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Coefficient bounds for a subclass of starlike functions of complex order
Applied Mathematics and Computation 218 (2011) 693–698
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Coefficient bounds for a subclass of starlike functions of complex order Murat Çag˘lar ⇑, Erhan Deniz, Halit Orhan Atatürk University, Faculty of Science, Department of Mathematics, 25240 Erzurum, Turkey
a r t i c l e
i n f o
Dedicated to Professor H. M. Srivastava on the Occasion of his Seventieth Birth Anniversary Keywords: Analytic functions Coefficient bounds Starlike functions of complex order Differential operator
a b s t r a c t In the present work, we determine coefficient bounds for functions in certain subclass of starlike functions of complex order b, which are introduced here by means of a multiplier differential operator. Several corollaries and consequences of the main results are also considered. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Let A denote the family of functions f of the form
f ðzÞ ¼ z þ
1 X
ak zk
ð1:1Þ
k¼2
that are analytic in the open unit disk U ¼ fz : jzj < 1g and X denote the class of bounded analytic functions (Schwarz functions) w(z) in U which satisfy the conditions w(0) = 0 and jw(z)j < 1 for z 2 U. If f and g are in A, we say that f is subordinate to g, written f g, if there exists a function w 2 X such that f ðzÞ ¼ gðwðzÞÞ ðz 2 UÞ. For a function f(z) in A, the multiplier differential operator Dna;d f was extended by Deniz and Orhan in [8] as follows:
D0a;d f ðzÞ ¼ f ðzÞ; D1a;d f ðzÞ ¼ Da;d f ðzÞ ¼ adz2 ðf ðzÞÞ00 þ ða dÞzðf ðzÞÞ0 þ ð1 a þ dÞf ðzÞ; ð1:2Þ
.. .
Dna;d f ðzÞ ¼ Da;d Dn1 a;d f ðzÞ ; where a P d P 0 and n 2 N0 ¼ N [ f0g. If f is given by (1.1) then from the definition of the operator Dna;d f ðzÞ it is easy to see that
Dna;d f ðzÞ ¼ z þ
1 X
Unk ak zk ;
k¼2
where Uk ¼ ½1 þ ðadk þ a dÞðk 1Þ; ðUnk ¼ ½Uk n Þ; a P d P 0 and n 2 N0 . ⇑ Corresponding author. E-mail addresses: [email protected] (M. Çag˘lar), [email protected] (E. Deniz), [email protected] (H. Orhan). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.01.085
ð1:3Þ
M. Çag˘lar et al. / Applied Mathematics and Computation 218 (2011) 693–698
694
It should be remarked that the Dna;d is a generalization of many other linear operators considered earlier. In particular, for f 2 A we have the following: Dn1;0 f ðzÞ Dn f ðzÞ the operator defined by Sa˘la˘gean (see [16]). Dna;0 f ðzÞ Dna f ðzÞ the operator studied by Al-Oboudi (see [2]). Dna;d f ðzÞ the same operator considered for 0 6 d 6 a 6 1, by Ra˘ducanu and Orhan (see [14]). By using the extended operator Dna;d we define a new subclass of functions belonging to the class A. Definition 1.1. Let b – 0 be any complex number, a P d P 0; 0 6 k 6 1; n 2 N0 and for the parameters A and B such that 1 6 B < A 6 1, we say that a function f ðzÞ 2 A is in the class Hnb;k;a;d ðA; BÞ if it satisfies the following subordination condition:
0 1 0 n 1 B F k;a;d ðzÞ C 1 þ Az 1A 1 þ @z n ; b 1 þ Bz F k;a;d ðzÞ
z 2 U;
ð1:4Þ
n where F nk;a;d ðzÞ ¼ ð1 kÞDnþ1 a;d f ðzÞ þ kDa;d f ðzÞ. Recently, Altintas et al. [3], Aouf and Al-Yamy [4], Orhan and Kamali [13], Sohi and Singh [17], Wiatrowski [18], Aouf and Owa [6], Aouf et al. [5], Murugusundaramoorthy and Srivastava [11], Nasr and Aouf [12], Attiya [7], Goel and Mehrok [9] and others, introduced and studied some properties of certain classes of analytic functions of complex order.
Remark 1.1. Throughout our present investigation, we tacitly assume that the parametric constraints listed in (1.3) and Definition 1.1 are satisfied. Our basic tool is the following lemma known as Rogosinski Inequality for majorization. P P1 k k Lemma 1.1 [15]. Let gðzÞ ¼ 1 k¼q dk z and GðzÞ ¼ k¼q Dk z ; q P 0. If g(z) = w(z)G(z) where w(z) 2 X, then dq = 0 and Pn Pn1 2 2 jd j 6 jD j (n = q + 1, q + 2, . . .). k k¼qþ1 k k¼q 2. Coefficient inequalities Firstly, for functions in the class Hnb;k;a;d ðA; BÞ we establish the following theorem. Theorem 2.1. Let the function f(z) defined by (1.1) be in the class Hnb;k;a;d ðA; BÞ; z 2 U. (i) If (A B)2jbj2 > (k 1){2B(A B)Re{b} + (1 B2)(k 1)}, let
G¼
ðA BÞ2 jbj2 ðk 1Þf2BðA BÞRefbg þ ð1 B2 Þðk 1Þg
;
k ¼ 2; 3; . . . ; n 1;
M = [G] (Gauss Symbol), and [G] is the greatest integer not greater than G. Then
jaj j 6
j Y
1 n j ½ð1
kÞUj þ kðj 1Þ!
jðA BÞb ðk 2ÞBj for j ¼ 2; 3; . . . ; M þ 2
ð2:1Þ
n j ½ð1
Mþ3 Y 1 jðA BÞb ðk 2ÞBj for j > M þ 2: kÞUj þ kðj 1ÞðM þ 1Þ! k¼2
ð2:2Þ
U
k¼2
and
jaj j 6
U
(ii) If (A B)2jbj2 6 (k 1){2B(A B)Re{b} + (1 B2)(k 1)}, then
jaj j 6
ðA BÞjbj ðj 1ÞUnj ½ð1 kÞUj þ k
for j P 2:
ð2:3Þ
Proof. If f ðzÞ 2 Hnb;k;a;d ðA; BÞ, then there is a Schwarz function w(z) = w1z + 2 X such that
0
F nk;a;d ðzÞ
0
1
1B C 1 þ AwðzÞ 1A ¼ ; 1 þ @z n b 1 þ BwðzÞ F k;a;d ðzÞ
z 2 U:
ð2:4Þ
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By definition of Dna;d f ðzÞ and F nk;a;d ðzÞ, we can write
F nk;a;d ðzÞ ¼ z þ
1 X
Unk ½ð1 kÞUk þ kak zk :
ð2:5Þ
k¼2
From (2.4), we obtain
n 0 0 o z F nk;a;d ðzÞ F nk;a;d ðzÞ ¼ F nk;a;d ðzÞ½ðA BÞb þ B zB F nk;a;d ðzÞ wðzÞ;
wðzÞ 2 X:
ð2:6Þ
Now, by using (2.5) Eq. (2.6) can be written as 1 X
( n k ½ð1
U
k
kÞUk þ kðk 1Þak z ¼
ðA BÞbz þ
k¼1
1 X
) n k ½ð1
U
k
kÞUk þ k½ðA BÞb Bðk 1Þak z
wðzÞ:
k¼1
By using Lemma 1.1, we have j X
2 2 2 2 2 U2n k ½ð1 kÞUk þ k ðk 1Þ jak j 6 ðA BÞ jbj þ
k¼2
j1 X
2 2 2 U2n k ½ð1 kÞUk þ k jðA BÞb Bðk 1Þj jak j :
k¼1
After simple calculation from last inequality, we obtain j1 X
2 2 2 2 2 2 2 2 2n U2n k ½ð1 kÞUk þ k ðk 1Þ jak j þ Uj ½ð1 kÞUj þ k ðj 1Þ jaj j 6 ðA BÞ jbj
k¼2
þ
j1 X
2 2 2 U2n k ½ð1 kÞUk þ k jðA BÞb Bðk 1Þj jak j
k¼1
and for j P 2 j1 X
2 2 2 2 ðj 1Þ2 U2n j ½ð1 kÞUj þ k jaj j 6 ðA BÞ jbj þ
2 2 2 2 U2n k ½ð1 kÞUk þ k fjðA BÞb Bðk 1Þj ðk 1Þ gjak j :
ð2:7Þ
k¼2
Now there may be following two cases: (i) Let (A B)2jbj2 > (k 1){2B(A B)Re{b} + (1 B2)(k 1)}, suppose that j 6 M + 2, then for j = 2, (2.7) gives
ja2 j 6
ðA BÞjbj
Un2 ½ð1 kÞU2 þ k
;
which gives (2.1) for j = 2. We establish (2.1) for j 6 M + 2, from (2.7), by mathematical induction. Suppose (2.1) is valid for j = 2, 3, . . . , (k 1). Then it follows from (2.7) j1 X fjðA BÞb Bðk 1Þj2 ðk 1Þ2 g
2 2 2 2 ðj 1Þ2 U2n j ½ð1 kÞUj þ k jaj j 6 ðA BÞ jbj þ
k¼2
(
1
k Y
ððk 1Þ!Þ2
p¼2
) jðA BÞb ðp 2ÞBj
2
¼
1
j Y
ððj 2Þ!Þ2
k¼2
jðA BÞb ðk 2ÞBj2 :
Thus, we get the inequality (2.1) which completes the proof of (2.1). Next, we suppose n > M + 2. Then (2.7) gives 2 2 2 2 ðj 1Þ2 U2n j ½ð1 kÞUj þ k jaj j 6 ðA BÞ jbj þ
Mþ2 X
2 2 2 2 U2n k ½ð1 kÞUk þ k fjðA BÞb Bðk 1Þj ðk 1Þ gjak j
k¼2 j1 X
þ
2 2 2 2 U2n k ½ð1 kÞUk þ k fjðA BÞb Bðk 1Þj ðk 1Þ gjak j
k¼Mþ3
6 ðA BÞ2 jbj2 þ
M þ2 X
2 2 2 2 U2n k ½ð1 kÞUk þ k fjðA BÞb Bðk 1Þj ðk 1Þ gjak j :
k¼2
On substituting upper estimates for a2, . . . , aM+2 obtained above and simplifying, we obtain (2.2). (ii) Let (A B)2jbj2 6 (k 1){2B(A B)Re{b} + (1 B2)(k 1)}, then from (2.7) we have 2 2 2 2 ðj 1Þ2 U2n j ½ð1 kÞUj þ k jaj j 6 ðA BÞ jbj
which completes the proof of (2.3).
h
ðj P 2Þ;
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696
Remark 2.2 (1) For a = k = 1, d = 0, in Theorem 2.1, we get the result due to Attiya [7]. (2) For a = k = b = 1, d = n = 0, in Theorem 2.1, we get the result of Goel and Mehrok [9]. (3) For B ¼ 1M ; A ¼ a ¼ k ¼ 1; d ¼ 0, in Theorem 2.1, we get the result of Aouf et al. [5]. M Next, we will obtain sharp upper bounds for ja4j for the class Hnb;k;a;d ðA; BÞ. Theorem 2.3. Let f ðzÞ 2 Hnb;k;a;d ðA; BÞ and a2, a3 and a4 are real, then
8 jcþBj ; if 1 6 jcj 6 Bþ4 ; > 3 > > 3M3 > > < jcþBjðj3cBj2 16Þ3 ð3M 22 M 3 2M32 Þ ðj3cBj2 8jcjjcBjÞ jcþBj ; if 4B 6 jcj 6 Bþ4 þ 3M32M ; ja4 j 6 3M3 þ 324M4 ðj3cBj2 8jcjjcBjÞ 2 3 3 2 ðj3cBjþ4Þ 3 > > > > > : jcjjcþBjjcBj ðj3cBj2 8jcjjcBjÞ ; if Bþ4 þ 3M32M 6 jcj 6 B þ 2; 3 3!M3 2 ðj3cBjþ4jÞ m where M k ¼ ðUnk ½ð1 kÞUk þ kÞ; ðMm k ¼ ½M k Þ and c = (A B)b B.
Proof. Setting wðzÞ ¼ zðb1 þhðzÞÞ ; hðzÞ ¼ c1 z þ c2 z2 þ 2 X. After some computation, (2.4) yields 1þb hðzÞ
1 ( ) 1 1 X X ½Mk ðk 1Þak b1 M k1 ½ðA BÞb Bðk 2Þak1 zk ¼ hðzÞ ½M k1 ½ðA BÞb Bðk 2Þak1 b1 Mk ðk 1Þak zk :
k¼2
k¼2
ð2:8Þ Equating the coefficients of z2,
a2 ¼
M1 ðA BÞbb1 M2
and a2 ¼
M1 ðA BÞbb1 : M2
ð2:9Þ
Also, Eq. (2.8) can be written as ( ) 1 1 X X ½M k ðk 1Þak b1 M k1 ½ðA BÞb Bðk 2Þak1 zk ¼ hðzÞ ½M k1 ½ðA BÞb Bðk 2Þak1 b1 M k ðk 1Þak zk : k¼3
Putting hðzÞ ¼ z 1 n X
h
ð2:10Þ
k¼2
i ; b 2 X. By (2.10) and definition of h(z), we have
c1 þbðzÞ 1þc1 bðzÞ
o ðM k ðk 1Þak b1 M k1 ½ðA BÞb Bðk 2ÞÞak1 c1 ðM k2 ½ðA BÞb Bðk 3Þak2 b1 M k1 ðk 2Þak1 Þ zk
k¼3
¼ bðzÞ
1 n X
o Mk2 ½ðA BÞb Bðk 3Þak2 b1 ðk 2ÞM k1 ak1 c1 ððk 1ÞMk ak b1 M k1 ½ðA BÞb Bðk 2Þak1 Þ zk :
k¼3
ð2:11Þ Equating the coefficients of z
3
2M 3 a3 b1 M 2 ½ðA BÞb Ba2 ¼ c1 ½ðA BÞb b1 M 2 a2 :
ð2:12Þ
By (2.9) and (2.12), we obtain
c1 ¼
" #, ! 2M3 M2 ½ðA BÞb Ba22 M 22 ja2 j2 1 : a3 2 ðA BÞb 2M 3 ðA BÞb ðA BÞ2 jbj2
ð2:13Þ
Now jc1j 6 1 and it follows from (2.13) that
! M 22 ½ðA BÞb Ba22 ðA BÞjbj M 22 ja2 j2 : 1 6 a3 2M3 2M3 ðA BÞb ðA BÞ2 jbj2
ð2:14Þ
By using Lemma 1.1, for n = 4 in the equality (2.11) we obtain
jð3M3 a4 b1 M 3 ½ðA BÞb 2Ba3 Þ c1 ðM 2 ½ðA BÞb Ba2 2b1 M3 a3 Þj 6 jðA BÞb b1 M2 a2 c1 ð2M 3 a3 b1 M 2 ½ðA BÞb Ba2 Þj:
ð2:15Þ
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697
Using (2.9), (2.12) and (2.13), (2.15) can be written as
M 3 ½ðA BÞb 2B½ðA BÞb Ba32 M 3 M 2 ½3ðA BÞb 4Ba2 3M 3 a4 2 2 ðA BÞb 2ðA BÞ2 b ! " # " #2 , M 22 ½ðA BÞb Ba22 4M 23 M 2 a2 M 22 ½ðA BÞb Ba22 M 22 ja2 j2 a3 a3 1 þ 2M3 ðA BÞb 2M3 ðA BÞb ðA BÞ2 jbj2 ðA BÞ2 jbj2 , ! ! M 22 ja2 j2 4M23 M 22 ½ðA BÞb Ba22 M22 ja2 j2 1 6 ðA BÞjbj 1 a : 3 ðA BÞjbj 2M 3 ðA BÞb ðA BÞ2 jbj2 ðA BÞ2 jbj2
ð2:16Þ
Assuming a2, a3 and a4 to be real, (2.14) and (2.16) can be written as
M 2 jðA BÞb Bja22 ðA BÞjbj M22 ja2 j2 a3 2 1 ¼l 2M 3 2M3 ðA BÞjbj ðA BÞ2 jbj2
! ð1 6 l 6 1Þ
ð2:17Þ
and
! M 2 j3ðA BÞb 4Bja2 M 22 ja2 j2 3a4 ¼ þl 1 2M3 2M 3 ðA BÞ2 jbj2 ðA BÞ2 jbj2 ! ! M22 ja2 j2 M 22 ja2 j2 2 M 2 a2 2 ðA BÞjbj þ ðg l Þ ð1 6 g 6 1Þ: 1 1 l M3 M3 ðA BÞ2 jbj2 ðA BÞ2 jbj2 M32 jðA BÞb 2BjjðA BÞb Bja32
The right-hand side is maximum when g = 1. Thus,
! ðA BÞjbj M22 ja2 j2 3a4 6 þ 1 M3 2M 3 ðA BÞ2 jbj2 ðA BÞ2 jbj2 M 2 a2 M 2 j3ðA BÞb 4Bja2 1 l2 l2 þl ðA BÞjbj 2ðA BÞjbj ! 3 2 2 3 l2 M2 x lM2 j3c Bjx M cjc Bjx jc þ Bj M2 x 2 ¼ 2 þ ¼ hðl; xÞ; þ 1 l 1 M3 jc þ Bj 2jc þ Bj 2M 3 jc þ Bj2 jc þ Bj2 M32 jðA BÞb 2BjjðA BÞb Bja32
ð2:18Þ
say (where (A B)b B = c and a2 = x). For extreme values, @@hl ¼ 0 ¼ @h which yield @x
l¼
M 2 j3c Bjx ; 4ðjc þ Bj þ M 2 xÞ
x¼
2jc þ Bjðj3c Bj2 16Þ 3M 3 ðj3c Bj2 8cjc BjÞ
;
x ¼ 0:
ð2:19Þ
It is easy to verify that (0, 0) gives the maximum value of h(l, x) provided j3c Bj 6 4. From (2.18), we obtain first part of Theorem 2.3 provided 1 6 jcj 6 Bþ4 . 3 Also
2
16Þ M2 j3cBjx0 l0 ¼ 4ðjcþBjþM ; x0 ¼ 2jcþBjðj3cBj gives maximum value of h(l, x) 2 x0 Þ 3M ðj3cBj2 8cjcBjÞ 3
ðj3c Bj2 16Þ3 3M22 M 3 2M32 jc þ Bj jc þ Bj hðl0 ; x0 Þ ¼ þ M3 108M 43 ðj3ðA BÞb 4Bj2 8jðA BÞb BjjðA BÞb 2BjÞ2 2
ðj3cBj 8cjcBjÞ provided j3c Bj P 4 and jcj 6 Bþ4 þ 3M32M . From (2.18), we have second part of Theorem 2.3 provided 3 2 ðj3cBjþ4Þ 4B 3
2
ðj3cBj 8cjcBjÞ 6 jcj 6 Bþ4 þ 3M32M . 3 2 ðj3cBjþ4Þ
ðABÞjbj When x0 P ðABÞjbj M 2 , maximum value of h(l,x) occurs at x0 ¼ M 2 . Thus, from (2.18) we obtain last part of Theorem 2.3 if Bþ4 3
2
ðj3cBj 8cjcBjÞ þ 3M32M 6 jcj 6 B þ 2. h 2 ðj3cBjþ4Þ
Finally, we give sharp upper bounds for ja3 ma22 j for the class Hnb;k;a;d ðA; BÞ. To prove our main result, we need the following lemmas. Lemma 2.1 [1]. If w 2 X, then
8 > < t w2 tw2 6 1 1 > : t
if t 6 1; if 1 6 t 6 1; if t P 1:
M. Çag˘lar et al. / Applied Mathematics and Computation 218 (2011) 693–698
698
When t < 1 or t > 1, the equality holds if and only if w(z) = z or one of its rotations. If 1 < t < 1, then equality holds if and only if w(z) = z2 or one of its rotations. Equality holds for t =1 if and only if wðzÞ ¼ zðzþkÞ ð0 6 k 6 1Þ or one of its rotations, while for t = 1 1þkz the equality holds if and only if wðzÞ ¼ zðzþkÞ ð0 6 k 6 1Þ or one of its rotations. 1þkz Lemma 2.2 [10]. If w 2 X, then for any complex number t
w2 tw2 6 maxf1; jtjg: 1 The result is sharp for the functions w(z) = z or w(z) = z2. Theorem 2.4. If f(z) given by (1.1) belongs to Hnb;k;a;d ðA; BÞ, then
8 Un3 ½ð1kÞU3 þk > ðABÞjbj > B if m 6 r1 ; ðA BÞjbj 1 2 m > n 2U3 ½ð1kÞU3 þk > U2n ½ð1kÞU2 þk2 > 2 > < ðABÞjbj a3 ma2 6 if r1 6 m 6 r2 ; 2 2Un3 ½ð1kÞU3 þk > > > > ðABÞjbj > Un ½ð1kÞU þk > if m P r2 : 2Un ½ð1kÞU3 þk ðA BÞjbj 1 þ 2m U2n3½ð1kÞU 3þk2 þ B 3
2
2
and for m complex,
a3 ma2 6 2
where
) ( ! ðA BÞjbj Un3 ½ð1 kÞU3 þ k max 1; ðA BÞb 1 þ 2m 2n þ B ; n 2 2U3 ½ð1 kÞU3 þ k U2 ½ð1 kÞU2 þ k
2n
2
2n
2
U2 þk ððABÞbB1Þ U ½ð1kÞU2 þk ððABÞbBþ1Þ r1 ¼ U2 ½ð1kÞ , r2 ¼ 2 2Un ½ð1kÞ . The results are sharp. U3 þkðABÞb 2Un3 ½ð1kÞU3 þkðABÞb 3
Proof. By making use of Lemmas 2.1 and 2.2, we prove Theorem 2.4. h Acknowledgement The present investigation was supported by Atatürk University Rectorship under ‘‘The Scientific and Research Project of Atatürk University’’, Project No.: 2010/28. References [1] R.M. Ali, V. Ravichandran, N. Seenivasagan, Coefficient bounds for p-valent functions, Appl. Math. Comput. 187 (2007) 35–46. [2] F.M. Al-Oboudi, On univalent functions defined by a generalized Sa˘la˘gean operator, Int. J. Math. Math. Sci. 27 (2004) 1429–1436. [3] O. Altintas, H. Irmak, S. Owa, H.M. Srivastava, Coefficient bounds for some families of starlike and convex functions of complex order, Appl. Math. Lett. 20 (2007) 1218–1222. [4] M.K. Aouf, M.A. Al-Yamy, Coefficient estimates for certain class of analytic functions of complex order and type beta, Demonstratio Math. 38 (2005) 313–322. [5] M.K. Aouf, H.E. Darwish, A.A. Attiya, On a certain class of analytic functions defined by using Hadamard product and complex order, Proc. Pakistan Acad. Sci. 37 (1) (2000) 71–77. [6] M.K. Aouf, S. Owa, M. Obradovic, Certain classes of analytic functions of complex order and type beta, Rend. Mat. 11 (1991) 691–714. [7] A.A. Attiya, On a generalization class of bounded starlike functions of complex order, Appl. Math. Comput. 187 (2007) 62–67. [8] E. Deniz, H. Orhan, The Fekete–Szegö problem for a generalized subclass of analytic functions, Kyungpook Math. J. 50 (2010) 37–47. [9] R.M. Goel, B.S. Mehrok, On the coefficients of a subclass of starlike functions, Indian J. Pure Appl. Math. 12 (5) (1981) 634–647. [10] F.R. Keogh, E.P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Am. Math. Soc. 20 (1969) 8–12. [11] G. Murugusundaramoorthy, H.M. Srivastava, Neighborhoods of certain classes of analytic functions of complex order, J. Inequal. Pure Appl. Math. 5 (2) (2004) 1–8. Article 24, electronic. [12] M.A. Nasr, M.K. Aouf, Starlike functions of complex order, J. Nat. Sci. Math. 25 (1985) 1–12. [13] H. Orhan, M. Kamali, Starlike, convex and close-to-convex functions of complex order, Appl. Math. Comput. 135 (2003) 251–262. [14] D. Ra˘ducanu, H. Orhan, Subclasses of analytic functions defined by a generalized differential operator, Int. J. Math. Anal. 4 (1) (2010) 1–15. [15] M.S. Robertson, Quasi-subordinate functions, Mathematical Essays Dedicated to A.J. MacIntyre, Ohio University Press, Athens, Ohio, 1967, pp. 311–330. [16] G.S. Sa˘la˘gean, Subclasses of univalent functions, in: Complex Analysis, Proc. 5th Rom. Finn. Semin., Part 1, Bucharest 1981, Lect. Notes Math., vol. 1013, 1983, pp. 362–372. [17] N.S. Sohi, L.P. Singh, A class of bounded starlike functions of complex order, Indian J. Pure Appl. Math. 33 (1991) 29–35. [18] P. Wiatrowski, The coefficients of a certain family of holomorphic functions, Zeszyty Nauk. Univ. Lódz Nauk. Mat. Przyrod (Ser. 2) 39 (1971) 75–85.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Coefficient bounds for a subclass of starlike functions of complex order Murat Çag˘lar ⇑, Erhan Deniz, Halit Orhan Atatürk University, Faculty of Science, Department of Mathematics, 25240 Erzurum, Turkey
a r t i c l e
i n f o
Dedicated to Professor H. M. Srivastava on the Occasion of his Seventieth Birth Anniversary Keywords: Analytic functions Coefficient bounds Starlike functions of complex order Differential operator
a b s t r a c t In the present work, we determine coefficient bounds for functions in certain subclass of starlike functions of complex order b, which are introduced here by means of a multiplier differential operator. Several corollaries and consequences of the main results are also considered. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Let A denote the family of functions f of the form
f ðzÞ ¼ z þ
1 X
ak zk
ð1:1Þ
k¼2
that are analytic in the open unit disk U ¼ fz : jzj < 1g and X denote the class of bounded analytic functions (Schwarz functions) w(z) in U which satisfy the conditions w(0) = 0 and jw(z)j < 1 for z 2 U. If f and g are in A, we say that f is subordinate to g, written f g, if there exists a function w 2 X such that f ðzÞ ¼ gðwðzÞÞ ðz 2 UÞ. For a function f(z) in A, the multiplier differential operator Dna;d f was extended by Deniz and Orhan in [8] as follows:
D0a;d f ðzÞ ¼ f ðzÞ; D1a;d f ðzÞ ¼ Da;d f ðzÞ ¼ adz2 ðf ðzÞÞ00 þ ða dÞzðf ðzÞÞ0 þ ð1 a þ dÞf ðzÞ; ð1:2Þ
.. .
Dna;d f ðzÞ ¼ Da;d Dn1 a;d f ðzÞ ; where a P d P 0 and n 2 N0 ¼ N [ f0g. If f is given by (1.1) then from the definition of the operator Dna;d f ðzÞ it is easy to see that
Dna;d f ðzÞ ¼ z þ
1 X
Unk ak zk ;
k¼2
where Uk ¼ ½1 þ ðadk þ a dÞðk 1Þ; ðUnk ¼ ½Uk n Þ; a P d P 0 and n 2 N0 . ⇑ Corresponding author. E-mail addresses: [email protected] (M. Çag˘lar), [email protected] (E. Deniz), [email protected] (H. Orhan). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.01.085
ð1:3Þ
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694
It should be remarked that the Dna;d is a generalization of many other linear operators considered earlier. In particular, for f 2 A we have the following: Dn1;0 f ðzÞ Dn f ðzÞ the operator defined by Sa˘la˘gean (see [16]). Dna;0 f ðzÞ Dna f ðzÞ the operator studied by Al-Oboudi (see [2]). Dna;d f ðzÞ the same operator considered for 0 6 d 6 a 6 1, by Ra˘ducanu and Orhan (see [14]). By using the extended operator Dna;d we define a new subclass of functions belonging to the class A. Definition 1.1. Let b – 0 be any complex number, a P d P 0; 0 6 k 6 1; n 2 N0 and for the parameters A and B such that 1 6 B < A 6 1, we say that a function f ðzÞ 2 A is in the class Hnb;k;a;d ðA; BÞ if it satisfies the following subordination condition:
0 1 0 n 1 B F k;a;d ðzÞ C 1 þ Az 1A 1 þ @z n ; b 1 þ Bz F k;a;d ðzÞ
z 2 U;
ð1:4Þ
n where F nk;a;d ðzÞ ¼ ð1 kÞDnþ1 a;d f ðzÞ þ kDa;d f ðzÞ. Recently, Altintas et al. [3], Aouf and Al-Yamy [4], Orhan and Kamali [13], Sohi and Singh [17], Wiatrowski [18], Aouf and Owa [6], Aouf et al. [5], Murugusundaramoorthy and Srivastava [11], Nasr and Aouf [12], Attiya [7], Goel and Mehrok [9] and others, introduced and studied some properties of certain classes of analytic functions of complex order.
Remark 1.1. Throughout our present investigation, we tacitly assume that the parametric constraints listed in (1.3) and Definition 1.1 are satisfied. Our basic tool is the following lemma known as Rogosinski Inequality for majorization. P P1 k k Lemma 1.1 [15]. Let gðzÞ ¼ 1 k¼q dk z and GðzÞ ¼ k¼q Dk z ; q P 0. If g(z) = w(z)G(z) where w(z) 2 X, then dq = 0 and Pn Pn1 2 2 jd j 6 jD j (n = q + 1, q + 2, . . .). k k¼qþ1 k k¼q 2. Coefficient inequalities Firstly, for functions in the class Hnb;k;a;d ðA; BÞ we establish the following theorem. Theorem 2.1. Let the function f(z) defined by (1.1) be in the class Hnb;k;a;d ðA; BÞ; z 2 U. (i) If (A B)2jbj2 > (k 1){2B(A B)Re{b} + (1 B2)(k 1)}, let
G¼
ðA BÞ2 jbj2 ðk 1Þf2BðA BÞRefbg þ ð1 B2 Þðk 1Þg
;
k ¼ 2; 3; . . . ; n 1;
M = [G] (Gauss Symbol), and [G] is the greatest integer not greater than G. Then
jaj j 6
j Y
1 n j ½ð1
kÞUj þ kðj 1Þ!
jðA BÞb ðk 2ÞBj for j ¼ 2; 3; . . . ; M þ 2
ð2:1Þ
n j ½ð1
Mþ3 Y 1 jðA BÞb ðk 2ÞBj for j > M þ 2: kÞUj þ kðj 1ÞðM þ 1Þ! k¼2
ð2:2Þ
U
k¼2
and
jaj j 6
U
(ii) If (A B)2jbj2 6 (k 1){2B(A B)Re{b} + (1 B2)(k 1)}, then
jaj j 6
ðA BÞjbj ðj 1ÞUnj ½ð1 kÞUj þ k
for j P 2:
ð2:3Þ
Proof. If f ðzÞ 2 Hnb;k;a;d ðA; BÞ, then there is a Schwarz function w(z) = w1z + 2 X such that
0
F nk;a;d ðzÞ
0
1
1B C 1 þ AwðzÞ 1A ¼ ; 1 þ @z n b 1 þ BwðzÞ F k;a;d ðzÞ
z 2 U:
ð2:4Þ
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By definition of Dna;d f ðzÞ and F nk;a;d ðzÞ, we can write
F nk;a;d ðzÞ ¼ z þ
1 X
Unk ½ð1 kÞUk þ kak zk :
ð2:5Þ
k¼2
From (2.4), we obtain
n 0 0 o z F nk;a;d ðzÞ F nk;a;d ðzÞ ¼ F nk;a;d ðzÞ½ðA BÞb þ B zB F nk;a;d ðzÞ wðzÞ;
wðzÞ 2 X:
ð2:6Þ
Now, by using (2.5) Eq. (2.6) can be written as 1 X
( n k ½ð1
U
k
kÞUk þ kðk 1Þak z ¼
ðA BÞbz þ
k¼1
1 X
) n k ½ð1
U
k
kÞUk þ k½ðA BÞb Bðk 1Þak z
wðzÞ:
k¼1
By using Lemma 1.1, we have j X
2 2 2 2 2 U2n k ½ð1 kÞUk þ k ðk 1Þ jak j 6 ðA BÞ jbj þ
k¼2
j1 X
2 2 2 U2n k ½ð1 kÞUk þ k jðA BÞb Bðk 1Þj jak j :
k¼1
After simple calculation from last inequality, we obtain j1 X
2 2 2 2 2 2 2 2 2n U2n k ½ð1 kÞUk þ k ðk 1Þ jak j þ Uj ½ð1 kÞUj þ k ðj 1Þ jaj j 6 ðA BÞ jbj
k¼2
þ
j1 X
2 2 2 U2n k ½ð1 kÞUk þ k jðA BÞb Bðk 1Þj jak j
k¼1
and for j P 2 j1 X
2 2 2 2 ðj 1Þ2 U2n j ½ð1 kÞUj þ k jaj j 6 ðA BÞ jbj þ
2 2 2 2 U2n k ½ð1 kÞUk þ k fjðA BÞb Bðk 1Þj ðk 1Þ gjak j :
ð2:7Þ
k¼2
Now there may be following two cases: (i) Let (A B)2jbj2 > (k 1){2B(A B)Re{b} + (1 B2)(k 1)}, suppose that j 6 M + 2, then for j = 2, (2.7) gives
ja2 j 6
ðA BÞjbj
Un2 ½ð1 kÞU2 þ k
;
which gives (2.1) for j = 2. We establish (2.1) for j 6 M + 2, from (2.7), by mathematical induction. Suppose (2.1) is valid for j = 2, 3, . . . , (k 1). Then it follows from (2.7) j1 X fjðA BÞb Bðk 1Þj2 ðk 1Þ2 g
2 2 2 2 ðj 1Þ2 U2n j ½ð1 kÞUj þ k jaj j 6 ðA BÞ jbj þ
k¼2
(
1
k Y
ððk 1Þ!Þ2
p¼2
) jðA BÞb ðp 2ÞBj
2
¼
1
j Y
ððj 2Þ!Þ2
k¼2
jðA BÞb ðk 2ÞBj2 :
Thus, we get the inequality (2.1) which completes the proof of (2.1). Next, we suppose n > M + 2. Then (2.7) gives 2 2 2 2 ðj 1Þ2 U2n j ½ð1 kÞUj þ k jaj j 6 ðA BÞ jbj þ
Mþ2 X
2 2 2 2 U2n k ½ð1 kÞUk þ k fjðA BÞb Bðk 1Þj ðk 1Þ gjak j
k¼2 j1 X
þ
2 2 2 2 U2n k ½ð1 kÞUk þ k fjðA BÞb Bðk 1Þj ðk 1Þ gjak j
k¼Mþ3
6 ðA BÞ2 jbj2 þ
M þ2 X
2 2 2 2 U2n k ½ð1 kÞUk þ k fjðA BÞb Bðk 1Þj ðk 1Þ gjak j :
k¼2
On substituting upper estimates for a2, . . . , aM+2 obtained above and simplifying, we obtain (2.2). (ii) Let (A B)2jbj2 6 (k 1){2B(A B)Re{b} + (1 B2)(k 1)}, then from (2.7) we have 2 2 2 2 ðj 1Þ2 U2n j ½ð1 kÞUj þ k jaj j 6 ðA BÞ jbj
which completes the proof of (2.3).
h
ðj P 2Þ;
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Remark 2.2 (1) For a = k = 1, d = 0, in Theorem 2.1, we get the result due to Attiya [7]. (2) For a = k = b = 1, d = n = 0, in Theorem 2.1, we get the result of Goel and Mehrok [9]. (3) For B ¼ 1M ; A ¼ a ¼ k ¼ 1; d ¼ 0, in Theorem 2.1, we get the result of Aouf et al. [5]. M Next, we will obtain sharp upper bounds for ja4j for the class Hnb;k;a;d ðA; BÞ. Theorem 2.3. Let f ðzÞ 2 Hnb;k;a;d ðA; BÞ and a2, a3 and a4 are real, then
8 jcþBj ; if 1 6 jcj 6 Bþ4 ; > 3 > > 3M3 > > < jcþBjðj3cBj2 16Þ3 ð3M 22 M 3 2M32 Þ ðj3cBj2 8jcjjcBjÞ jcþBj ; if 4B 6 jcj 6 Bþ4 þ 3M32M ; ja4 j 6 3M3 þ 324M4 ðj3cBj2 8jcjjcBjÞ 2 3 3 2 ðj3cBjþ4Þ 3 > > > > > : jcjjcþBjjcBj ðj3cBj2 8jcjjcBjÞ ; if Bþ4 þ 3M32M 6 jcj 6 B þ 2; 3 3!M3 2 ðj3cBjþ4jÞ m where M k ¼ ðUnk ½ð1 kÞUk þ kÞ; ðMm k ¼ ½M k Þ and c = (A B)b B.
Proof. Setting wðzÞ ¼ zðb1 þhðzÞÞ ; hðzÞ ¼ c1 z þ c2 z2 þ 2 X. After some computation, (2.4) yields 1þb hðzÞ
1 ( ) 1 1 X X ½Mk ðk 1Þak b1 M k1 ½ðA BÞb Bðk 2Þak1 zk ¼ hðzÞ ½M k1 ½ðA BÞb Bðk 2Þak1 b1 Mk ðk 1Þak zk :
k¼2
k¼2
ð2:8Þ Equating the coefficients of z2,
a2 ¼
M1 ðA BÞbb1 M2
and a2 ¼
M1 ðA BÞbb1 : M2
ð2:9Þ
Also, Eq. (2.8) can be written as ( ) 1 1 X X ½M k ðk 1Þak b1 M k1 ½ðA BÞb Bðk 2Þak1 zk ¼ hðzÞ ½M k1 ½ðA BÞb Bðk 2Þak1 b1 M k ðk 1Þak zk : k¼3
Putting hðzÞ ¼ z 1 n X
h
ð2:10Þ
k¼2
i ; b 2 X. By (2.10) and definition of h(z), we have
c1 þbðzÞ 1þc1 bðzÞ
o ðM k ðk 1Þak b1 M k1 ½ðA BÞb Bðk 2ÞÞak1 c1 ðM k2 ½ðA BÞb Bðk 3Þak2 b1 M k1 ðk 2Þak1 Þ zk
k¼3
¼ bðzÞ
1 n X
o Mk2 ½ðA BÞb Bðk 3Þak2 b1 ðk 2ÞM k1 ak1 c1 ððk 1ÞMk ak b1 M k1 ½ðA BÞb Bðk 2Þak1 Þ zk :
k¼3
ð2:11Þ Equating the coefficients of z
3
2M 3 a3 b1 M 2 ½ðA BÞb Ba2 ¼ c1 ½ðA BÞb b1 M 2 a2 :
ð2:12Þ
By (2.9) and (2.12), we obtain
c1 ¼
" #, ! 2M3 M2 ½ðA BÞb Ba22 M 22 ja2 j2 1 : a3 2 ðA BÞb 2M 3 ðA BÞb ðA BÞ2 jbj2
ð2:13Þ
Now jc1j 6 1 and it follows from (2.13) that
! M 22 ½ðA BÞb Ba22 ðA BÞjbj M 22 ja2 j2 : 1 6 a3 2M3 2M3 ðA BÞb ðA BÞ2 jbj2
ð2:14Þ
By using Lemma 1.1, for n = 4 in the equality (2.11) we obtain
jð3M3 a4 b1 M 3 ½ðA BÞb 2Ba3 Þ c1 ðM 2 ½ðA BÞb Ba2 2b1 M3 a3 Þj 6 jðA BÞb b1 M2 a2 c1 ð2M 3 a3 b1 M 2 ½ðA BÞb Ba2 Þj:
ð2:15Þ
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Using (2.9), (2.12) and (2.13), (2.15) can be written as
M 3 ½ðA BÞb 2B½ðA BÞb Ba32 M 3 M 2 ½3ðA BÞb 4Ba2 3M 3 a4 2 2 ðA BÞb 2ðA BÞ2 b ! " # " #2 , M 22 ½ðA BÞb Ba22 4M 23 M 2 a2 M 22 ½ðA BÞb Ba22 M 22 ja2 j2 a3 a3 1 þ 2M3 ðA BÞb 2M3 ðA BÞb ðA BÞ2 jbj2 ðA BÞ2 jbj2 , ! ! M 22 ja2 j2 4M23 M 22 ½ðA BÞb Ba22 M22 ja2 j2 1 6 ðA BÞjbj 1 a : 3 ðA BÞjbj 2M 3 ðA BÞb ðA BÞ2 jbj2 ðA BÞ2 jbj2
ð2:16Þ
Assuming a2, a3 and a4 to be real, (2.14) and (2.16) can be written as
M 2 jðA BÞb Bja22 ðA BÞjbj M22 ja2 j2 a3 2 1 ¼l 2M 3 2M3 ðA BÞjbj ðA BÞ2 jbj2
! ð1 6 l 6 1Þ
ð2:17Þ
and
! M 2 j3ðA BÞb 4Bja2 M 22 ja2 j2 3a4 ¼ þl 1 2M3 2M 3 ðA BÞ2 jbj2 ðA BÞ2 jbj2 ! ! M22 ja2 j2 M 22 ja2 j2 2 M 2 a2 2 ðA BÞjbj þ ðg l Þ ð1 6 g 6 1Þ: 1 1 l M3 M3 ðA BÞ2 jbj2 ðA BÞ2 jbj2 M32 jðA BÞb 2BjjðA BÞb Bja32
The right-hand side is maximum when g = 1. Thus,
! ðA BÞjbj M22 ja2 j2 3a4 6 þ 1 M3 2M 3 ðA BÞ2 jbj2 ðA BÞ2 jbj2 M 2 a2 M 2 j3ðA BÞb 4Bja2 1 l2 l2 þl ðA BÞjbj 2ðA BÞjbj ! 3 2 2 3 l2 M2 x lM2 j3c Bjx M cjc Bjx jc þ Bj M2 x 2 ¼ 2 þ ¼ hðl; xÞ; þ 1 l 1 M3 jc þ Bj 2jc þ Bj 2M 3 jc þ Bj2 jc þ Bj2 M32 jðA BÞb 2BjjðA BÞb Bja32
ð2:18Þ
say (where (A B)b B = c and a2 = x). For extreme values, @@hl ¼ 0 ¼ @h which yield @x
l¼
M 2 j3c Bjx ; 4ðjc þ Bj þ M 2 xÞ
x¼
2jc þ Bjðj3c Bj2 16Þ 3M 3 ðj3c Bj2 8cjc BjÞ
;
x ¼ 0:
ð2:19Þ
It is easy to verify that (0, 0) gives the maximum value of h(l, x) provided j3c Bj 6 4. From (2.18), we obtain first part of Theorem 2.3 provided 1 6 jcj 6 Bþ4 . 3 Also
2
16Þ M2 j3cBjx0 l0 ¼ 4ðjcþBjþM ; x0 ¼ 2jcþBjðj3cBj gives maximum value of h(l, x) 2 x0 Þ 3M ðj3cBj2 8cjcBjÞ 3
ðj3c Bj2 16Þ3 3M22 M 3 2M32 jc þ Bj jc þ Bj hðl0 ; x0 Þ ¼ þ M3 108M 43 ðj3ðA BÞb 4Bj2 8jðA BÞb BjjðA BÞb 2BjÞ2 2
ðj3cBj 8cjcBjÞ provided j3c Bj P 4 and jcj 6 Bþ4 þ 3M32M . From (2.18), we have second part of Theorem 2.3 provided 3 2 ðj3cBjþ4Þ 4B 3
2
ðj3cBj 8cjcBjÞ 6 jcj 6 Bþ4 þ 3M32M . 3 2 ðj3cBjþ4Þ
ðABÞjbj When x0 P ðABÞjbj M 2 , maximum value of h(l,x) occurs at x0 ¼ M 2 . Thus, from (2.18) we obtain last part of Theorem 2.3 if Bþ4 3
2
ðj3cBj 8cjcBjÞ þ 3M32M 6 jcj 6 B þ 2. h 2 ðj3cBjþ4Þ
Finally, we give sharp upper bounds for ja3 ma22 j for the class Hnb;k;a;d ðA; BÞ. To prove our main result, we need the following lemmas. Lemma 2.1 [1]. If w 2 X, then
8 > < t w2 tw2 6 1 1 > : t
if t 6 1; if 1 6 t 6 1; if t P 1:
M. Çag˘lar et al. / Applied Mathematics and Computation 218 (2011) 693–698
698
When t < 1 or t > 1, the equality holds if and only if w(z) = z or one of its rotations. If 1 < t < 1, then equality holds if and only if w(z) = z2 or one of its rotations. Equality holds for t =1 if and only if wðzÞ ¼ zðzþkÞ ð0 6 k 6 1Þ or one of its rotations, while for t = 1 1þkz the equality holds if and only if wðzÞ ¼ zðzþkÞ ð0 6 k 6 1Þ or one of its rotations. 1þkz Lemma 2.2 [10]. If w 2 X, then for any complex number t
w2 tw2 6 maxf1; jtjg: 1 The result is sharp for the functions w(z) = z or w(z) = z2. Theorem 2.4. If f(z) given by (1.1) belongs to Hnb;k;a;d ðA; BÞ, then
8 Un3 ½ð1kÞU3 þk > ðABÞjbj > B if m 6 r1 ; ðA BÞjbj 1 2 m > n 2U3 ½ð1kÞU3 þk > U2n ½ð1kÞU2 þk2 > 2 > < ðABÞjbj a3 ma2 6 if r1 6 m 6 r2 ; 2 2Un3 ½ð1kÞU3 þk > > > > ðABÞjbj > Un ½ð1kÞU þk > if m P r2 : 2Un ½ð1kÞU3 þk ðA BÞjbj 1 þ 2m U2n3½ð1kÞU 3þk2 þ B 3
2
2
and for m complex,
a3 ma2 6 2
where
) ( ! ðA BÞjbj Un3 ½ð1 kÞU3 þ k max 1; ðA BÞb 1 þ 2m 2n þ B ; n 2 2U3 ½ð1 kÞU3 þ k U2 ½ð1 kÞU2 þ k
2n
2
2n
2
U2 þk ððABÞbB1Þ U ½ð1kÞU2 þk ððABÞbBþ1Þ r1 ¼ U2 ½ð1kÞ , r2 ¼ 2 2Un ½ð1kÞ . The results are sharp. U3 þkðABÞb 2Un3 ½ð1kÞU3 þkðABÞb 3
Proof. By making use of Lemmas 2.1 and 2.2, we prove Theorem 2.4. h Acknowledgement The present investigation was supported by Atatürk University Rectorship under ‘‘The Scientific and Research Project of Atatürk University’’, Project No.: 2010/28. References [1] R.M. Ali, V. Ravichandran, N. Seenivasagan, Coefficient bounds for p-valent functions, Appl. Math. Comput. 187 (2007) 35–46. [2] F.M. Al-Oboudi, On univalent functions defined by a generalized Sa˘la˘gean operator, Int. J. Math. Math. Sci. 27 (2004) 1429–1436. [3] O. Altintas, H. Irmak, S. Owa, H.M. Srivastava, Coefficient bounds for some families of starlike and convex functions of complex order, Appl. Math. Lett. 20 (2007) 1218–1222. [4] M.K. Aouf, M.A. Al-Yamy, Coefficient estimates for certain class of analytic functions of complex order and type beta, Demonstratio Math. 38 (2005) 313–322. [5] M.K. Aouf, H.E. Darwish, A.A. Attiya, On a certain class of analytic functions defined by using Hadamard product and complex order, Proc. Pakistan Acad. Sci. 37 (1) (2000) 71–77. [6] M.K. Aouf, S. Owa, M. Obradovic, Certain classes of analytic functions of complex order and type beta, Rend. Mat. 11 (1991) 691–714. [7] A.A. Attiya, On a generalization class of bounded starlike functions of complex order, Appl. Math. Comput. 187 (2007) 62–67. [8] E. Deniz, H. Orhan, The Fekete–Szegö problem for a generalized subclass of analytic functions, Kyungpook Math. J. 50 (2010) 37–47. [9] R.M. Goel, B.S. Mehrok, On the coefficients of a subclass of starlike functions, Indian J. Pure Appl. Math. 12 (5) (1981) 634–647. [10] F.R. Keogh, E.P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Am. Math. Soc. 20 (1969) 8–12. [11] G. Murugusundaramoorthy, H.M. Srivastava, Neighborhoods of certain classes of analytic functions of complex order, J. Inequal. Pure Appl. Math. 5 (2) (2004) 1–8. Article 24, electronic. [12] M.A. Nasr, M.K. Aouf, Starlike functions of complex order, J. Nat. Sci. Math. 25 (1985) 1–12. [13] H. Orhan, M. Kamali, Starlike, convex and close-to-convex functions of complex order, Appl. Math. Comput. 135 (2003) 251–262. [14] D. Ra˘ducanu, H. Orhan, Subclasses of analytic functions defined by a generalized differential operator, Int. J. Math. Anal. 4 (1) (2010) 1–15. [15] M.S. Robertson, Quasi-subordinate functions, Mathematical Essays Dedicated to A.J. MacIntyre, Ohio University Press, Athens, Ohio, 1967, pp. 311–330. [16] G.S. Sa˘la˘gean, Subclasses of univalent functions, in: Complex Analysis, Proc. 5th Rom. Finn. Semin., Part 1, Bucharest 1981, Lect. Notes Math., vol. 1013, 1983, pp. 362–372. [17] N.S. Sohi, L.P. Singh, A class of bounded starlike functions of complex order, Indian J. Pure Appl. Math. 33 (1991) 29–35. [18] P. Wiatrowski, The coefficients of a certain family of holomorphic functions, Zeszyty Nauk. Univ. Lódz Nauk. Mat. Przyrod (Ser. 2) 39 (1971) 75–85.