Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 205845, 19 pages
doi:10.1155/2011/205845
Research Article
Differential Subordinations of
Arithmetic and Geometric Means of Some
Functionals Related to a Sector
A. Lecko1 and M. Lecko2
1
Department of Analysis and Differential Equations, University of Warmia and Mazury in Olsztyn,
Street Zolnierska 14, 10-561 Olsztyn, Poland
2
Department of Mathematics, Rzeszów University of Technology, Street W. Pola 2,
35-959 Rzeszów, Poland
Correspondence should be addressed to A. Lecko, alecko@matman.uwm.edu.pl
Received 30 December 2010; Accepted 30 March 2011
Academic Editor: Hans Keiding
Copyright q 2011 A. Lecko and M. Lecko. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Some general theorems on differential subordinations of some functionals connected with
arithmetic and geometric means related to a sector are proved. These results unify a number of
well known results concerning inclusion relation between the classes of analytic functions built
with using arithmetic and geometric means.
1. Introduction
For r > 0 let r {z ∈ : |z| < r}. Let 1 .
Let the functions f and F be analytic in the unit disc . A function f is called
subordinate to F, written f ≺ F, if F is univalent in , f0 F0 and f ⊂ F .
Let D be a domain in 2 and ψ : 2 ⊃ D → be an analytic function, and let p be
and h be a function analytic and
a function analytic in with pz, zp z ∈ D, z ∈
univalent in . The function p is said to satisfy the first-order differential subordination if
ψ pz, zp z ≺ hz,
ψ p0, 0 h0,
z∈ .
1.1
The general theory of the differential subordinations has been studied intensively by
many authors. A survey of this theory can by found in the monograph by Miller and Mocanu
1.
2
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For β ∈ 0, 2 let
hβ z 1z
1−z
β
,
z∈ .
1.2
onto the sector of the angle βπ symmetrical with
It is clear that hβ maps univalently
respect to the real axis with the vertex at the origin.
In this paper we are interested in the following problem referring to 1.1 to find the
constant ck n, γ, α, β so that to the following relation is true:
zp z
pz 1 α k
p z
γ
≺ hck n,α,γ,β z, z ∈
⇒ p ≺ hβ ,
1.3
with suitable assumptions on function p and constants n, α, γ, β. For selected parameters
n, γ, α, β the theorems presented here reduce to the well-known theorems proved by
various authors. Particularly, results of this type can be applied to examine inclusion relation
between subclasses of analytic functions defined with using arithmetic or geometric means
of some functionals, for example, the class of α-convex functions or γ -starlike functions.
The lemma below that slightly generalizes a lemma proved by Miller and Mocanu 2
will be required in our investigation.
Lemma 1.1 see 2. Let q :
q0 1. Let
→ be a function analytic and univalent on , injective on ∂ and
pz 1 ∞
ck zk ,
z∈ ,
1.4
kn
be analytic in , p /
≡ 1. Suppose that there exists a point z0 ∈
such that pz0 ∈ ∂q and
p{z ∈ : |z| < |z0 |} ⊂ q .
1.5
If ξ0 q−1 pz0 and q ξ0 exists, then there exists an m ≥ n for which
z0 p z0 mξ0 q ξ0 .
1.6
2. Main Results
In the first theorem which follows directly from Theorem 2.2 3 we prove that ck n, α, γ, β ≥
β. Let us start with the following definition.
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Definition 2.1 see 3. Let γ ∈ 0, 1 and Φ be a function analytic in domain D ⊂ . By Hγ,
≡ 1 and p ⊂ D
Φ will be denoted the class of functions p analytic in with p0 1, p /
such that the function
γ
zp z
Φpz ,
Qz pz 1 pz
Q0 1, z ∈ ,
2.1
is well defined in .
Theorem 2.2 see 3. Let γ ∈ 0, 1, h a convex function such that 0 ∈ h , h0 1, Φ a
function analytic in a domain D ⊂ such that h ⊂ D, and Re Φhz > 0 for z ∈ . If
p ∈ Hγ, Φ and
γ
zp z pz 1 Φ pz
≺ hz,
pz
z∈ ,
2.2
then
p ≺ h.
2.3
Definition 2.3. Let k, n ∈ , α ≥ 0 and γ ∈ 0, 1. By Hk n, α, γ will be denoted the class of
functions p analytic in of the form 1.4 such that the function
zp z
Qz pz 1 α k
p z
γ
,
Q0 1, z ∈ ,
2.4
w ∈ \ {0},
2.5
is well defined in .
Remark 2.4. 1 Setting
Φw Φk,α w α
,
wk−1
we see that
Hk 1, α, γ H γ, Φk,α .
2.6
2 For each k, n, α, γ as in Definition 2.3 the class Hk n, α, γ is nonempty. To see
this take
pz 1 cn zn ,
z∈ ,
2.7
for sufficiently small cn ∈ .
3 Clearly, for γ 0 the class Hk n, α, 0 contains all analytic functions p of the form
1.4.
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4 Let k γ 1. Then
Qz pz αzp z,
2.8
z∈ .
Therefore the class H1 n, α, 1 contains all analytic functions p of the form 1.4.
5 Let p be analytic function in of the form 1.4. Suppose that pz0 0 for
some z0 ∈ . Then
pz z − z0 m p1 z,
with p1 z0 / 0 for z ∈ . Then we have
where m ≥ 1 and p1 is analytic function in
z
2.9
z∈ ,
mz − z0 m−1 p1 z z − z0 m p1 z
p z
z
k
pk z
z − z0 m p1 z
mp1 z z − z0 p1 z
z − z0 mk−11 p1k z
2.10
.
Hence we see that for k 1 and γ ∈ 0, 1 or for k ≥ 2 and γ ∈ 0, 1 the function
zp z
Qz pz 1 α k
p z
γ
,
z∈ ,
2.11
has a pole at z0 . Therefore for such k and γ we see that every p ∈ Hk n, α, γ is nonvanishing in D.
Theorem 2.5. Let k ∈ , α ≥ 0, γ ∈ 0, 1 and β ∈ 0, 1 be such that k − 1β ≤ 1. If p ∈
Hk 1, α, γ and
zp z
pz 1 α k
p z
γ
≺ hβ z,
2.12
z∈ ,
then
2.13
p ≺ hβ .
Proof. The case α 0 is evident so we assume that α > 0. For k ∈ N and α > 0 let Φ Φk,α be defined by 2.5. For β ∈ 0, 1 the function hβ is convex with 0 ∈ ∂h . Since k −
1β ≤ 1, we have
Re Φ hβ z Re Φ
1z
1−z
β α Re
1−z
1z
k−1β Applying Theorem 2.2 with hβ instead of h we get the assertion.
> 0,
z∈ .
2.14
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5
Now we prove two theorems were we improve the result of Theorem 2.5. The problem
1.3 will be divided into two cases: k 1 and k > 1.
First we consider the case k 1. The theorem below was proved in 4. To be selfcontained we include its proof.
Theorem 2.6. Fix n ∈ , α ≥ 0 and γ ∈ 0, 1. Let β ∈ 0, β1 n, α, γ , where β β1 n, α, γ is
the solution of the equation
β c1 n, α, γ, β 4 − γ,
2.15
2γ
arctan nαβ .
c1 n, α, γ, β β π
2.16
with
If p ∈ H1 n, α, γ and
zp z γ
≺ hc1 n,α,γ,β z,
pz 1 α
pz
z∈ ,
2.17
then
p ≺ hβ .
2.18
Proof. 1 Assume that α > 0 and γ ∈ 0, 1 since the cases γ 0 or α 0 are evident.
Suppose, on the contrary, that p is not subordinate to hβ . Then, by the minimum
principle for harmonic mappings there exists r0 ∈ 0, 1 such that
r0 p
⊂ hβ ,
2.19
and one of the following cases hold:
max Arg pz : z ∈
r0
π
max Arg pz : |z| r0 β ,
2
2.20
π
min Arg pz : |z| r0 −β ,
2
2.21
pz0 0,
2.22
or
min Arg pz : z ∈
r0
or
for some z0 ∈ ∂
r0 .
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2 Assume that 2.20 holds. Then there exists z0 ∈ ∂
r0
such that
π
Arg pz0 β .
2
2.23
Let ξ0 h−1
pz0 . Thus
β
pz0 hβ ξ0 1 ξ0
1 − ξ0
β
0.
/
2.24
± 1 and
Therefore ξ0 /
1 ξ0
xi
1 − ξ0
2.25
xi − 1
.
xi 1
2.26
for x > 0, that is,
ξ0 Since ξ0 /
± 1, so hβ ξ0 exists. Hence and by Lemma 1.1 there exists an m ≥ n for which
z0 p z0 mξ0 hβ ξ0 .
2.27
3 Consequently,
ξ0 hβ ξ0 β
z0 p z0 γ
hβ ξ0 1 mα
pz0 1 α
pz0 hβ ξ0 xiβ 1 mαβ 1 x
2x
2
2.28
γ
i
.
In view of the fact that x > 0 let us take
mαβ 1 x2
π
i ∈ 0,
.
arg 1 2x
2
2.29
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Hence and from 2.28 we have
γ mαβ 1 x2
z0 p z0 γ
β
i
arg xi 1 arg pz0 1 α
pz0 2x
mαβ 1 x2
π
i
β γ arg 1 2
2x
mαβ 1 x2
π
β γ arctan
.
2
2x
2.30
By the above and by the fact that m ≥ n we have
nαβ 1 x2
z0 p z0 γ
π
arg pz0 1 α
≥ β γ arctan
pz0 2
2x
2.31
π
π
≥ β γ arctan nαβ c1 n, α, γ, β .
2
2
On the other hand, 2.30 yields
mαβ 1 x2
π
z0 p z0 γ
π
arg pz0 1 α
β γ arctan
i ≤ βγ
.
pz0 2
2x
2
2.32
Finally, the above and 2.31 lead to
π
π
π
z0 p z0 γ
c1 n, α, γ, β
≤ arg pz0 1 α
≤ 2π − c1 n, α, γ, β ,
≤ βγ
2
pz0 2
2
2.33
for all β ∈ 0, β1 n, α, γ .
Thus we arrive at a contradiction with 2.17 so p ≺ hβ .
4 When 2.21 holds, we see that x < 0 in 2.25. Next we finish the proof by similar
argumentations like in the above.
5 Assume now that 2.22 holds. In view of Remark 2.4 this is possible only when k γ 1.
a For β < 1 the boundary ∂hβ has the corner at 0 of the angle βπ < π. Since p∂
is an analytic curve, in view of 2.19 the case pz0 0 does not hold for β < 1.
r0 b Let now β ≥ 1.
Assume that p z0 /
0. Since z0 p z0 is an outer normal to the curve p∂
by 2.19 we see that
π 3
π
π π π
3−β
π − β ≤ arg z0 p z0 ≤ β β 1 .
2
2
2
2
2
2
r0 at pz0 ,
2.34
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Hence taking into account that
3 − β ≤ c1 n, α, 1, β ,
β 1 ≤ 4 − c1 n, α, 1, β ,
pz0 αz0 p z0 αz0 p z0 ,
2.35
we deduce that
π
π
π
≤ arg pz0 αz0 p z0 ≤ β γ
≤ 2π − c1 n, α, 1, β ,
c1 n, α, 1, β
2
2
2
2.36
for all β ∈ 0, β1 n, α, γ . In this way we arrive at a contradiction with 2.17 so p ≺ hβ .
If p z0 0, then
pz0 αz0 p z0 0 ∈
/ hβ ,
2.37
and once again we contradict 2.17.
Special Cases
1 The case n 1, α 1 was proved in 5.
2 The case γ 1 was proved in 6.
Corollary 2.7 see 6. Let n ∈ , α ≥ 0. Let β ∈ 0, β1 n, α, 1, where β β1 n, α, 1 is the
solution of the equation
with
If p is analytic function in
β c1 n, α, 1, β 3,
2.38
2
c1 n, α, 1, β β arctan nαβ .
π
2.39
of the form 1.4 and
pz αzp z ≺ hc1 n,α,1,β z,
then
z∈ ,
2.40
2.41
p ≺ hβ .
3 The case γ 1, α 1 was remarked 7.
4 The case γ 1, α 1, n 1 was proved in detail in 8.
Now we consider the problem 1.3 for k ≥ 2.
Theorem 2.8. Let k ≥ 2, n ∈ , α ≥ 0, γ ∈ 0, 1 and let β ∈ 0, 1/k − 1. If p ∈
Hk n, α, γ and
zp z
pz 1 α k
p z
γ
≺ hck n,α,γ,β z,
z∈ ,
2.42
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then
2.43
p ≺ hβ ,
where
ck n, α, γ, β
nαβ cos k − 1βπ/2
2γ
× arctan β
1k−1β/2 1−k−1β/2
.
π
1 k − 1β
1 − k − 1β
nαβ sin k − 1βπ/2
2.44
Proof. 1 We repeat argumentation from Parts 1 and 2 of the proof of Theorem 2.6.
2 We have
z0 p z0 pz0 1 α k
p z0 ⎡
γ
hβ ξ0 ⎣1 mα
xi
β
xi
β
ξ0 hβ ξ0 hkβ ξ0 ⎤β
⎦
γ
mαβ 1 x2
i
1
2x
xik−1β
1
mαβ 1 x2
2x1k−1β
2.45
γ
1−k−1β
i
.
Since x > 0 and 1 − k − 1β ≥ 0, we can take
arg 1 mαβ 1 x2
2x1k−1β
1−k−1β
i
π
.
∈ 0,
2
2.46
Hence
γ z0 p z0 arg pz0 1 α k
p z0 γ mαβ 1 x2 1−k−1β
β
arg xi 1 i
2x1k−1β
mαβ 1 x2
mαβ 1 x2
k − 1βπ
k − 1βπ
π
β γ arg 1 sin
cos
i
2
2
2
2x1k−1β
2x1k−1β
mαβ 1 x2 /2x1k−1β cos k − 1βπ/2
π
β γ arctan
.
2
1 mαβ1 x2 /2x1k−1β sin k − 1βπ/2
2.47
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Thus, from 2.46 and by the fact that m ≥ n we obtain
z0 p z0 arg pz0 1 α k
p z0 γ nαβax cos k − 1βπ/2
π
≥ β γ arctan
,
2
1 nαβax sin k − 1βπ/2
2.48
where
ax 1 x2
,
2x1k−1β
x/
0.
2.49
We have
a x 1 − k − 1β x2 − 1 k − 1β
2x2k−1β
,
x/
0.
2.50
3 Assume now that k − 1β < 1. Observe that the function a attains its minimum at
the point
x0 1 k − 1β
.
1 − k − 1β
2.51
Moreover
1
ax0 1k−1β/2 1−k−1β/2 .
1 k − 1β
1 − k − 1β
2.52
Hence, and from 2.48, we have
γ z0 p z0 arg pz0 1 α k
p z0 nαβax0 cos k − 1βπ/2
π
π
≥ β γ arctan
β
2
2
1 nαβax0 sin k − 1βπ/2
nαβ cos k − 1βπ/2
γ arctan 1k−1β/2 1−k−1β/2
nαβ sin k − 1βπ/2
1 k − 1β
1 − k − 1β
π
ck n, α, γ, β .
2
2.53
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On the other hand, using the fact that 0 ≤ k − 1βπ/2 < π/2, from 2.47 we obtain
γ z0 p z0 arg pz0 1 α k
p z0 2.54
mαβ 1 x2 /2x1k−1β cos k − 1βπ/2
π
π
.
≤ βγ
β γ arctan
2
2
1 mαβ1 x2 /2x1k−1β sin k − 1βπ/2
Finally, the above and 2.53 yield
γ π
π
π
z0 p z0 ck n, α, γ, β
≤ βγ
≤ arg pz0 1 α k
≤ 2π − ck n, α, γ, β ,
2
2
2
p z0 2.55
for all β ∈ 0, 1/k − 1.
Thus we arrive at a contradiction with 2.17 so p ≺ hβ .
4 For k − 1β 1 we have ck n, α, γ, β β.
This ends the proof of the theorem for the case x > 0.
5 When 2.21 holds, we see that x < 0 in 2.25. Next we finish the proof by similar
argumentations like in the above.
6 Since β ≤ 1/k − 1 < 1, arguing as in Part 5a of the proof of Theorem 2.6 we see
that the case 2.22 does not hold.
Special Cases
1 β 1/k − 1.
Then ck n, α, γ, β β.
Corollary 2.9. Let k ≥ 2, n ∈ , α ≥ 0 and γ ∈ 0, 1. If p ∈ Hk n, α, γ and
zp z
pz 1 α k
p z
γ
≺ h1/k−1 z ,
z∈ ,
2.56
then
p ≺ h1/k−1 .
2.57
2 k 2, β 1.
Corollary 2.10. Let n ∈ , α ≥ 0 and γ ∈ 0, 1. If p ∈ H2 n, α, γ and
zp z γ
> 0,
Re pz 1 α 2
p z
z∈ ,
2.58
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then
Re pz > 0,
2.59
z∈ .
3 The case n 1, α 1 was proved in 9.
4 γ 1.
Corollary 2.11. Let k ≥ 2, n ∈ , α ≥ 0 and β ∈ 0, 1/k − 1. If p is a function analytic in
of the form 1.4 nonvanishing in
and
pz α
zp z
≺ hck n,α,1,β z,
pk−1 z
2.60
z∈ ,
then
p ≺ hβ ,
2.61
where ck n, α, 1, β is given by 2.44.
5 γ 1, k 2.
Corollary 2.12. Let n ∈ , α ≥ 0, and β ∈ 0, 1. If p is a function analytic in
nonvanishing in
and
pz α
zp z
≺ hc2 n,α,1,β z,
pz
of the form 1.4
z∈ ,
2.62
where
nαβ cos βπ/2
2
c2 n, α, 1, β β arctan 1β/2
,
π
1β
1 − β1−β/2 nαβ sin βπ/2
2.63
then
2.64
p ≺ hβ .
6 The case γ 1, k 2, n 1, α 1 was proved in 10. The same result was
reproved in 11 and once again in 12.
7 γ 1, k 2, β 1.
Corollary 2.13. Let n ∈ , α ≥ 0, and β ∈ 0, 1. If p is an analytic function in
1.4 nonvanishing in
and
zp z
Re pz α
> 0,
pz
z∈ ,
of the form
2.65
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13
then
Re pz > 0,
z∈ .
2.66
3. Applications
All this type results can be applied in the theory of analytic functions. Some results concerning the inclusion relations between subclasses of analytic functions can be formulated.
Let An, n ∈ N, denote the class of functions of the form
fz z ∞
3.1
ak zk ,
kn1
which is analytic in . For short, let A A1. Also let S denote the class of all functions
in A which are univalent in .
To use theorems and corollaries listed in the previous section we put instead
of the function p some functionals over the class An, such as pz fz/z, pz zf z/fz or the others. In this way the inclusion relations between selected subclasses
of analytic functions can be obtained.
3.1. Arithmetic Means
I γ 1, k 1.
i pz fz/z, z ∈ , f ∈ An, n ∈ N.
For n ∈ , α ≥ 0, β ∈ 0, 1 let Rn α, β denote class of functions f ∈ An such that
1 − α
or, equivalently,
fz
αf z ≺ hβ z,
z
z∈ ,
arg 1 − α fz αf z < β π ,
z
2
z∈ .
3.2
3.3
Using Corollary 2.7 we have the following.
Corollary 3.1. Let n ∈ , α ≥ 0, and β ∈ 0, β1 n, α, 1. If f ∈ An and
arg 1 − α fz αf z < c1 n, α, 1, β π ,
z
2
3.4
then
arg fz < β π ,
z
2
The above result we can write in the following form.
z∈ .
3.5
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Corollary 3.2.
Rn α, c1 n, α, 1, β ⊂ Rn 0, β .
3.6
ii pz f z, z ∈ , f ∈ An, n ∈ For n ∈ , α ≥ 0, β ∈ 0, 1 let Tn α, β denote class of functions f ∈ An such that
f z αzf z ≺ hβ z,
3.7
z∈ ,
or, equivalently,
arg f z αzf z < β π ,
2
z∈ .
3.8
Remark 3.3. 1 The class T1 1, 1 was introduced in 13.
2 The class T1 α, 1 coincides with the class Hα, 1, −1 studied in 14.
Observe that f ∈ Tn α, β if and only if zf ∈ Rn α, β.
Using Corollary 2.7 we have the following.
Corollary 3.4. Let n ∈ , α ≥ 0, and β ∈ 0, β1 n, α, 1. If f ∈ An and
arg f z αzf z < c1 n, α, 1, β π ,
2
z∈ ,
3.9
then
arg f z < β π ,
2
z∈ .
3.10
Hence we have the following.
Corollary 3.5.
Tn α, c1 n, α, 1, β ⊂ Tn 0 , β .
3.11
II γ 1, k 2.
ii pz zf z/fz, z ∈ , f ∈ An, n ∈ .
For n ∈ , α ≥ 0, β ∈ 0, 1 let Mn α, β denote class of functions f ∈ An such
0 for z ∈ and
that fzf z /
zf z
zf z
α 1 ≺ hβ z,
1 − α
fz
f z
z∈ ,
3.12
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or, equivalently,
< βπ ,
arg 1 − α zf z α 1 zf z
fz
f z
2
z∈ .
3.13
Remark 3.6. 1 The class M1 α, 1, that is, the class of so-called α-convex functions was
introduced by Mocanu 15.
2 The class M1 0, 1 is identical with the class S∗ of starlike functions. The
class M1 1, 1 is identical with the class C of convex functions.
3 The class M1 0, β denoted by S∗ β were defined by Brannan and Kirwan 16
and, independently, by Stankiewicz 17, 18. Functions in this class are called strongly starlike
of order β.
The class M1 1, β denoted by C∗ β contains functions called strongly convex of order
β.
Using Corollary 2.12 we have the following result proved by Marjono and Thomas
19.
Corollary 3.7. Let n ∈ , α ≥ 0, and β ∈ 0, 1. If f ∈ An and
< c2 n, α, 1, β π ,
arg 1 − α zf z α 1 zf z
fz
f z
2
z∈ ,
3.14
then
arg zf z < β π ,
fz 2
z∈ .
3.15
For α 1 one has the result due to Nunokawa and Thomas [12]:
Corollary 3.8. Let n ∈ and β ∈ 0, 1. If f ∈ An and
arg 1 zf z < c2 n, 1, 1, β π ,
f z
2
z∈ ,
3.16
then
arg zf z < β,
fz z∈ .
3.17
Corollary 3.9.
Mn α, c2 n, α, 1, β ⊂ Mn 0, β ,
M1 α, 1 ⊂ S∗ ,
C ⊂ S∗ .
3.18
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3.2. Geometric Mean
I α 1, k 1.
i pz fz/z, z ∈ , f ∈ An, n ∈ .
For n ∈ , β ∈ 0, 1, and γ ∈ 0, 1 let Ln γ, β denote class of functions f ∈
An such that
fz
z
1−γ
γ
f z ≺ hβ z,
3.19
z∈ ,
or equivalently
fz 1−γ γ π
f z
arg
<β ,
z
2
z∈ .
3.20
Remark 3.10. The class L1 γ, 1 was introduced in 20.
Applying Theorem 2.6 with α 1 we have the following.
Corollary 3.11. Let n ∈ , γ ∈ 0, 1, and β ∈ 0, β1 n, 1, γ . If f ∈ A(1) and
π
fz 1−γ γ f z
arg
< c1 n, 1, γ, β ,
z
2
z∈ ,
3.21
then
arg fz < β π ,
z
2
z∈ .
3.22
Corollary 3.12.
Ln γ, c1 n, 1, γ, β ⊂ Ln 0, β .
3.23
II α 1, k 2.
ii pz zf z/fz, z ∈ , f ∈ An, n ∈ .
For n ∈ , β ∈ 0, 1, and γ ∈ 0, 1 let Sn∗ γ, β denote class of functions f ∈
An such that
zf z
fz
1−γ zf z
1 f z
γ
≺ hβ z,
z∈ ,
3.24
International Journal of Mathematics and Mathematical Sciences
17
or, equivalently,
zf z 1−γ
zf z γ π
1 arg
<β ,
fz
f z
2
z∈ .
3.25
Remark 3.13. 1 The class Sn∗ γ, 1, that is, the class of so-called γ -starlike functions was
introduced by Lewandowski et al. 21.
2 Clearly,
S1∗ 0, 1 S∗ ,
S1∗ 1, 1 C,
S1∗ 0, β M1 0, β S∗ β ,
3.26
S1∗ 1, β M1 1, β C∗ β .
Using Theorem 2.8 we obtain results due to Darus and Thomas 22.
Theorem 3.14. Let n ∈ , γ ∈ 0, 1, and β ∈ 0, 1. If f ∈ An and
π
zf z 1−γ
zf z γ 1 arg
< c2 n, 1, γ, β ,
fz
f z
2
z∈ ,
3.27
then
arg zf z < β π ,
fz 2
z∈ .
3.28
Corollary 3.15.
Sn∗ γ ; c2 n, 1, γ, β ⊂ Sn∗ β ,
S1∗ γ, 1 ⊂ S∗ .
3.29
As further applications of Theorems 2.6 and 2.8 we can use arbitrary well-defined
functionals over the class An. We recall two examples:
1
pz Kaδ1 fz
,
z
z ∈ , f ∈ An, n ∈ , a > 0, δ ≥ 0,
3.30
18
International Journal of Mathematics and Mathematical Sciences
where the integral operator Kaδ over the class An was defined by Komatu 23 as follows:
Kaδ fz
aδ
Γδ
1
t
a−2
0
1 δ−1
log
fztdt,
t
z∈ ,
3.31
where Γ is the Gamma function;
2
pz Lλ fz
,
z
z ∈ , f ∈ A, λ ≥ −1,
3.32
where the operator Lλ over the class A called Ruscheweyh derivative 24 was defined as
follows:
Lλ fz z
1 − zλ1
∗ fz,
z∈ .
3.33
References
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18 J. Stankiewicz, “Quelques problèmes extrémaux dans les classes des fonctions -angulairement
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