Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 583972, 11 pages
doi:10.1155/2011/583972
Research Article
The Fekete-Szegö Problem for p-Valently Janowski
Starlike and Convex Functions
Toshio Hayami1 and Shigeyoshi Owa2
1
2
School of Science and Technology, Kwansei Gakuin University, Sanda, Hyogo 669-1337, Japan
Department of Mathematics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan
Correspondence should be addressed to Shigeyoshi Owa, owa@math.kindai.ac.jp
Received 13 January 2011; Accepted 4 May 2011
Academic Editor: A. Zayed
Copyright q 2011 T. Hayami and S. Owa. 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.
For p-valently Janowski starlike and convex functions defined by applying subordination for the
generalized Janowski function, the sharp upper bounds of a functional |ap2 − μa2p1 | related to the
Fekete-Szegö problem are given.
1. Introduction
Let Ap denote the family of functions fz normalized by
fz zp ∞
p 1, 2, 3, . . . ,
an zn
1.1
np1
which are analytic in the open unit disk U {z ∈ C : |z| < 1}. Furtheremore, let W be the class
of functions wz of the form
wz ∞
wk zk ,
1.2
k1
which are analytic and satisfy |wz| < 1 in U. Then, a function wz ∈ W is called the Schwarz
function. If fz ∈ Ap satisfies the following condition
1
Re 1 b
zf z
−p
fz
> 0 z ∈ U,
1.3
2
International Journal of Mathematics and Mathematical Sciences
for some complex number b b /
0, then fz is said to be p-valently starlike function of
complex order b. We denote by Sb∗ p the subclass of Ap consisting of all functions fz which
are p-valently starlike functions of complex order b. Similarly, we say that fz is a member
of the class Kb p of p-valently convex functions of complex order b in U if fz ∈ Ap satisfies
1 zf z −
p
−
1
> 0 z ∈ U,
Re 1 b f z
1.4
for some complex number b b /
0.
Next, let Fz zf z/fz u iv and b ρeiϕ ρ > 0, 0 ϕ < 2π. Then, the
condition of the definition of Sb∗ p is equivalent to
sin ϕ
cos ϕ 1 zf z
Re 1 −p
1
v > 0.
u−p b fz
ρ
ρ
1.5
We denote by dl1 , p the distance between the boundary line l1 : cos ϕu sin ϕv ρ − p cos ϕ 0 of the half plane satisfying the condition 1.5 and the point F0 p. A simple
computation gives us that
cos ϕ × p sin ϕ × 0 ρ − p cos ϕ
ρ,
d l1 , p cos2 ϕ sin2 ϕ
1.6
that is, that dl1 , p is always equal to |b| ρ regardless of ϕ. Thus, if we consider the circle
C1 with center at p and radius ρ, then we can know the definition of Sb∗ p means that FU is
covered by the half plane separated by a tangent line of C1 and containing C1 . For p 1, the
same things are discussed by Hayami and Owa 3.
Then, we introduce the following function:
pz 1 Az
1 Bz
−1 B < A 1,
1.7
which has been investigated by Janowski 4. Therefore, the function pz given by 1.7 is
said to be the Janowski function. Furthermore, as a generalization of the Janowski function,
Kuroki et al. 6 have investigated the Janowski function for some complex parameters A and
B which satisfy one of the following conditions:
i A / B,
|B| < 1,
|A| 1,
Re 1 − AB |A − B|;
A/
B,
|B| 1,
|A| 1,
1 − AB > 0.
ii
1.8
Here, we note that the Janowski function generalized by the conditions 1.8 is analytic and
univalent in U, and satisfies Repz > 0 z ∈ U. Moreover, Kuroki and Owa 5 discussed
the fact that the condition |A| 1 can be omitted from among the conditions in 1.8-i as
International Journal of Mathematics and Mathematical Sciences
3
the conditions for A and B to satisfy Repz > 0. In the present paper, we consider the more
general Janowski function pz as follows:
pz p Az
1 Bz
p 1, 2, 3, . . . ,
1.9
for some complex parameter A and some real parameter B A /
pB, −1 B 0. Then, we
don’t need to discuss the other cases because for the function:
qz p A1 z
1 B1 z
pB1 , |B1 | 1 ,
A1 , B1 ∈ C, A1 /
1.10
letting B1 |B1 |eiθ and replacing z by −e−iθ z in 1.10, we see that
p − A1 e−iθ z p Az
pz q −e−iθ z ≡
1 − |B1 |z
1 Bz
A −A1 e−iθ , B −|B1 | ,
1.11
maps U onto the same circular domain as qU.
Remark 1.1. For the case B −1 in 1.9, we know that pz maps U onto the following half
plane:
p2 − |A|2
Re p A pz >
,
2
1.12
and for the case −1 < B 0 in 1.9, pz maps U onto the circular domain
A pB
pz − p AB <
.
1 − B2 1 − B2
1.13
Let pz and qz be analytic in U. Then, we say that the function pz is subordinate
to qz in U, written by
pz ≺ qz
z ∈ U,
1.14
if there exists a function wz ∈ W such that pz qwz z ∈ U. In particular, if qz is
univalent in U, then pz ≺ qz if and only if
p0 q0,
pU ⊂ qU.
1.15
4
International Journal of Mathematics and Mathematical Sciences
We next define the subclasses of Ap by applying the subordination as follows:
zf z p Az
≺
fz ∈ Ap :
z ∈ U ,
fz
1 Bz
zf z p Az
≺
Kp A, B fz ∈ Ap : 1 z ∈ U ,
f z
1 Bz
Sp∗ A, B
1.16
where A /
pB, −1 B 0. We immediately know that
fz ∈ Kp A, B
iff
zf z
∈ Sp∗ A, B.
p
1.17
Then, we have the next theorem.
Theorem 1.2. If fz ∈ Sp∗ A, B −1 < B 0, then fz ∈ Sb∗ p, where
B −pB ReA cos ϕ B ImA sin ϕ A − pB
b
eiϕ
1 − B2
0 ϕ < 2π .
1.18
Espesially, fz ∈ Sp∗ A, −1 if and only if fz ∈ Sb∗ p where b p A/2.
Proof. Supposing that zf z/fz ≺ p Az/1 − z, it follows from Remark 1.1 that
zf z p2 − |A|2
>
,
Re p A
fz
2
1.19
zf z 2
Re 2 p A
> Re 2p p A − p A .
fz
1.20
that is, that
This means that
⎡ ⎤
2 p A zf z
⎢
⎥
Re⎣ − p ⎦ > −1,
fz
p A2
1.21
which implies that
zf z
1
Re
> −1.
−p
fz
1/2 p A
Therefore, fz ∈ Sb∗ , where b p A/2. The converse is also completed.
1.22
International Journal of Mathematics and Mathematical Sciences
5
Next, for the case −1 < B 0, by the definition of the class Sb∗ A, B, if a tangent line
l2 of the circle C2 containing the point p is parallel to the straight line L : cos θu sin θv 0 −π ∃ θ < π, and the image FU by Fz zf z/fz is covered by the circle C2 , then
there exists a non-zero complex number b with argb θ π and |b| dl2 , p such that
fz ∈ Sb∗ p, where dl2 , p is the distance between the tangent line l2 and the point p. Now,
for the function pz p Az/1 Bz A /
pB, −1 < B 0, the image pU is equivalent
to
C2 p − AB A − pB
<
,
ω ∈ C : ω −
1 − B2 1 − B2
1.23
and the point ξ on ∂C2 {ω ∈ C : |ω − p − AB/1 − B 2 | |A − pB|/1 − B2 } can be written
by
A − pB
p − AB
ξ : ξθ eiθ 2
1−B
1 − B2
−π ∃ θ < π .
1.24
Further, the tangent line l2 of the circle C2 through each point ξθ is parallel to the straight
line L : cos θu sin θv 0. Namely, l2 can be represented by
l2 : cos θ u −
A − pB cos θ p − B ReA
1 − B2
sin θ v −
A − pB sin θ − B ImA
1 − B2
0,
1.25
which implies that
l2 : cos θu sin θv −
A − pB p − B ReA cos θ − B ImA sin θ
1 − B2
0.
1.26
Then, we see that the distance dl2 , p between the point p and the above tangent line l2 of the
circle C2 is
A − pB p − B ReA cos θ − B ImA sin θ cos θ × p sin θ × 0 −
1 − B2
−B −pB ReA cos θ − B ImA sin θ A − pB
.
1 − B2
1.27
Therefore, if the subordination
zf z p Az
≺
fz
1 Bz
A/
pB, −1 < B 0
1.28
6
International Journal of Mathematics and Mathematical Sciences
holds true, then fz ∈ Sb∗ where
b
−B −pB ReA cos θ − B ImA sin θ A − pB
1 − B2
eiθπ .
1.29
Finally, setting ϕ θ π 0 ϕ < 2π, the proof of the theorem is completed.
Noonan and Thomas 8, 9 have stated the qth Hankel determinant as
⎛
an
an1 · · · anq−1
⎞
⎟
⎜
⎜ an1 an2 · · · anq ⎟
⎟
⎜
⎟
Hq n det⎜
⎜ ..
.
.
.
..
..
.. ⎟
⎟
⎜ .
⎠
⎝
anq−1 anq · · · an2q−2
n, q ∈ N {1, 2, 3, . . .} .
1.30
This determinant is discussed by several authors with q 2. For example, we can know that
the functional |H2 1| |a3 − a22 | is known as the Fekete-Szegö problem, and they consider
the further generalized functional |a3 − μa22 |, where a1 1 and μ is some real number see,
1. The purpose of this investigation is to find the sharp upper bounds of the functional
|ap2 − μa2p1 | for functions fz ∈ Sp∗ A, B or Kp A, B.
2. Preliminary Results
We need some lemmas to establish our results. Applying the Schwarz lemma or subordination principle.
Lemma 2.1. If a function wz ∈ W, then
|w1 | 1.
2.1
Equality is attained for wz eiθ z for any θ ∈ R.
The following lemma is obtained by applying the Schwarz-Pick lemma see, e.g., 7.
Lemma 2.2. For any functions wz ∈ W,
|w2 | 1 − |w1 |2
2.2
holds true. Namely, this gives us the following representation:
w2 1 − |w1 |2 ζ,
for some ζ |ζ| 1.
2.3
International Journal of Mathematics and Mathematical Sciences
7
3. p-Valently Janowski Starlike Functions
Our first main result is contained in
Theorem 3.1. If fz ∈ Sp∗ A, B, then
ap2 − μa2p1 ⎧ 1 − 2μ A − p 1 − 2pμ B ⎪
⎪ A − pB
⎪
⎨
2
⎪
⎪
A − pB
⎪
⎩
2
1 − 2μ A − p 1 − 2pμ B 1 ,
1 − 2μ A − p 1 − 2pμ B 1 ,
3.1
with equality for
fz ⎧
zp
⎪
⎪
or zp eAz B 0
⎪
⎪
⎨ 1 BzpB−A/B
1 − 2μ A − p 1 − 2pμ B 1 ,
⎪
⎪
⎪
⎪
⎩
1 − 2μ A − p 1 − 2pμ B 1 .
zp
1 or zp eA/2z B 0
2
Bz2 pB−A/2B
3.2
Proof. Let fz ∈ Sp∗ A, B. Then, there exists the function wz ∈ W such that
zf z p Awz
,
fz
1 Bwz
3.3
which means that
n−1
n − p an A − kBak wn−k
np1 ,
3.4
kp
where ap 1. Thus, by the help of the relation in Lemma 2.2, we see that
1 2 *
)
ap2 − μa2p1 A − pB w2 A − p 1 B w12 − μ A − pB w12 2
A − pB 1 − w12 ζ A − p 1 B − 2μ A − pB w12 .
2
3.5
8
International Journal of Mathematics and Mathematical Sciences
Then, by Lemma 2.1, supposing that 0 w1 1 without loss of generality, and applying the
triangle inequality, it follows that
1 − w12 ζ A − p 1 B − 2μ A − pB w12 1 A − p 1 B − 2μ A − pB − 1 w12
⎧
⎨ A − p 1 B − 2μ A − pB ⎩
1
A − p 1 B − 2μ A − pB 1; w1 1 ,
A − p 1 B − 2μ A − pB 1; w1 0 .
3.6
Especially, taking μ p 1/2p in Theorem 3.1, we obtain the following corollary.
Corollary 3.2. If fz ∈ Sp∗ A, B, then
⎧ A A − pB ⎪
⎪
⎪
⎨
2p
ap2 − p 1 a2 p1 ⎪
2p
⎪
A − pB
⎪
⎩
2
|A| p ,
3.7
|A| p ,
with equality for
fz ⎧
zp
⎪
⎪
or zp eAz B 0
⎪
⎪
⎨ 1 BzpB−A/B
⎪
⎪
⎪
⎪
⎩
zp
1 Bz2 pB−A/2B
p A/2z2
or z e
B 0
|A| p ,
|A| p .
3.8
Furthermore, putting A p − 2α and B −1 for some α 0 α < p in Theorem 3.1,
we arrive at the following result by Hayami and Owa 2, Theorem 3.
Corollary 3.3. If fz ∈ Sp∗ α, then
⎧
1
⎪
⎪
,
p
−
α
2
p
−
α
1
−
4
p
−
α
μ
μ
⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
p
−
α
1
1
μ ,
ap2 − μa2p1 p − α
2
⎪
2 p−α
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
p−α1
⎪
⎪
⎪
μ ,
⎩ p−α 4 p−α μ− 2 p−α 1
2 p−α
3.9
International Journal of Mathematics and Mathematical Sciences
9
with equality for
fz ⎧
z
⎪
⎪
⎪
⎪
⎪
⎨ 1 − z2p−α
⎪
⎪
⎪
⎪
⎪
⎩
z
1 − z2 p−α
p−α1
1
μ or μ 2
2 p−α
p−α1
1
μ 2
2 p−α
,
3.10
.
4. p-Valently Janowski Convex Functions
Similarly, we consider the functional |ap2 − μa2p1 | for p-valently Janowski convex functions.
Theorem 4.1. If fz ∈ Kp A, B, then
ap2 − μa2p1 ⎧ *
3
)
2
⎪
p A − pB
p 1 − 2p p 2 μ A − p 1 − 2p2 p 2 μ B ⎪
⎪
⎪
,
⎪
2 ⎪
⎪
⎪
2 p1 p2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
2
2 3
⎪
⎪
⎪
p 1 − 2p p 2 μ A − p 1 − 2p2 p 2 μ B p 1 ,
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
pA − pB
⎪
⎪
⎪
,
⎪
⎪
⎪ 2 p2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩ p 12 − 2pp 2μ A − p 13 − 2p2 p 2μ B p 12 ,
4.1
with equality for
fz ⎧
A
⎪
p
⎪
;
p
1;
−Bz
or zp 1 F1 p, p 1; Az B 0,
F
p,
p
−
z
⎪ 2 1
⎪
B
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
2
2 3
⎪
⎪
p 1 − 2p p 2 μ A − p 1 − 2p2 p 2 μ B p 1 ,
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
p
p A 2
p pB − A
p
⎪
p
2
p
⎪
⎪
,
;
1
;
−Bz
,
1
;
z
z
or
z
F
F
B 0,
2 1
1 1
⎪
⎪
2
2B
2
2
2 2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩ p 12 − 2pp 2μ A − p 13 − 2p2 p 2μ B p 12 ,
4.2
10
International Journal of Mathematics and Mathematical Sciences
where 2 F1 a, b; c; z represents the ordinary hypergeometric function and 1 F1 a, b; z represents the
confluent hypergeometric function.
Proof. By the help of the relation 1.17 and Theorem 3.1, if fz ∈ Kp A, B, then
2
p 2
p 2
p1
2 a −μ
ap1 ap2 −
p p2
p p2
2
p1
2 μap1 ,
p p2
4.3
C μ ,
where Cμ is one of the values in Theorem 3.1. Then, dividing the both sides by p 2/p
and replacing p 12 /pp 2μ by μ, we obtain the theorem.
Now, letting μ p 13 /2p2 p 2 in Theorem 4.1, we have the following corollary.
Corollary 4.2. If fz ∈ Kp A, B, then
⎧ A A − pB ⎪
⎪
⎪
⎪
⎪
⎨ 2 p2
3
p1
a2p1 ap2 − 2 ⎪
2p p 2
⎪
⎪ pA − pB
⎪
⎪
⎩ 2 p2
|A| p ,
4.4
|A| p ,
wiht equality for
fz ⎧
A
⎪
p
⎪
z 2 F1 p, p − ; p 1; −Bz or zp 1 F1 p, p 1; Az B 0
⎪
⎪
B
⎨
|A| p ,
⎪
⎪
p
p A
p pB − A
p
⎪
⎪
⎩zp 2 F1
,
; 1 ; −Bz2 or zp 1 F1
, 1 ; z2 B 0
2
2B
2
2
2 2
|A| p ,
4.5
where 2 F1 a, b; c; z represents the ordinary hypergeometric function and 1 F1 a, b; z represents the
confluent hypergeometric function.
Moreover, we suppose that A p − 2α and B −1 for some α 0 α < p. Then, we
arrive at the result by Hayami and Owa 2, Theorem 4.
International Journal of Mathematics and Mathematical Sciences
11
Corollary 4.3. If fz ∈ Kp α, then
ap2 − μa2p1 ⎧ *
)
2 ⎪
p p−α
p 1 2 p − α 1 − 4p p 2 p − α μ
⎪
⎪
⎪
⎪
⎪
2 ⎪
⎪
⎪
p1 p2
⎪
⎪
2 ⎪
⎪
⎪
p
1
⎪
⎪
μ ,
⎪
⎪
⎪
2p p 2
⎪
⎪
⎪
⎪
⎪
2
2 ⎪
⎨ pp − α
p1
p1 p−α1
μ
,
⎪
p2
2p p 2
2p p 2 p − α
⎪
⎪
⎪
⎪
⎪
⎪
) *
⎪
⎪ p p − α 4p p 2 p − α μ − p 1 2 2 p − α 1
⎪
⎪
⎪
⎪
⎪
2 ⎪
⎪
p1 p2
⎪
⎪
⎪
2 ⎪
⎪
⎪
p1 p−α1
⎪
⎪
μ
,
⎪
⎩
2p p 2 p − α
4.6
with equality for
fz ⎧
⎪
⎪
⎪
zp 2 F1 p, 2 p − α ; p 1; z
⎪
⎪
⎨
⎪
p
⎪
⎪
p ⎪
⎪
⎩zp 2 F1 , p − α; 1 ; z2
2
2
2
2 p1
p1 p−α1
μ or μ ,
2p p 2
2p p 2 p − α
2 2
p1 p−α1
p1
μ
,
2p p 2
2p p 2 p − α
4.7
where 2 F1 a, b; c; z represents the ordinary hypergeometric function.
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