e{;(6=):Then, applying the Lemma 2.1, we have
z0w0(z0) =cw(z0); c1 (8)
From (7)-(8), we obtain
Re (In+2f(z0)
In+1f(z0)1
In+1f(z0)
Inf(z0)1) = Re (1 +z0w0(z0)
1
2w(z0)(2 +w(z0))) =
= Re (1 + 2 c1
(2 +w(z0))) = 1 + 2cRe (1
(2 +w(z0))) =
= 1 + 2cRe (1
(2 +e{)) (6=) =
= 1 + 2c1
31 +2
3=5
3
witch contradicts the hypothesis (5).
Thus, we conclude that jw(z)j<1 for allz2U; and equation (6) yields the
inequality
jIn+1f(z)
Inf(z)1j<1
2;(z2U)
which implies thatIn+1f(z)
Inf(z)lie in a circle which is centered at 1 and whose
radius is1
2which means thatIn+1f(z)
Inf(z)2
;and so
RefIn+1f(z)
Inf(z)gjIn+1f(z)
Inf(z)1j (9)
i.e.Inf(z) is uniformly starlike in U.2
Using (9), we introduce a sucient coecient bound for uniformly star-
like functions in the following theorem:
Theorem 3.2 Letf2A; n2N[f0g;and the dierential operator Inf.
If
1X
k=2(2k+ 1)jak+1j<1
thenInf(z)2SP().
Proof. Let1X
k=2(2k+ 1)jak+1j<1:
4

It is sucient to show that
jIn+1f(z)
Inf(z)(1 +)j<1 +
2:
We nd that
(10)
jIn+1f(z)
Inf(z)(1+)j=j+P1
k=2(k)ak+1zk1
1 +P1
k=2ak+1zk1j<+P1
k=2(k)jak+1j
1P1
k=2jak+1j;
2+1X
k=2(2k+ 1)jak+1j<1 +: (11)
This shows that the values of the function
(z) =In+1f(z)
Inf(z)(12)
lie in a circle which is centered at (1 + ) and whose radius is1+
2;which
means thatIn+1f(z)
Inf(z)2
. HenceInf(z)2SP().2
We determine the sucient coecient bound for uniformly convex func-
tions in the next theorem:
Theorem 3.3 Letf2A; n2N[f0g:If the dierential operator Inf
satises the following inequality:
Re [In+3f(z)In+2f(z)
In+2f(z)In+1f(z)2
In+2f(z)
In+1f(z)1]<3; (13)
thenInf(z)is uniformly convex in U.
Proof. If we dene w(z) by
In+2f(z)
In+1f(z)1 =1
2w(z);(z2U); (14)
thenw(z) satises the conditions of Jack's Lemma. Making use of the same
technique as in the proof of Theorem 3.1, we can easily get the desired proof
of Theorem 3.3. 2
Theorem 3.4 Letf2A; n2N[f0g;and the dierential operator Inf.
If
1X
k=2(k+ 1)(2k+ 1)jak+1j<1; (15)
thenInf(z)2UCV ().
5

Proof. It is sucient to show that
jIn+2f(z)
In+1f(z)1j<1 +
2:
Making use of the same technique as in the proof of Theorem 3.2, we can
prove the inequality (15). 2
The following theorems give the sucient conditions for uniformly close-
to-convex functions.
Theorem 3.5 Letf2A; n2N[f0g:If the dierential operator Inf
satises the following inequality:
Re (In+2f(z)
In+1f(z)1)<1
3; (16)
thenInf(z)is uniformly close-to-convex in U.
Proof. If we dene w(z) by
(Inf)0(z)1 =1
2w(z);(z2U); (17)
then clearly, w(z) is analytic in Uandw(0) = 0. Furthermore, by logarith-
mically dierentiating (17), we obtain
In+2f(z)
In+1f(z)1 =zw0(z)
2 +w(z);(z2U): (18)
Therefore, by using the conditions of Jack's Lemma and (18), we have
Re [In+2f(z0)
In+1f(z0)1] =cRe (w(z0)
2 +w(z0)) =
=c
3>1
3
which contradicts the hypotheses (16). Thus, we conclude that jw(z)j<1
for allz2U; and equation (17) yields the inequality
j(Inf)0(z)1j<1
2;(z2U)
which implies that ( Inf)0(z)2
, which means
Re ((Inf)0(z))j(Inf)0(z)1j
and, hence ( Inf)(z) is uniformly close-to-convex in U.2
Theorem 3.6 Letf2A; n2N[f0g;and the dierential operator Inf.
IfInf(z)satises the following inequality:
1X
k=2(k+ 1)jak+1j<1
2; (19)
thenInf(z)2UCC ().
6

References
[1] J.W.Alexander, Functions which map the interior of the unit circle upon
simple regions , Ann. of Math., 17(1915 – 1916), 12 – 22.
[2] A.W.Goodman, On the Schwarz-Cristoel transformation and p-valent
functions , Trans. Amer. Math. Soc., 68,(1950), 204-223.
[3] A.W.Goodman, On uniformly starlike function , J.Math.Anal.Appl.,
155,(1991), 364-370.
[4] I.S.Jack, Functions starlike and convex of order , J.London Math.Soc.,
3,(1971), 469-474.
[5] W.Kaplan, Close-to-convex schlich functions , Michigan Math.J.,
1,(1952), 169-185.
[6] G.S. S al agean, Subclasses of univalent functions , Complex Analysis-Fift
Romanian-Finnish Seminar, Part 1 (Bucharest, 1981), 362-372, Lecture
Notes in Math., 1013, Springer, Berlin 1983.
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