General Mathematics, Vol. 16, Nr. 2 (2008), 19–28
Multivalued starlike functions of complex
order1
Y. Polatog̃lu, M. Çag̃lar, E. Yavuz, S. Owa
Abstract
Let S∗λ (1−b) (b 6= 0, complex) denote the class of functions f (z) =
z+a2 z 2 +· · · analytic in the open unit disc D = {z ∈ C||z| < 1} which
¶¸
·
µ
(Dλ f (z))′
f (z)
1
iθ
−1
>
satisfy for z = re ∈ D, z 6= 0 and Re 1 + b z λ
D f (z)
0 (0 ≤ λ < 1), where Dλ f (z) = Γ(2 − λ)z λ Dλz f (z) and Dλz f (z) is the
fractional derivative of f (z).
The aim of this paper is to investigate certain properties of the
mentioned above class.
2000 Mathematics Subject Classification: 30C45.
Key words and phrases: starlike functions of complex order, fractional
integral, fractional derivative, distortion theorem, coefficient inequality
1
Introduction
Let A denote the class of functions f (z) = z + a2 z 2 + · · · which are analytic
in D. Let Ω be the family of functions w(z) regular in D, and satisfying the
conditions w(0) = 0, |w(z)| < 1 for all z ∈ D. We use P to denote the class
of functions p(z) = 1 + p1 z + p2 z 2 + · · · which are analytic in D and have a
1
Received 30 August, 2007
Accepted for publication (in revised form) 4 January, 2008
19
20
Y. Polatog̃lu, M. Çag̃lar, E. Yavuz, S. Owa
positive real part in D. Let h(z) = z + b2 z 2 + · · · and s(z) = z + c2 z 2 + · · ·
be analytic functions in D. We say that h(z) is subordinate to s(z), written
h ≺ s, if
(1.1)
h(z) = s(w(z)), for some w(z) ∈ Ω, and for all z ∈ D.
In particular, if s(z) is univalent in D, then h ≺ s if and only if h(0) =
s(0), and h(D) ⊂ s(D) (see [2]).
The fractional integral of order λ is defined for a function f (z), by
Z z
f (ζ)
1
−λ
dζ (λ > 0),
(1.2)
Dz f (z) =
Γ(λ) 0 (z − ζ)1−λ
where the function f (z) is analytic in a simply connected region of the
complex plane containing the origin and the multiplicity of (z − ζ)λ−1 is
removed by requiring log(z − ζ) to be real when (z − ζ) > 0 ([7], [8], [10]).
The fractional derivative of order λ is defined for a function f (z), by
Z z
1
d
f (ζ)
λ
(1.3)
Dz f (z) =
dζ (0 ≤ λ < 1),
Γ(1 − λ) dz 0 (z − ζ)λ
where the function f (z) is analytic in a simply connected region of the
complex plane containing the origin and the multiplicity of (z − ζ)−λ is
removed as in definition of the fractional integral ([7], [8], [10]).
Under the hypotheses of the fractional derivative of order λ, the fractional derivative of (n + λ) is defined, for a function f (z), by
(1.4)
Dn+λ
f (z) =
z
dn λ
D f (z) (0 ≤ λ < 1, n ∈ N0 = N ∪ {0}).
dz n z
By means of the definitions above, we see that
k
D−λ
z z =
Γ(k + 1) k+λ
Γ(k + 1) k−λ
z
(λ > 0); Dλz z k =
z
(0 ≤ λ < 1)
Γ(k + 1 + λ)
Γ(k + 1 − λ)
and
Dn+λ
zk =
z
Γ(k + 1)
z k−n−λ (0 ≤ λ < 1, n ∈ N0 ).
Γ(k + 1 − n − λ)
21
Multivalued starlike functions of complex order
Therefore, we conclude that, for any real λ
Dλz z k =
Γ(k + 1) k−λ
z .
Γ(k + 1 − λ)
In this paper we will study some of the properties of the class S∗λ (1 − b)
(b 6= 0, complex) defined as follows.
Definition 1.1. A function f ∈ A is said to be multivalued starlike of
complex order (1 − b) (b 6= 0, complex), that is f ∈ S∗λ (1 − b) if and only if
¶
µ
1
(Dλ f (z))′
− 1 = p(z)
(z ∈ D)
(1.5)
1+
z λ
b
D f (z)
for some p(z) ∈ P and 0 ≤ λ < 1.
Using (1.3), Owa and Srivastava [9] introduced the operator
Dλ :A −→ A defined by Dλ f (z) = γ(2 − λ)z λ Dzλ f (z) (0 ≤ λ < 1).
We note that S∗0 (1 − b) = S(1 − b), where S(1 − b) is the class of starlike
functions of complex order introduced and studied by Nasr and Aouf [5].
The following lemma due to Jack [3] (also [4]) plays an important role
in our proofs.
Lemma 1.2. Let w(z) be a non-constant and analytic function in the open
unit disc D with w(0) = 0. If |w(z)| attains its maximum value on the circle
|z| = r at the point z1 , then z1 w′ (z1 ) = kw(z1 ) for some real k ≥ 1.
2
Main Results
Lemma 2.1. Let f (z) be an element of A. Then the fractional λ−operator
(2.1)
Dλ f (z) = Γ(2 − λ)z λ Dλz f (z) satisfies the following equalities
(i) Dλ f (z) = Γ(2 − λ)z λ Dλz f (z) = z +
∞
X
n=2
an
Γ(n + 1)Γ(2 − λ) n
z
Γ(n + 1 − λ)
22
Y. Polatog̃lu, M. Çag̃lar, E. Yavuz, S. Owa
(ii) For λ = 1, Df (z) = lim Dλ f (z) = zf ′ (z)
λ→1
(iii) For λ < 1, α < 1,
α
λ
λ
α
D (D f (z)) = D (D f (z)) = z+
∞
X
n=2
(iv) D(Dλ f (z)) = z +
∞
X
nan
n=2
an
Γ(2 − α)Γ(2 − λ)(Γ(n + 1))2 n
z
Γ(n + 1 − λ)Γ(n + 1 − α)
Γ(2 − λ)Γ(n + 1) n
z = z(Dλ f (z))′
Γ(n + 1 − λ)
= Γ(2 − λ)z λ (λDλz f (z) + zDλ+1
f (z)).
z
Proof. The proof of this lemma follows from the definition of Dλ f (z).
Lemma 2.2. Let f (z) = z + a2 z 2 + · · · and g(z) = z + b2 z 2 + · · · be analytic functions in the open unit disc D. Then the solution of the fractional
differential equation
(2.2)
(2.3)
Dλz f (z) =
f (z) = z +
1
z −λ g(z) is ,
Γ(2 − λ)
∞
X
n=2
bn
Γ(n + 1 − λ)
zn.
Γ(2 − λ)Γ(n + 1)
Proof. Using the definitions of fractional integral, fractional derivative and
fractional derivative of order (n + λ), we get
(2.4)
Γ(3) 2−λ
Γ(n + 1) n−λ
Γ(2) 1−λ
Dλz f (z) =
z
+ a2
z
+ · · · + an
z
+···
Γ(2 − λ)
Γ(3 − λ)
Γ(n + 1 − λ)
1
1
1
z −λ g(z) =
z −λ (z + b2 z 2 + · · · ) =
z 1−λ +
Γ(2 − λ)
Γ(2 − λ)
Γ(2 − λ)
(2.5)
1
1
b2
z 2−λ + · · · + bn
z n−λ + · · · .
Γ(2 − λ)
Γ(2 − λ)
Therefore we have
1
Dzλ f (z) =
z −λ g(z) which implies
Γ(2 − λ)
Multivalued starlike functions of complex order
23
Γ(3) 2−λ
Γ(n + 1) n−λ
Γ(2) 1−λ
z
+ a2
z
+ · · · + an
z
+ ···
Γ(2 − λ)
Γ(3 − λ)
Γ(n + 1 − λ)
(2.6)
=
1
1
1
z 1−λ + b2
z 2−λ + · · · + bn
z n−λ + · · · .
Γ(2 − λ)
Γ(2 − λ)
Γ(2 − λ)
Comparing the coefficient of z n−λ in both sides of (2.5) we obtain
(2.7)
an =
Γ(n + 1 − λ)
bn
Γ(2 − λ)Γ(n + 1)
Theorem 2.3. Let f (z) ∈ A and satisfies the condition
(2.8)
µ
¶
(Dλ f (z))′
2bz
z λ
−1 ≺
= F (z),
1−z
D f (z)
then f (z) ∈ Sλ∗ (1 − b). This result is sharp since the function f (z) satisfies
1
z 1−λ (1 − z)−2b .
the fractional differential equation Dλz f (z) =
Γ(2 − λ)
Proof. We define the function w(z) by
Dλ f (z)
= (1 − w(z))−2b ,
z
where the value of (1 − w(z))−2b is 1 at z = 0 (i.e, we consider the corresponding Riemann branch), then w(z) is analytic in D, w(0) = 0, and
(2.9)
¶
µ
2bzw′ (z)
(Dλ f (z))′
−1 =
.
z λ
1 − w(z)
D f (z)
Now, it is easy to realize that the subordination (2.9) is equivalent to
|w(z)| < 1 for all z ∈ D. Indeed, assume the contrary: then, there exists
z1 ∈ D such that |w(z)| attains its maximum value on the circle |z| = r at the
point z1 , that is |w(z1 )| = 1. Then, by I.S. Jack’s lemma, z1 w′ (z1 ) = kw(z1 )
for some real k ≥ 1. For such z1 we have
µ
¶
(Dλ f (z1 ))′
2kbw(z1 )
z1 λ
−1 =
= F (w(z1 )) ∈
/ F (D),
1 − w(z1 )
D f (z1 )
24
Y. Polatog̃lu, M. Çag̃lar, E. Yavuz, S. Owa
because |w(z1 )| = 1 and k ≥ 1. But this contradicts (2.9), so the assumption
is wrong, i.e, |w(z)| < 1 for all z ∈ D.
The sharpness of this result follows from the fact that
1
1
z 1−λ (1 − z)−2b =
z −λ z(1 − z)−2b ⇒
Γ(2 − λ)
Γ(2 − λ)
¶
µ
2bz
(Dλ f (z))′
λ
λ λ
−2b
−1 =
.
D f (z) = Γ(2 − λ)z Dz f (z) = z(1 − z)
⇒ z λ
1−z
D f (z)
Dλz f (z) =
On the other hand we have
µ
¶
¶
µ
2bz
1 (Dλ f (z))′
1 + w(z)
(Dλ f (z))′
−1 ≺
⇒1+
−1 =
.
z λ
λ
1−z
b
1 − w(z)
D f (z)
D f (z)
Corollary 2.4. If f (z) ∈ S∗λ (1 − b), then
¯µ
¯
¶1/2b
¯
¯
1−λ
z
¯
¯
− 1¯ < 1.
¯
¯ Γ(2 − λ)Dλz f (z)
¯
This corollary is a simple consequence of Theorem 2.3.
Remark 1. Putting λ = 0, we obtain the result obtained in [6].
Theorem 2.5. If f (z) ∈ S∗λ (1 − b), then
(2.11)
1−λ
|b|−Reb
¯ λ
¯
r1−λ (1 − r)|b|−Reb
¯Dz f (z)¯ ≤ r (1 + r)
≤
.
Γ(2 − λ)(1 + r)|b|+Reb
Γ(2 − λ)(1 − r)|b|+Reb
This result is sharp since the function satisfies the fractional differential
1
equation Dλz f (z) =
z 1−λ (1 − z)−2b .
Γ(2 − λ)
Proof. If p(z) ∈ P, then we have ([2])
¯
¯
2¯
¯
1
+
r
¯ ≤ 2r .
¯p(z) −
(2.12)
¯
1 − r2 ¯ 1 − r2
Using the definition of the class S∗λ (1 − b), then we can write
¯·
¯
¶¸
µ
λ
2¯
′
¯
1
1
+
r
(D
f
(z))
¯ 1+
¯ ≤ 2r .
(2.13)
−1 −
z λ
¯
2
b
1 − r ¯ 1 − r2
D f (z)
Multivalued starlike functions of complex order
25
It follows from (2.16) that
¯
¯µ
¶
¯ (Dλ f (z))′
2|b|r
2br2 ¯¯
¯ z
≤
−1 −
¯
¯
λ
2
1−r
1 − r2
D f (z)
which implies that
¶
µ
(Dλ f (z))′
2r2 Reb + 2r|b|
2r2 Reb − 2r|b|
≤
Re
z
−
1
≤
.
1 − r2
1 − r2
Dλ f (z)
Thus we obtain that
(2.16)
µ
¶
1 − 2|b|r − (1 − 2Reb)r2
(Dλ f (z))′
1 + 2|b|r − (1 − 2Reb)r2
≤
Re
z
≤
.
1 − r2
1 − r2
Dλ f (z)
On the other hand, since
µ
¶
(Dλ f (z))′
∂
Re z λ
= r log |Dλ f (z)|,
∂r
D f (z)
then, the inequality (2.16) can be written in the form
(2.17)
∂
1 + 2|b|r − (1 − 2Reb)r2
1 − 2|b|r − (1 − 2Reb)r2
≤
log |Dλ f (z)| ≤
.
r(1 − r)(1 + r)
∂r
r(1 − r)(1 + r)
Integrating both sides of the inequality (2.17) from 0 to r we obtain (2.11).
Remark 2. Putting λ = 0, we obtain the result obtained in [1].
Theorem 2.6. If f (z) ∈ S∗λ (1 − b), then
(2.18)
|an | ≤
n−2
Y
Γ(n + 1 − λ)
(k + 2|b|) n = 2, 3, · · · .
(n − 1)!Γ(2 − λ)Γ(n + 1) k=0
This inequality is sharp because the extremal function satisfies the fractional
1
z 1−λ (1 − z)−2b .
differential equation Dλz f (z) =
Γ(2 − λ)
Proof. Using the definition of the class S∗λ (1 − b), we can write
(2.19)
z(Dλ f (z))′ = (Dλ f (z))(1 + bp1 z + bp2 z 2 + · · · + bpn z n + · · · ).
26
Y. Polatog̃lu, M. Çag̃lar, E. Yavuz, S. Owa
Comparison of the coefficient of z n in both sides (2.19) gives
n−1
(2.20)
1 Γ(n + 1 − λ) X Γ(k + 1)
an =
ak bpn−k , a1 ≡ 1.
(n − 1) Γ(n + 1) k=1 Γ(k + 1 − λ)
Therefore we have
n−1
(2.21)
2|b| Γ(n + 1 − λ) X Γ(k + 1)
|an | ≤
|ak |, |a1 | ≡ 1
(n − 1) Γ(n + 1) k=1 Γ(k + 1 − λ)
because |pn | ≤ 2 for all n ≥ 1 whenever p(z) ∈ P ([2]).
It follows that
γ(3 − λ)γ(2)
2|b| = (2 − λ)|b|,
|a2 | ≤
γ(3)γ(2 − λ)
¶
µ
2|b|γ(4 − λ)
γ(2)
γ(3)
|a3 | ≤
|a1 | +
|a2 |
2γ(4)
γ(2 − λ)
γ(3 − λ)
γ(4 − λ)
≤
2|b| (1 + 2|b|) ,
2!γ(4)γ(2 − λ)
and
¶
µ
2|b|γ(5 − λ)
γ(3)
γ(4)
γ(2)
|a4 | ≤
|a1 | +
|a2 | +
|a3 |
3γ(5)
γ(2 − λ)
γ(3 − λ)
γ(4 − λ)
γ(5 − λ)
2|b| (1 + 2|b|) (2 + 2|b|) .
3!γ(5)γ(2 − λ)
Therefore, applying the mathematical induction, we complete the proof of
the theorem.
Now we consider the fractional differential equation
1
Dλz f (z) =
z 1−λ (1 − z)−2b .
Γ(2 − λ)
Using by Lemma 2.2 we can write
≤
(2.22)
Γ(2) 1−λ
Γ(3) 2−λ
Γ(n + 1) n−λ
z
+ a2
z
+ · · · + an
z
Γ(2 − λ)
Γ(3 − λ)
Γ(n + 1 − λ)
"
! #
Ãn−2
∞
X
Y 2b + k
1
z −λ z +
=
z n , which implies
Γ(2 − λ)
k
+
1
n=2
k=0
Dλz f (z) =
27
Multivalued starlike functions of complex order
f (z) = z +
∞
X
n=2
Γ(n + 1 − λ)
Γ(2 − λ)Γ(n + 1)
Ãn−2
!
Y 2b + k
k=0
k+1
zn.
Remark 3. Putting λ = 0, we obtain the result obtained by Nasr and Aouf
[5].
References
[1] M. K. Aouf, S. Owa and M. Obradović, Certain classes of analytic
function of complex order, Rend. di Mat. 11(1991), 691-714.
[2] A. W. Goodman,Univalent Functions, volume I and volume II, Mariner
Publishing Comp. Inc., Tampa, Florida, 1983.
[3] I. S. Jack, Functions starlike and convex of order α, J. London Math.
Soc. 3(1971), no. 2, 469-474.
[4] S. S. Miller and P. T. Mocanu, Differential Subordinations, Theory and
Application, Marcel Dekker Inc., 2000.
[5] M. A. Nasr and M. K. Aouf, Starlike function of complex order, J.
Natur. Sci. Math. 25(1985), No.1, 1-12.
[6] M. Obradović, M. K. Aouf & S. Owa, On some results for starlike
functions of complex order, Publications Inst. Math. 46(20)(1989), 7985.
[7] S. Owa, Univalent and Geometric Function Theory Seminar Notes, TC
İstanbul Kültür University Pub., İstanbul, 2006.
[8] S. Owa, On the distortion theorems I., Kyunpook Math. J., 18(1978),
53-59.
[9] S. Owa and H. M. Srivastava, Univalent and starlike generalized hypergeometric functions, Canad. J. Math. 39(1987), 1057-1077.
28
Y. Polatog̃lu, M. Çag̃lar, E. Yavuz, S. Owa
[10] H. M. Srivastava & S. Owa (Editors), “Univalent Functions, Fractional
Calculus and Their Applications”, Halsted Press (Ellis Horword Limited, Chichester), Jhon Wiley and Sons, New York, Chichester, Bribane
and Toront,1989.
Yaşar Polatog̃lu
Department of Mathematics and Computer Science,
TC İstanbul Kültür University, 34156 İstanbul, Turkey
E-mail: y.polatoglu@iku.edu.tr
Mert Çag̃lar
Department of Mathematics and Computer Science,
TC İstanbul Kültür University, 34156 İstanbul, Turkey
E-mail: m.caglar@iku.edu.tr
Emel Yavuz
Department of Mathematics and Computer Science,
TC İstanbul Kültür University, 34156 İstanbul, Turkey
E-mail: e.yavuz@iku.edu.tr
Shigeyoshi Owa
Department of Mathematics,
Kinki University, Higashi-Osaka, Osaka 577-8502, Japan
E-mail: owa@math.kindai.ac.jp